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is a monic first-order #DO. 5.3.7. Proposition. The operator L is a solution to Eqs. (5.1.3). Proof. It is similar to that of Proposition 1.7.5(ii). For any m we have fcCiJ) • 0. Applying this to exp£(t, z), we get (7.3.6). Let I < 0, then i-e- dupp'q9 + ( s e e 1 9 3 9 ) { ^ P - 1 , 9 } that w ' = Su}P-i,q+i = o There are such forms dipP,q-i
There is a differential operator on the left-hand side of the equality and an \I>DO of order less than N on the right-hand side. So, both of them are differential operators of order less than N. Now, 0 = dm{$yk)
= dm$-yk
+ $- dmyk = dm$yk
+ $-
dmyk
= 3 m $ - yk + L™$yk + L™$yk = (dm$ + L™
$d§-lL™
L™} = [L, L™} = [L?, L].
D
5.3.8. Proposition. If {ak} and {0k} are connected by f3k — eak where en = 1 then L™ = 0 and L satisfies the equations of the nth GD hierarchy. Proof. It was actually proven in 1.7.5(i).
•
5.3.9. Remark. One can take as yk a sum not necessarily of two exponentials, as in (5.3.3), but of any number of them, as well as an integral with respect to the parameter a, with any weight. For the nth reduction of the hierarchy (nth GD) this does not work.
The KP
Hierarchy
81
5.3.10. Exercise. Find the one-soliton (N = 1) solution to the KP hierarchy. Write a formula at least for u\. Answer. 4cosh2((£(*,/?) -i{t,a)
+lna)/2) '
The general formula for {uk} can be written, e.g., in the following way. Let x = ~d In y. The Faa di Bruno polynomials [Faa] are differential polynomials Pfc(x) of variable x> satisfying the recurrence equations P0(X) = 1.
Pk+i(x) = (d + x)Pk(x),
(5-3.11)
i.e. Pfe(x) = (d + x) fc l. For example, PQ = 1, Px = x , -P2 = x' + X2- We have $ = d + x, L = (0 + X)d(d + x ) - 1 - Let (d + X ) _ 1 = E f «fc9_fcMultiplying by d + \ o n the left, it is easy to obtain the recurrence formula a,k+i = — (d + x)ak where from a^ = (—l)fc_1Pfc_i(x) and
L = d + X'Y.?(-l)kPk-i(x)d-k.
Now, ufe = ( - l ) V f t - i ( x ) -
5.3.12. Proposition. For the iV-soliton solution the formula res L = a 2 In A holds. Proof. Let $ = dN + aidN-x + a2dN~2 + ••• which implies $ _ 1 = d~N a1d~N~1 + (a 2 - a2)d~N-2 + • • •. We have L = QdQ-1 = {dN + (nd"-1 + (a 2 - a2)d-N~2
+ a2dN~2 + • • -)d{d-N -
-
a18-N~1
+ . . . ) = 0 - a\d~l + •••
whence res L = —a[. It is easy to see from (5.3.5) that ai = —A'/A = -d In A, then res L = d2 In A. D 5.4
Hamiltonian Structure
5.4.1. We are going to discuss the Hamiltonian structure of Eqs. (5.1.3). The first structure was suggested by Watanabe [Wat] by analogy with that for the GD hierarchies (Chap. 3). The second structure was added to that in [Di87]. The complete solution to this problem was obtained by Radul [Rad87] who noticed that the above-mentioned pair of Hamiltonian
82
Soliton
Equations
and Hamiltonian
Systems
structures is nothing more than the beginning of an infinite series of the Hamiltonian pairs. More precisely, as we have seen, each GD hierarchy is a reduction of the KP hierarchy. There is proven in [Rad87] that for any n there is a pair of Hamiltonian structures that goes exactly to the Hamiltonian pair of the nth GD hierarchy by this reduction. As to Eqs. (5.1.5) involving three independent variables, there is no grounds to distinguish one of them as an evolutionary time, it is more natural to consider them in a field theory (see Chap. 19 below). 5.4.2. Let L — d + u0 4- ui<9 _1 + u2d~2 + • • • . (In comparison to Chap. 5 we added here the term uo.) Let Ln = dn + vn^dn'x
+ vn-2dn-2
+ ••• .
Lemma 5.2.7 relates variables {it,} and {vi}. The derivations n —1
oo
i= — oo .7=0
form the Lie algebra V of vector fields being in one-to-one correspondence with the * D O ra-l
The module 0 ° is the space of functionals / = f f(v)dx. fi 1 consists of operators
The dual space
n-l
X = d - " X n _ i + d - n + 1 X „ _ ! + • • • = J^
d'^Xi
6
R/d~nR„
i=—oo
(the sums are supposed to be finite). The above written series for X is but one representative of X in R/d~nR-, addition of "tails" X^^~*~ 1 X i is irrelevant. The pairing is -
/
71—1
res aXdx = / \ J diXidx .
The KP
Hierarchy
83
Further, n-l
df = Sf/SLn
= ]T
d-^Sf/Svi.
—oo
The Hamiltonian mapping is X H-» Hn(X) A n (X) = (LnX)+Ln
- Ln(XLn)+
= c>A"(X) where e dnR-
It is easy to see that this is a mapping R/d~nRX (see above) are insignificant. We have
Ln = Ln -
,
zn.
—> dnR~, the "tails" of
A"(X) = A" ( 0 ) (X) + z M " ( o o ) ( X ) , where i4"W(X) = {LnX)+Ln
- Ln(XLn)+
(5.4.3)
and Anl°°HX)
= [Ln,X+] - [L",X]+ = [L n ,X + ]+ + [L",X+]_ - [L n ,X] + = -[Z/\X_]+ + [I/\X+]_
which can be written as An^(X)
= -[L1,X.]+
+ [Ll,X+\-
.
(5.4.4)
5.4.5. In the reduction to the submanifold L" = 0 we have L™ = Ln and X = X_. The Hamiltonian mapping turns into that for the nth GD hierarchy. The following propositions can be proved in the same way as the corresponding ones for the GD hierarchies. 5.4.6. Proposition. The relation [dA"{X),dA"(Y)}
=
dAn(dAnix)Y-dAniY)X+[X,Y]L)
holds where [X,Y]L = (-X(LnY)+
+ (XL")_Y)_
5.4.7. Proposition. The form W ( 9 A " ( X ) , 5 A " ( Y ) ) = {dA*(X)>Y)
is closed.
-(X^Y).
84
Soliton Equations
and Hamiltonian
Systems
The correspondence between Hamiltonians and vector fields is, as usual, ft H4 dh = ^ - ( j / i / j i n ) , and the Poisson bracket is An(Sf/6Ln)Sg/SLndx.
{/,~9}=df~g = J res
5.5
Resolvent
5.5.1. Extending the mapping Hn to series in z_1, we shall find its kernel. The formal series oo
T{z)=
z~i~1Li
^
= (z-L)-1
+
(L~z)-1
i=—oo
where (z — L)~l is considered as a series in positive powers of z~nL and (L — z)~l as a series in positive powers of znL~l is called a resolvent. T(z) = E a + M ^ C 2 " ^ ' • Thus, for all a the powers of d are bounded from above, and for all j3 the same is with the powers of z. We call the space of such series i?((z - 1 )). The same T but considered as an element of R/d~nR-((z~1)), i.e. up to terms of order 0(d~n~1) will be symbolized byT". 5.5.2. Lemma. {LnTn)+
= 0.
Proof. The tails 0(d-n~1) do not affect (LnTn)+ n defining T it is quite obvious: Lnrn
=
and for the series
oo oo ^0-i-lLi+n_^J2,-i-l+nLi=o_ — oo
n
—oo
5.5.3. Proposition. An(Tn(ez))
= 0,
where en = 1. Proof. A"(T"(ez)) = (LnTn)+Ln
- Ln(TnLn)+
= 0.
If ez is substituted for z the operator L™ remains the same.
•
The KP
85
Hierarchy
5.5.4. Proposition. An{d~n) Proof. An(d~n)
= (Lnd-n)+Ln
= 0.
- Ln(d~nLn)+
= Ln - Ln = 0.
•
5.5.5. Proposition. The kernel of the mapping An : Rld-nR((z-x))
dnR-{{z-1))
->•
consists of linear combinations Y,CeTn(ez) (0
+
bd-n,
where ce and b are series in z _ 1 with constant coefficients. Proof. We start with lemmas. 5.5.6. Lemma. If Y = Y,o J"*9*- Vk G const.
• A and
[-&"> Y\-
=°
then y
= 2/0 =
In the expansion of the commutator [L n ,Y] the coefficient of QN-I-I c o n t a m s the variable Vj with the smallest index j = —i, it is involved in the term —iy'NV-i — NyNv'_t. This expression must vanish for alii > N. This yields yN = 0 unless N = 0 and y'0 = 0. D Proof.
5.5.7. Lemma. If oo
oo n —1
X = J2 xiz'^X = Z] E *taZ~i_10a e R/d-nR„ , An(X) = 0 and the constants in all Xj._ n , . X ^ - n + i , . . . , XJO are zero, then X = 0. Proof. The equation A"(X) = 0 can be written as a recurrence relation: A n ( 0 ) (Xt) + An^
(Xi+n)
= 0.
(5.5.8)
Let Xjj be the first nonvanishing term in a recurrence chain. Then it must be An(-°°\Xh) = 0, i.e. [Ln,(Xh)+]- [Ln,(Xh)-]+ = 0 from where we have [Ln, (Xh)+]_ = 0 and [Ln, (Xh)-]+ = 0. The first of these equalities implies (Xil)+ = const., according to the previous lemma. The assumptions of the present lemma imply that the constant is zero, (Xi1)+ — 0.
86
Soliton Equations and Hamiltonian
Systems
The second of the equalities, together with 1.8.10, yields (-X^)- = ad n, the recurrence relation (5.5.8) gives An^(Xh) + An^co\Xu+n) = 0, i.e. (Lnad~n)+Ln
- Ln(ad-nLn)+
= -[Ln, ( X l l + n ) + ] _ + [Ln, (Xil+n)-]+
.
Taking the projection on R+, we get [a,Ln]+ =
[Ln,(Xil+n)-]+.
On the right-hand side stands an operator of order < n — 2 while on the left-hand side there is a term na'dn~l hence a' = 0. According to the assumption, a = 0, i.e. X^ = 0 . • Return to the proof of Proposition 5.5.5. We must show that a linear combination of the resolvents and d~n can have any set of constants in the coefficients of d~n,...,d° since, according to Lemma 5.5.7, these coefficients uniquely determine a solution. All the constants in Tn(ez) we are talking about form the operator T™ = ^-n{tz)~%~1dl. Let e be a primitive root of 1. All the roots are 1, e, e 2 , . . . , e" _ 1 . Denote for simplicity of writing o Ti = (l/n)f £ l = (1/n) £ ( e * * ) - ' - ^ . i=—n
It is easy to see that n-l
] T tPzTj - znd~n = 1, j=o n-l
J2enjTj=dj=0
]T e^-^z-% = d~2 , j=0
n-l
£
f?jz3-nTj
= d-n+2 ,
3=0
J2e2jz2~nTj=d-n+1. j=o
(5.5.9)
87
The KP Hierarchy
In the reduction to the nth GD, when Ln is a purely differential operator, all X e R+ automatically belong to the kernel of the Hamilton mapping An. Therefore, (T")_ belongs to the kernel. This is why we called this object "a resolvent" in Chap. 1. Moreover, the first of equalities (5.5.9) reads now YllZo &zTj = znd~n modulo irrelevant nonnegative powers of d, i.e. d~n can also be expressed in a form of a linear combination of the basic resolvents and is not needed as an independent element of the basis of the kernel of the Hamiltonian mapping. This explains the difference between Propositions 1.8.9 and 5.5.5. 5.5.10. Definition. The negative part of the resolvent, T-(z) is called the principal part. We have seen that the negative part (more exactly, the constants in the negative part) uniquely determines the resolvent.
5.6
Hamiltonians of the K P Hierarchy
5.6.1. The coefficients of the expansion in powers of z~x of the expression [ res Tndx = ] T z^'1
f res Vc
are taken as Hamiltonians. 5.6.2. Proposition. The relation res Lrdx = -Lr~n — n 5L» J /
e
R/d~nR-
holds. This can also be written as
5L"J
res Tndx=
--^— T™. d(zn)
The proof is the same as of Propositions 3.5.2 and 3.5.3.
•
5.6.3. Proposition. The Hamiltonians hr = J res Udx generate, by virtue of the structures Hn^ and ffn(°°), the vector fields T -d[v-,L«] l
n
+
J
and
r -( n
88
Soliton Equations
and Hamiltonian
Systems
i.e. the equations (Ln)t = l[Lr+,Ln]
and
(Ln)t =
^[Ln,Lr+~n]
which are equivalent to Lt = -[Lr+,L] n
and
Lt =
n
-[L,Lr+n].
These equations differ from Eqs. (5.1.3) by nonimportant constants. Proof. According to (5.4.3) and (5.4.4), An^\dhr)
= -{{LnLr-n)+Ln
A^^idhi)
r n rn = -(\L - ( _,L - )_ n
- Ln(Lr-nLn)+) -
= - [-[Ln,Lr+],
[Ll,Lr_-n}+)
r - I'-({Ln,Lr+-n}--{Ln,LrSnU) n =
r
-{[Ln,Lr-n\„
+ [Ln,Lr+-n]+) =
T
-[Ln,Lr-n].
U
5.6.4. Proposition. The Hamiltonians hr = J res Lr are in involution with respect to all Poisson brackets. Proof. This follows from the recurrence relations (5.5.8) in the same manner as in 3.6.3. • 5.6.5. Finally, we come to the reduction to the submanifold Mo = 0, or u n _i = 0. In 5f/6Ln = 53-TO d~l~xXi the coefficient Xn_i becomes indefinite. The situation is similar to that in Chap. 3. Namely, this coefficient is not involved in An^°°\8f/5L), besides, the last expression is an operator of order not greater than n — 2. Thus, the first Hamiltonian mapping can be reduced to the submanifold UQ = 0. As to An^, the requirement that _4™(°) (Sf/5L) should be of order not greater than n — 2 yields the equation res[Ln,6f/6Ln}=0
(5.6.6)
from where X„_i can be expressed in terms of X, with i < n — 1. As we know, res[A, B], where A and B are operators, is an exact derivative of an element of A, i.e. Eq. (5.6.6) can be integrated. After that, X„_i enters the equation as a linear term and can be found explicitly. The expression is too cumbersome, and we skip it.
Chapter 6
Baker Function, r-Function
6.1
Dressing
6.1.1. The material of this chapter significantly uses works by Sato and other mathematicians of the Kyoto school (see the survey [DJKM]). In many cases it is convenient to represent the KP operator L = d + u\d~l + u2d~2 + • • • or its nth reduction (GD) in a "formal dressing" (or "sandwich") form: L = <$>d<$>-x,
(6.1.2)
where (f> is a $ D O (j> = ^ ^ ° Wid~l and w0 = 1. For example, we found in this form the soliton-type solutions in Chaps. 1 and 5 (where the role of <j> is played by $d~N). Equation (6.1.2) implies expressing all {ui} in terms of differential polynomials in {wi}. For example, Ui = —w'^, ui = — w^+wiw'^. This gives an embedding of the differential algebra Au into the algebra of differential polynomials in w^ which is called Aw • It is not difficult to show: 6.1.3. Lemma. The expressions of {u,} in terms of {u;,} have the form Ui = -w'i +
Qi(wi,...,Wi-i),
where Qi are differential polynomials. 6.1.4. Corollary. Any differential polynomial in {u>i} has a unique representation as a differential polynomial in {«j} with coefficients being ordinary polynomials in {uii}. The coefficients do not contain the generators {wi} if and only if the element of Aw belongs to the image of Au when it is embedded into Aw. The dressing operator
90
Soliton Equations
and Hamiltonian
Systems
6.1.5. Proposition. The vector fields dm can be extended to Aw by formulas dm<j)=-L^(f>.
(6.1.6)
They remain commutative. Proof. We have to prove that the vector field defined by (6.1.6) on Aw being restricted to Au coincides with the former dm. dmL = dm^d^1) = -L™^-1
= {dmCfyd^-1 + (frd^L™
^-'(dm^-1
= [L, L™\ = [L™, L].
Now we prove the commutativity. dm(dn
+ UlL™
= [L™, Ln]-<j> + Ln_L™4> = [L™, Ln_}-
6.1.7. Important remark about the notion of the integral. Any differential polynomial from Au is at the same time an element of Aw, f(u) = fi(w). This, however, does not enable us to write f f(u)dx = J f\{w)dx. The integral on the left is an integral in Au, i.e. a class of equivalence of f(u) with respect to addition of exact derivatives in Au, on the right there is the same with respect to Aw. The element which was not an exact derivative in Au can become that in Aw. For example, u\ = —w^. This does not mean that J u\dx = — J w[dx = 0. In the analytical terms, a fast decreasing at ±oo function can be a derivative of a function that is not decreasing, or a periodic function be a derivative of a non-periodic function. The bottom line is that, taking an integral, we always must remember with respect to which differential algebra this integral is considered. By default, integrals will be in Au unless otherwise stated.
6.2
Baker Function
Tne 6.2.1. Let £(t,z) = J2Tti*zk* D 0 a r e n o t r e a l operators: their action on functions is not defined unless they are not purely differential. However, their action on a special function £ will be defined by dm^(t, z) = zm and dmexp£(t,z) = z m exp£(i, z) for m both positive and negative
•
Baker Function,
91
T-Function
(recall that ii is identified with x). Then oo
w(t,z) = (j>exp^(t,z) = y^Wj2 l exp^(t, z) =
w(t,z)exp£(t,z)
i=0
is called the (formal) Baker, or wave, function (also called the BakerAkhiezer function). Let
z)
= wd~x exp£(i, z) - w'd~2 exp£(t, z) + w"d~3 exp£(i, z) + •. It is easy to prove 6.2.2. Proposition. The Baker function satisfies the equations Lkw = zkw ,
dmw = L™w ,
k e Z.
Indeed, the first equation is obtained applying both sides of the equation Lk(f) = <j>dk to exp £{t, z). Now, dmw = (9m) exp £(i, z) +
•
The Baker function is determined up to the multiplication by a constant series in z~x starting with 1. 6.2.3. Proposition. The adjoint Baker function satisfies the equations L*w* = zw*,
dmw* = -L™*w*.
The proof is similar to that of Proposition 6.2.2. 6.2.4. Proposition. Coefficients of the series in z~x: w^/w
belong to Au.
Proof. From Ln = dn + Un-id"" 1 + M n _ 2 d"- 2 + • • • one can find dn = Ln + a n _ i L n _ 1 + a„_2L™~"2 + • • • where a* e Au. Applying this to w and dividing by w, we obtain ui^^/w = zn + a n _ 2 -z n _ 1 + a„_2;z™~2 + • • •. • res
We will consider two types of residues, those of formal series in d: 9 S a »^' = a - i ' a n d those of formal series in z: res z Yl aizl — a-i-
92
Soliton Equations
and Hamiltonian
Systems
6.2.5. Very simple and extremely useful lemma. Let P and Q be two ^ D O , then res,[(Pe")-(Qe-")]=resaPQ*, where Q* is the formal adjoint of Q. Proof. The left-hand side is r e s z [ ( P e - ) . ( Q e — ) ] = res z ( £ > z * £ > ( - * ) ' ' ) =
£
(-lfr>ifc ,
and the right-hand side is res e (P-Q*) = r e s a 5 ] p i 5 i ( - d ) ^ = ij
£
(-l)J'p<<&.
•
»+j'=—1
6.2.6. Proposition (bilinear identity). The identity res2(9j1---^w)-u;* = 0 holds for any (i\,...,
im).
Proof. Since dmw = L™w, it suffices to consider only the case when all ik for k > 1 vanish. Then res z (a'w) • w* =
ieax(did^t'z))(
= r e s ^ d V e ^ X ^ T V * 2 ) = res 9 d*
/(*') = E ( * i - * i ) < 1 " - ( C - * m ) < m a i 1 - - - ^ / ( * ) A i ! - - - i m ! . It is amazing that all the equations of the hierarchy are contained in this bilinear identity: the following converse theorem holds.
•
Baker Function,
93
T-Function
6.2.7. Proposition. Let oo
oo
0
0
be formal expansions where Wi and w* are functions of variables U, and WQ
=
WQ
= 1. L e t
res 2 (d'u;) • w* — 0 for all integers i > 0. Then letting <j) = ^ o ° w^1 andw*(i,2) = («^*)- 1 exp(-^(t,z)). Moreover, let
we nave w =
0 e x P f (*> z )
Tesz(di1---d%w)-w*=0 hold for any multi-index, ik > 0. Then u; and w* are the Baker and the adjoint Baker functions of an operator L satisfying the equations of the KP hierarchy. Proof. Let oo
oo
0
0
whence w =
Then
resg di(fytp* = Tesz(di4>ei)(tpe"i) = res z dlw • w* = 0 by assumption. This is true for all i. Taking into account that <jnp* = 1 + X where X £ R-, i.e. it is an integral operator, we have (fnp* = 1 and ip = Let L = 4>d<j> 1. We have ((5 m 0) + L™0)e« = (dm-d>-
= {dm-
= 0.
94
Soliton Equations and Hamiltonian
Systems
This yields dm
• The corollary explains why the KP hierarchy is so ubiquitous.
6.3
Shift Operator and r-Function
6.3.1. The remarkable bilinear identity of the previous section has important ramifications. Among its corollaries are two equalities we shall obtain in this section. We start with a lemma. 6.3.2. Lemma, then
(i) If f(z) = Y^a aiz~%
ls a
formal series where ao = 1,
res^zXl-z/O-^CWO-l)More general, if /(£, z) = Y^oo ai(0z~i
then
ieSz f((,Z)(i - z/tr1 = a-(Cz)u=<, where f-{C,,z) = YX'ai(Oz~i(ii) I f / ( z ) = i;-oo «<«"'. then r e s ^ r ^ l - 2/C)- 1 + z~\l
- C / z ) " 1 ] / ^ ) = /(C) •
(Here (1 — -z/C) -1 is understood as a series in £ - 1 while (1 — C,/z)_1 is a series in z"1.)
Baker Function, T-Function
95
Proof, (i) oo
oo
res z £
oo
= 5 > ( C ) C i + 1 = C/-K,*)|z=C
0 , ( 0 * " ' • Y,(z/CY
j = —oo
j=0
i=l
and (ii) r e s ^ C ^ l - z/0'1
+ z-\±
OO
C/z)-1)!^)
~
OO
z_<
= res Y, °i i = —oo
•E
OO
ZJ 1
J
" C" ' = E
a _<
^
= /(0 •
D
i = —oo
j = —oo
Let G(C) be an operator acting on series in z~x with coefficients depending on variables t = (t\,ti, • • •) as 1
G(()f(t, z) =
1
1
f[t1--?,t2-^,t3-—,...,z
This is the shift operator. 6.3.3. Proposition. The following identities hold: w(t, z)-1 = G(z)w*(t, z)
(6.3.4)
and dlnw(t,z)
= [-G{z) + l}w1.
(6.3.5)
Proof. The bilinear identity implies res z w{t, z)G(C)w* {t, z) = 0. Taking into account the identity e x p £ ^ ° zk/k^k
= (1 - z/Q^1,
res z w(t, z)G(Qw* (t, z){\ - z/0'1
we have
= 0.
This expression can be transformed with the aid of the first part of the above lemma, we have the identity
<{u>(i,C)G(CK(t,0-i} = o whence Eq. (6.3.4) follows immediately. Similarly, res 2 dw(t, z)G{()w* (t, z) = 0
96
Soliton Equations and Hamiltonian
Systems
whence resz(dw(t, z) + zw(t, z))G(()w*(t, z)(l - z / C ) - 1 = 0. According to the lemma, this implies that [(dw(t, z) + zw(t, z))G(Qw'(t,
z)]-\z=i
= 0.
Calculate: [(dw(t, z) + zw(t, z))G(()w* (t, z)\= (dw(t, z) + zw(t, z))G(()w* (t, z) - {(dw(t, z) + zw(t, z))G(Ow* (t, z)}+ = (dw{t,z) + zw(t,z))G(Qw*(t,z)-z-w1+
G{C)wi.
Taking into account (6.3.4), we have
((w{tX)+w'{t,Q)w-\t,C,)-t-wi+G(QWl=Q, i.e. d\r\w{t,Cj = [-G(C) + l]wi.
•
6.3.6. One of the outstanding discoveries of Japanese mathematicians Hirota [Hir], Sato et al. was the so-called r-function. The idea is to replace infinitely many variables u0, U i , . . . or, alternatively, w\, W2,... by a single function r, describing the whole system. This is possible by the same reason why the n components of a potential vector field can be replaced by one single function, the potential function. 6.3.7. Lemma. Infinitely many functions a; = resL* which can be taken as independent generators of the differential algebra Au satisfy the equation res Ll = —diW\. In other words, a, form a potential vector field with the potential function — w\. Proof. We have 8$ = —Ll_(f>. Equating the residues on both sides, we obtain the required equality. D By the way, it is essential that the number of functions is equal to the dimension, i.e. the number of independent variables. If we considered n functions of one variable, we would fail to find one function representing the system. This is why it is so important to deal with the whole hierarchy and not with one, singled out, equation. Freezing all the time variables but a finite number of them, one can find only finite number of Mj's.
Baker Function,
97
r-Function
Let us look again at Eq. (6.3.5) that has the same effect: it expresses the whole function w in terms of one coefficient wi; unfortunately, only in a form of the derivative 91nw). Thus, there remains an indefinite factor that can depend on t2, h, It would be desirable to integrate this relation. Let us represent wi as a derivative, wi = — dlnr where r is determined up to a factor depending on t2, £3, Then (6.3.5) takes a form din w(t, z) = d[G(z) — 1] lnr(t). The next theorem states that it can be integrated without additional constants. 6.3.8. Theorem (Sato). A function r(ti, t2, • • •) can be chosen such that \nw{t,z)
= [G(z) -
l}lxiT(t)
or, in more detail, w(t,z)
T(h - l/z,t2
- l/{2z2),t3 - l / ( 3 z 3 ) , . . . ) T(tl,t2,t3,. ..)
(6.3.9)
and w = w(t,
z)ei{t>z).
The T-function is determined up to multiplication by cexp J3i° c«*j where c, c\, C2,... are arbitrary constants. One often writes this formula symbolically as w = r(t — [z _ 1 ])/r(i). Notice, that resL* = did In r. As a special case, we obtain resL = d2\n.r. This is the celebrated Hirota substitution which started the history of the T-function. The Hirota transform was prompted by soliton solutions (Sec. 5.3). Proposition 5.3.12 claims that resL = 9 2 l n A which coincides with the Hirota transform for r = A. Before we prove the theorem, let us discuss this example more carefully. A dressing operator was built there. In our present notations,
w(t,z) = — (N) 2/1
(N) IN
z-N Z-N+1
1
where A is the Wronskian of {yt}. Let us transform the determinant making zeros in the last column. If one subtracts from each row the next one divided
98
Soliton Equations and Hamiltonian
Systems
by z, then the entries of the last column vanish except the last one that is equal to 1. Other entries, except those of the last row, are Vk] ~ z-lyt+1)
= <4 ( l " ~)
ex?&><**) + ° * # (* - y ) ™P £(*.&),
where /?*. = eafc. Now,
and
This means that y^ (t) is replaced by ykl'(t— [z^1]) (see the above notation). The formula for w now reads: ., , "<''*>
=
A^-fz-1]) A(t) •
Thus, we proved: 6.3.10. Proposition. The Wronskian A of the functions y i , . . . , J/JV is the r-function of the ./V-soliton solution. 6.3.11. Proof of Theorem 6.3.8. Let us start with the following analysis. Suppose the formula lnw(t,z) = \G{z) — 1] lnr(t) is proven. The operator N(z) = — ^ ^ ° z~i~1dj + dz annihilates all functions of the form G{z)f{t). This is easy to check along with its other property: if / is a series E£° fa'*'1, then N(z)f = 0 implies / = 0. Apply N(z) to the supposed equality. We have N(z)\nw = —N(Z)ITIT = Y^Tz'^'1®]mr whence d j l n r = ieszz%N(z)kiw. On the other hand, the <9-derivative of the left-hand side must be equal to —diWi = r e s L \ Thus, we must do the following: introduce the quantities bi = res z zi ( - ^
z'^dj
+ d/dz J Inw
(6.3.12)
and prove: (i) dbi = a^ = resL 1 = —diW\ and (ii) dkbi = 3»6fe. After that we can put 6, = di l n r and, moving backward, recover the Sato equation.
Baker Function,
99
r-Function
(i) Equation (6.3.12) yields dh = res z zi ( - J T z^^dj
+ d/dz J w' • w'1.
Substituting w' • w-1 from (6.3.5), we get dbi = res z zi ( - ] T z-j-ldj
+ d/dz j (1 -
G(z))Wl
oo
= - r e s z zl Y^ z~j~ldjWi
= —diWi = resa L ! .
I
(ii) We have d(dkbi-dibk) = dkai-didk = 0. The expression d^bi-dibk is a differential polynomial in independent generators wi's (the derivatives diWk can be expressed as differential polynomials with the aid of hierarchy equations); it is easy to see that it has no constant term. Then it follows from the above equality that dkh — dibk = 0. • From (6.3.9) and (6.3.4) it is easy to obtain: * 2 ) ' h t1A323)' • • -} = g ^ I • (6.3.13) T(ti,t2,t3,...) T Notice a useful formula that immediately follows from (6.3.12):
iS'(t, z) =
T{h
+ l/Z t2
>
+ 1/(2
di l n r = res 2 zi ( - J T z'i^dj
+ d/dz j lniB.
(6.3.14)
6.3.15. Proposition. didilriT €
Au.
Proof. From (6.3.14): dtdilnr
= res z z i ( -JTz^^dj
+
d/dz)di\nw.
We have, dilnw = di ln(u;exp(-£(i,z))) = (Ll+w/w — zl). Ll+ is a differential operator hence, by virtue of 6.2.4, the coefficients of the expansion of Ll+w/w i n z - 1 belong to Au. •
100
Soliton Equations
and Hamiltonian
Systems
6.3.16. Remark. Even if we did not know that Ji = res^L1 = ddi\nr are first integrals of the KP hierarchy, we could get it now: di(ddi l n r ) = d{didi\ar) where (djdj l n r ) £ Au, so di f Jidx — 0. From a seemingly tautological fact: di{ddi\nr) = d(didi\nr) we have obtained a nontrivial conclusion. The nontrivial part of this proof is hidden in a chain of hardly noticeable assertions. It nevertheless requires considerable efforts: while di l n r is not a differential polynomial in {uk}, didi l n r are. 6.4
Resolvent and Baker Function. Fay Identities
6.4.1. The principal part of the resolvent was denned in 5.5.10: oo
T-(z)=
Ln_z-n~l.
^
(6.4.2)
n = —oo
6.4.3. Proposition. The principal part of the resolvent can be written as T^{z) = w(t,z)d~1w*(t,z).
(6.4.4)
Proof. We have oo
Ln_ = ( ^ S n 0 - 1 ) - = ^ a - ' r e s a ^ - 1 ^ " ^ " 1 . i=i
According to the "simple and useful lemma", this is oo
oo
5 3 a - < r e s z 5 i - 1 < ^ n e « * ' z ) ( ^ * ) - 1 e - £ ( t ' z ) = res z zn ^ a - * ^ " 1 * • w*) t=l
i=l
= res z
znwd-1-*
The residue of an operator is understood as j res ; \]aijz'd
ij
— 2_. I res0 \\aitjZl 3
\
*
J d* = V ^ a - i ^ d 5 . /
Now, veszzn(T-(z)-w(t,z)d-1w*(t,z)) for all n, which proves Eq. (6.4.4).
=0 •
6.4.5. Corollary. The coefficients of the series in z~x: ww* belong to Au.
Baker Function,
101
T-Function
There is another expression containing w which belongs to Au'6.4.6. Proposition. The coefficients of the series in z~x: x = w'/w belong to Au. Proof. The series L — d + u\d~x H can be converted: d — L + a\L~l + 02-L -2 + • • • with coefficients a* belonging to Au. Applying this to w, one getsu/ = (z + aiz-1 + a2z~2 -\ )w from where x = z + aiz~1+a2z~2-\ .
•
It is easy to obtain, as a corollary, that the expressions w^ /w have the same property since they can be expressed as differential polynomials of xThis follows either from the formula which is easy to prove by induction: ?/j( f c )
w
/w'\
,
= Pk[-), \ w
(6.4.7)
where Pk are the Faa di Bruno differential polynomials (see Sec. 5.3.10), or from the next lemma: 6.4.8. Lemma. The following formula holds / \
-—11
OO
°°
»-£a I - *—* 5>
-(fc) a
where a is a function. Proof of the lemma.
V
a \ 0
0
/v(fc)
°° =
aj
°°
a rv(fc+!)
y a -fc2Li_ya- f c - 1 ^—= 1.
n
a o o Let a = w. The coefficients of the expansion of (1 — w'/w)-1 of z~x belong to Au- On the other hand, they are ur f c '/w. One more expression for the principal part of the resolvent:
in powers
OO
T_(z) = ^ 5 - f e - 1 P f c ( x ) 5 , fc=0
x^w'/w,
S = ww*.
(6.4.9)
102
Soliton Equations and Hamiltonian
Systems
Indeed, oo
w k=0
= £ d ~ k ~ l w { k } w ~ l -ww* = J2d-k-1Pk{X)S. fc=0
(6.4.10)
k~0
6.4.11. Sato proved the following identities which are called Fay identities since they are analogous to the Fay identities in the theory of theta functions (we shall see later that in some special cases theta functions can play the role of tau functions). Let r(t+[s]±[z]) symbolize T(t1+s/l±z/l,t2 + s2/2±z2/2,t3 + s3/3± z ys,...).
6.4.12. Proposition (Fay identity). The r-function satisfies the identity (so - si)(*2 - s3)r(t + [s0] + [si])r(t + [s2] + [s3]) + cyclic(l, 2,3) = 0 where cyclic(l, 2,3) means all terms obtained by cyclic permutations s\, s2 and S3. Proof. We start with the bilinear identity res z r(t-y-
[2 _1 ])r(* + y+ [ z " 1 ] ^ - 2 ^VkZ"
= 0.
Let y = ([s 0 ]-[si]-[s 2 ]-[s 3 ])/2 and replace* by t+([s0} + [s1} + [s2} + [s3})/2. Using the formula exp(— XX a /^)V*) = 1 ~ a/b w e have res
—71—JI~Z8oli (1 - ZSi)(l
-ZS2)(1
r,^* -y - i * " 1 ^ ' + y + t z _ 1 ] ) = ° • -ZS3)
Applying Lemma 6.3.2, we obtain the needed identity. 6.4.13. holds
Proposition (Differential Fay identity).
•
The following identity
dr(t ~ [8l]) • r(t - [s 2 ]) - r(t - [ax]) • dr(t - [s 2 ])
+ (sT1 ~ *2 ^ M * - N ) • r(t - [s2]) - r(t) • r(t - M - [s2])} = 0. Proof. The Fay identity must be differentiated with respect to so letting s 0 = s 3 = 0, divided by sis2 and shifted t ^ t - [s\] - [s2]. •
Baker Function,
6.5
T-Function
103
Vertex Operators
6.5.1. Formulas (6.3.9) and (6.3.13) can be rewritten in terms of the so-called vertex operators. One can write f(t — [.z-1]) = exp(— E i ° dil («'))/(*)• Then w(t,z)-
^
Moreover, for every two non-commuting operators A and B an equation can be written eA•eB
= : e A + B :,
where the symbol of normal ordering ": :" means that in all monomials the operator A must be placed to the left of all J5's. Let ' di,
i> 0
Pi = \i\t\i\,
i<0.
These are Heisenberg generators. Then the last formula for w is w(t,z) =
— . (6.5.2) T(t) The symbol of normal ordering means that pi with negative i must be placed ope to the left of positive ones. The operator X{z) = : e x p E ^ o o P ' A ^ 1 ) • ^s called the vertex operator. Similarly, w (t,z) =
7-r T[t) Another vertex operator can be introduced: oo
/
— oo
v
.
(6.5.3)
\ r
'
It acts on T(£): X(A,M)r(ti,t 2 ,---) = exp ( - f > ( Y - A
exp ( f ) (±
- J L ) d}j rft.fa,...)
104
Soliton Equations
and Hamiltonian
Systems
= exp ( - ] T U(Xi - m) J r(t + [A"1] - [ M -1 ])
= exp ( - Y^ti(\i
- /!<) ) G*(A)G(/i)T(t).
6.5.4. Proposition. Operator X(A, fi) acts as an infinitesimal operator the space of r-functions, i.e. solving the differential equation dr/dt\ X\,nT where t\ is a variable, we obtain for each value of t\ a new function (which is called Sato's Backlund transform of the r-function). other words, this operator yields symmetries of the KP hierarchy.
in = TIn
Proof. A bilinear identity res z G(z)X(X,n)T(t)
• G*(z)T(i')e c ( t -''' 2 )
+ resz G(z)T(t) • G*{z)X'{X,^)T{t')ei{t-t''z)
=0
has to be proven. Here X'(X,fx) denotes X(X, pi) with t replaced by t'. Consider the first term. We have G(z) exp
- £ \
t4(A* - / ) ) = — ^
exp
- £
UQ? - /,<)
l
The first term is now xeszG(z)G*{X)G{lx)T{t)
• G*{z)T(t')e^t-t'^-^t^+^t^^—^-
• i — \l z
On the other hand, let us act by G*(A)G(/i) on the bilinear identity res z G(Z)T • G* (z)^*-*''z)
= 0.
We obtain res z G*(A)G( M )G(0)r(i) • G * ( Z ) r ( i ' ) e ? ( t - t ' ' z ) f = ^ - 0. 1 — ZJ A
Multiplying this by (/i/A) exp(-£(i, X)+£(t, /x)), subtracting from the above written first term, then applying the second part of the lemma in Sec. 2 we
Baker Function, r -Function
105
get this term in a form res z G(z)Gr(\)G{ji)T(t)
• G*(z)r(t')
_€(t-t',z)-«(t>A)+€(t,M)(', _ ,.\ [
1
N
[z(l-X/z)
l
, +
X(l-z/X)_
= (A - /x)G(/i)r(i)G* (A)r(t') exp(£(i, M) - £(i, A)). If the second term is treated in the same way, one can see that they cancel.
• 6.5.5. Remark. The fact that X(X,fi.) acts as an infinitesimal operator means that r + eX(X, y\)r is a r function when e is infinitesimal. It is easy to see that X(X,H)T is itself a T-function, since a shift of a r function is a r-function, and its multiplication by an exponent of a linear function in ti's is a gauge transformation which is insignificant at all. Then the bilinear identity holds even for finite e, and this is a T-function for an arbitrary e. This remark belongs to Adler, Shiota, and van Moerbeke [ASvM94]. 6.5.6. The operator X(X,fi) can be considered as a generator of infinitesimal symmetries if expanded in double series, in p, — X and A:
X(A,/i)r=X! m=0
m! '
A-"-"Xim)T.
E
(6.5.7)
n = —oo
Differential operators W„ ' are taken as generators of a Lie algebra which is called Wi+00. It is easy to calculate:
i-\-j=n
Wi3) =
]T
(6.5.8) : PiPj : +(n + l)(n + 2)pn .
: PiPjPk : - ( n + 2) ^
i+j+k=n
i+j=n
Operators Ln = (1/2) Yli+j=n '• PiPi '• Raina [KR], Sec. 2.3).
are
Virasoro generators (see Kac and
6.5.9. We give another formula for oo
W^m\X) = Y,
X~n-mW^
106
Soliton
Equations
= £ „ = o ( ( M - X)m/mi)Win)W)-
(i.e. X(\,»)
and Hamiltonian
Systems
Let
oo
9
W = E
jtv
then
*( A >M) = : exp(0(A) - 0( M )) : .
i— — oo
The normal ordering permits to handle all operators as if they commute. Then . ee(X)e~e(n)
X{\n)
P»(A)
^
m!
"
.
£x=A
Now, H^m>(A) = : 8^e~e^
• ee™ : = : Qm{\)
:.
Polynomials <3m(A) satisfy the recursion relations <2o(A) = 1,
Q m + i(A) = {dx -
e'(X))Qm(X).
They are Qm(X) = P m (-0'(A)) where P m are the so-called Faa di Bruno polynomials (see 5.3.10). It is easy to prove by induction using the above recursion formula that W(m\X)
=
E
mi4-2m2H
m! hkrrik—m
} • • ••m,k • -
i \m.n\ mi>m 2\-
x (~dxe/ll)mi(-dl0/2\)m2
• • • (-«9^/fc!) mfc : .
Some useful formulas for the Faa di Bruno polynomials can be found in the Appendix to this chapter. 6.6
r-Function and Fock Representation
6.6.1. In this section we show an example of a r-function which emerges literally out of thin air due to the universal property of the KP hierarchy. The example is based on the Fock representation of the Clifford algebra and was suggested by the mathematicians of the Kyoto school, (see [DJKM]). We consider an associative algebra A over C with generators 1 and {V'miV'm}' w G Z satisfying the denning relations
llpm,1pn}+ = Wm,i>n]+ = °»
[^m,<]
where [a, b]+ — ab + ba. This is the Clifford algebra.
Baker Function,
107
r-Function
We further consider a left A-module j4|vac) with a cyclic vector denoted as |vac) consisting of vectors a|vac), a € A, the defining relations V>„|vac) = 0(n < 0 ) ,
V£|vac) = 0(n > 0)
(6.6.2)
being imposed. In the same way the right A-module (vac|yl is defined consisting of vectors (vac|a, a € A with relations (vaclV'n = 0 (n > 0),
(vacIV'* = 0 (n < 0).
(6.6.3)
A pairing between (vac|yl and j4|vac) can thus be defined. Let (vac|a and 6|vac) be two elements. Then, using commutation rules and relations (6.6.2) and (6.6.3), the expression (vac|ab|vac) can be transformed to the form A(vac|l|vac) where A e C . It remains to put (vac|l|vac) = 1. Expressions (vac|a|vac) are called vacuum expectations and are denoted simply as (a). Let (vac | a • 61 vac) = {ab). This is the required pairing. 6.6.4. It is easy to see that
where [, ] is a usual commutator. This means that expressions ^ cm„i/>m'0n form a Lie algebra denoted as Q{V, V*). This algebra acts in the vector spaces V = £ m 6 Z C^m and V* = J2mez C d as [ V ' m C <Ap] = 5npi>m ,
[ t C . V£] = SmpVn
•
(6.6.5)
The spaces V and V* can be considered as dual with respect to the pairing (i/'m,V'n) = &mn- (This pairing differs from (V'mV'n) which was equal to 1 only if m = n < 0, it vanished otherwise.) It is easy to check that the actions of i>mipn m ^ a n ^ m ^ * a r e antiadjoint:
([«;,^],^1,<) = -W
(6-6.6)
The Lie algebra Q{V, V*) can be integrated in A up to a Lie group G(V,V*), e.g. a one-parametric group corresponding to ipmip^, m ^ n, is exp(ii/'mi/'n) = 1 + tymi>n- The action of the group G(V, V*) on V and V*
108
Soliton Equations and Hamiltonian
Systems
related to the action (6.6.6) of the algebra G(V, V*) is v H-> gvg~x ,v e V,
v* >-» g~xv*g, v* £ V*.
Prom (6.6.6) we have ( f f u , ^ - 1 , ^ ) = (vi,g~1V2g). In biorthogonal bases {ipm} and {VCJ, operators # • g~x and g _ 1 - g in V and V* have adjoint matrices: gipng"1
= ^2
a
mn^m
,
S'Vnfl' = E
a m
" V'm •
(6.6.7)
6.6.8. Definitions. (1) If t 1 ; <2, • • • is a set of parameters ("times") then the Hamiltonian is defined as oo
oo
#(*) = $ > £ ^»^+i/=1
n = —oo
(2) If
.
(3) The r-function related to g is r(t,g) = (g(t)) =
(eH^g)
(note that the last equality (g(t)) = (eH^g) follows from H(t)\vac) = 0, which is easy to verify). We further prove the following main proposition. 6.6.9. Proposition. r(t,g) is a r-function of the KP-hierarchy. The proof will be preceded by some preparations. A r-function generates Baker functions w and w*, namely, w(t, z) = G{z)T{t)/T{t),
w* (t, z) = G*
(z)r(t)/T(t),
where G(z)/(*i,<2, ...) = / ( * ! - * -1 >*2 - \Z~2M G*{z)f{UM, -..) = / (ti + *_1»*2 + \z~2^
- \Z~^
• • •) .
+ \z~3'
• • •)
(see Sec. 6.3). In order to prove Proposition 6.6.9 we must verify the bilinear identity 6.2.6.
Baker Function,
109
r-Function
Let OO
OO
i= — oo
i=—oo
6.6.10. Lemma. Relations
hold. Proof. We have OO
OO
[H(t),iP(z)}= £ HMUi= — oo OO
Y,
WW
i = — oo OO
OO
OO
= E E*< E ^MS-HUM'- E W*)* 4 i = —00 / = 1 oo
n = — oo
oo
i = — oo oo
oo
jz
5] 5 > ^ v = E* ' E ^^ = ^,^w-
i = —oo Z=l
i=l
i = — oo
Let u(a) = eaH^{z)e-aH. Then u(0) = VO2), du/da = eaH[H, ip(z)] aH aH = £(t,z)u{a) andu{a) = ea^z^{z). Letting e-aH = £(t,z)e il>{z)ea = 1 we obtain the required equation. The second relation can be obtained similarly. • 6.6.11. Lemma. Relations G(z)(eHMg) G*(z)(eH^g)
= (roeHW4>(z)g) =
r
1
•e ~ ^ ,
^ . / ' ' ' ^ ^ ) ^ ' '
hold. Proof. These relations can be proved for all g € A, not just for g G G(V,V*). From the very beginning it can be assumed that in every term of g the number of multipliers ipi is equal to that of ip*, otherwise the term vanishes. Thus one must prove the relation G(z)(eHV
. rph • . . ^ *
•••*) = e-^'Hr^Hiz)^
• • • V& • • •) •
110
Soliton Equations
and Hamiltonian
Systems
One can pass to generating functions: G(z)<e ff <'ty(Pi)" •" 1>
••r(qi)) • • • tf(p,Wfai) • • • r(qi)) •
According to Lemma 6.6.10, this is equivalent to G(z) exp[£(*,Pi) + • • - + £(i,pj) - £(t, qi) x (V»(pi) • • • ip(pi)ip*(qi)
£(i, Qi)}
• • • ip*(qi))
= exp[£(i,pi) + • • • + £(*,Pl) - £(i, 91)
£(i,«)]
x ( ^ ( ^ ( p ! ) • • • ^ W ( < ? i ) • • • V*(«R)> • Now we note that G(z)exp^(t,Pi)
= e x p j ] ( i, - y * " ' ) Pi = e x p ^ p ' f 1 - ~ ) i=i
= exp£(t,pi)
^
j=i
'
-G(z)exp[-f (*,&)] =
^
z
'
exp[-^(t,qi)}
z
z-qi
Thus, one has to verify the relation (roi>(z)ip(pi)---(piW(qi)---r(qi))
=
(z-pi)---(z-pi) (z-qi)---(z-qi)
x(ip(pi)---(pi)r(qi)---r(qi))^
On the left-hand side, tpo + ip-{z) = ^2_00 ipiZ1 can be substituted for tp(z). Then we note that [ip-(z),ip*(q)}+ = £ l L z^'* = q/(z - q). This implies that the left-hand side has the form a + J2s=i bs/(qs — z), where a and bs are independent of z. The term a is a = (VSlM>(Pi) • • • iPiWiqx)
• • • r(qi))
= mPi)
• • • (jPiWiqi) • ••riqi))
In other words, the left-hand side is a(zl + Pi-i(z))/(z — qi)- • • (z — qi), where P;_i is a polynomial of a degree < I — 1. It remains to add that this expression vanishes if z = p\,... ,pi. Indeed, ip(z)ip(pi) • • • ip(pi) = 0 if z=
Pi,---,
Pi-
The rest is clear: zl + P;_i = (z — p\) • • • (z — p{). The second relation can be proved in the same way. •
•
111
Baker Function, r-Function
Now the proof of Proposition 6.6.9: reszw(t,z)w*(t',z)
reszz-1{r0eH{t)ip{z)g)(iP-ieH^r9)/T(t,g)T(t',g)
=
Y,^oeH{t)i>n9)^-ieH^rng)/T(t,g)T(t',g).
= n
Recalling (6.6.7), we obtain Y^Y.^eH(tHng)^-ieH^grmanm)lT{t,g)T{t',g) n
m
J2WoeH{t)9i>m)(*l>-ieH{t,)grm)/T(t,9)T(t',g).
= m
This expression vanishes since one of ^ m |vac) and •i/'mlvac) vanishes for each m. • This fact seems to be very striking: solutions of the KP-hierarchy arising "from nothing" are obtained. We stop at this place, but there are many other remarkable facts in this direction, e.g. action of the group GL(oo) in the space of r-functions etc.
6A. Appendix. List of Useful Formulas for the Faa di Bruno Polynomials Faa di Bruno polynomials were defined in 5.3.10 as Pk(x) = (<9 + x) fc l. The following formulas involving Faa di Bruno polynomials hold: (d + x) fe = E Q W a ( x ) d Q=0 ^
a
,
(6.A.1)
'
(d ~ X)k = E t - 1 ) " " " ( * V
° ^-«(X).
Pk(x) = J2(~l)k-a(k)da(d-X)k-a, a=0
Pk(x) = E ^ " 1 ) " (^ a=0
(6-A.2)
(6.A.3)
^ '
( S + X)k~ada ,
(6.A.4)
Soliton Equations
112
and Hamiltonian
^ = E(a)Pa(x)(a"x)fc_a' a=0 ^
Systems
(6 A 5)
-"
'
dk = J2(-lT(ka)(d
+ x)k-a°Pa(x),
(6-A.6)
a=0
Pkix) =
k\
E x
(
s
r
(
, rai!ra2!-- -m*! $
)
- . . .
m
- .
(
,,7)
Chapter 7
Additional Symmetries, String Equation
7.1
Additional Symmetries
7.1.1. Commutativity of flows generated by the equations of the KP hierarchy means that each of them is a symmetry to all others. There are, however, symmetries which do not belong to the hierarchy itself. They are called additional symmetries. They do not commute between themselves. The hallmark of these symmetries is their explicit dependence on the variables £,. The additional symmetries were discovered independently at least twice, in quite different environments and philosophy. Only recently it became clear that they dealt with the same object (see Proposition 7.2.4 below). The additional symmetries were found by Sato et al. in the form of Backlund transformations of r-functions (see 6.5.4). They appeared also in works by Chen, Lee and Lin [CLL], Focas and Fuchssteiner [FF], and in the most explicit and handy form by Orlov and Schulman [OS], from the point of view of differential operators. In the below presentation of the theory we follow Orlov and Shulman's ideas. The additional symmetries form the so-called centerless Wi+oo-algebra, or the Woo algebra. They are the more important as they are involved in the so-called string equation and in the generalized Virasoro constraints in matrix models of the 2-d quantum gravity. We have seen that dk - Lk+ =
d")^1,
where L is the KP operator. Dressing an obvious relation [dk — dk,d] = 0 we obtain the hierarchy equation [dk — £+, £] = 0 . 113
114
Soliton Equations and Hamiltonian
Systems
There is another operator commuting with dk — dk. This is oo
r = 53*jia*-1. i
Dressing the relation [dk — dk,T] = 0 one obtains [dk — £+, M] = 0, or dkM = [Lk+, M],
where M = c ^ " 1 •
(7.1-2)
Applying both sides of the equality M
= 8™zlw,
MmLlw
= zld™w.
The commutation rule [L, M] = 1 (following from [d,T] = 1) implies that the correspondence Li->z and M <-^ dz can be extended to an anti-isomorphism of algebras generated by L, M and z,dz. 7.1.3. Definition. An additional symmetry is a solution to the differential equation 8tm^ =
-(MmLl)-4>,
where 9;*m symbolizes a derivative with respect to some additional variable l
lm-
7.1.4. Lemma. The equation dtmL =
-[(MmLl).,L]
holds which implies d\mLk = -[(MmLl)-,Lk\. d?mM =
Similarly,
-[(MmLl)_,M},
whence for any k and n d*lmMnLk
= -[(MmL')_,M"Lfc].
(7.1.5)
This is a simple corollary of the definition and the dressing formulas. 7.1.6. Proposition. The operators c?j* commute with all dk, i.e. they indeed determine symmetries.
Additional
Symmetries,
String Equation
115
Proof. We have [d*lm, dk)<j> = -d\mLk_4> + dk{MmLl)-<j> = [(MmL')_,Lfc]_ + Lk_(MmLl)-
{MmLl)-Lk_
([Lk+,Mm}Ll)-
+
= [ ( M m L ^ _ , L * ] _ 0 + [ L * , ( M m Z / ) _ ] _ c £ = O.
D
7.1.7. Proposition. The correspondence between generators
extends to an isomorphism between Lie algebras. Proof. We have
PfTO. dU
dln{MmL1)^
= {{MmLl)-,MnLk]„
{{MnLk)_,MmLl}-(t>
\{MnLk)-,{MmLl)-]
+
-[{MnLk)-,{MmLl)+]-
[MmLl,MnLk}-4>.
Let [zldm,Zkd?]
=J2Cp™){kn)zPdz
•
pq
Then
[MnLk, MmLL] = J2 C$™){kn)MqLp . pq
Therefore,
Mm, dln)
C#n>
pq
= ECwm)(fcn)5^-
D
pq
7.1.8. Remark. It may seem strange and confusing that dk commutes with dim, as we have seen before, while zk does not commute with zld™. The clue is that zk relates to d£ fc which cannot be identified with dk in this proposition, though their actions on <j> coincide. Their actions on M do
116
Soliton Equations and Hamiltonian
Systems
not: d^kM = - [ L * , M ] while dkM = [L+,M] and this is not the same. The reason for this distinction is the explicit dependence M on x. 7.1.9. Remark. There are some transformations of independent variables and of the dressing operator
r1
k
[M,L] = tf> J2ktkd
~\d
Lfe=i
—[M-,L].
-1.
Then fc-i
r 1 + fa^1 = ]T ktkLk~1 + 4>x$
fc=2
k=2
oo
oo
= Y,ktkLx +
^2wk(-k)d-k-1(f>-1
fc-i
fc=2
k=l
whence M+ = £ ~ fcffci/p1 + x and [M+,L] = Y% ktkdk-iL + [x,L]. Therefore, [M_,L] = - 1 + [L,x] — ^ 2 ° ktkdk~iL. The equation has the form OoiL = J2ukkd~k-1
+ I 5^tofe0fe-i
fc=0
\fc=2 OO
doiUi - (i-
l)«i-i + Y
ktkdk-im
.
fc=2
If, e.g. {ui} depend only on three variables, x,t2 = y,h = t (recall the KP equation, Sec. 5.1.8), then the additional symmetry is OQIUI
= 2yu[ + 4ttuiy,
d0iu2 = U\ + 2yu'2 + 4tu 2y •
Additional Symmetries, String Equation
117
7.1.11. Remark. The majority of additional symmetries suffer from a shortcoming: they can be expressed locally, as differential polynomials, only in terms of {wi} but not of {ui}. Some of them, however, are local even in terms of {ut}, as the above example shows. 7.2
Generating Function for Additional Symmetries
7.2.1. The double expansion °° In - \\m
Y(X,fi) = J2
°°
,
m=0
\-l-m-\MmLm+l).
J2 i=—oo
is the generator of the additional symmetries. 7.2.2. Proposition [Di95(a)]. w{t,n)d-1w*(t,\).
Y{\,fj,) =
(Notice that when A = n this formula goes to the formula for the principal part of the resolvent (6.4.4), i.e. for the generator of inner symmetries of the hierarchy.) Proof. It is similar to that in (6.4.4). oo
(MmLm+l)_
Y^d-iiesd{di-lMm(t>dm+l(f>-1)
= I oo
= ^ a ^ r e s ^ - ^ M " 1 ^ " ^ ) • f^e"5 l oo
= res z zm+l J2
d'\Mmw)^-l)w*
l oo
= res z zm+l Y, d-^d^w^-^w*
= res z zm+ld^wd-1wm
Now, ( \
ui
— rac
oo
oo
\
\
°°
°°
y(A,M)=res,^ £ 771=0 / = — OO
1 z(l-X/z)
2 m+l
\m+l+l
. .
I m\
——.-fr-xrtpwar^
1 A(l-z/A)
.
Soliton Equations
118
and Hamiltonian
Systems
1
x exp((/i - X)dz)w(t,z)d
w*(t,z)
7.2.3. Adler, Shiota and van Moerbeke [ASvM94] have shown that i+m m
'
m +1
for all m. This implies that the additional symmetries, in their action on r, coincide with the symmetries in Sec. 6.5.6. We shall give here a proof of this statement. It is equivalent to the following: 7.2.4. Proposition. The action of the infinitesimal operator X(X, n) on the Baker function w(t, z) generated by its action on T is connected with Y(A, fi) (see Sec. 6.5.6) by the formula
This means that the additional symmetries in Fuchssteiner-Chen, Lee and Lin-Orlov and Shulman sense are the same as Sato's vertex transformations. Proof. We have r(t - [z-1]) *(A,M) r(t)
r(t)Xr(t
s
= {r(t) exp^t
- [z-\
- [z- 1 ]) - r(t T*(t)
[z~l])XT(t)
- A + fi)r(t - [a" 1 ] + [A"1] - [M1])
- r ( t - [ z - 1 ] ) e x p ^ ! - A + M)r(< + [A"1] -
= | r ( t ) e x p ( - « i , A) + ^(t, H)) ( l ~ )
[M"1])}/^)
(l -
f)
x r M z l + ir1]-!/.1]) -r(t
- [z- 1 ])exp(-^(t, A) +i{t,n))r{t
+ [A"1] -
= exp(-£(i, A) + C(t, M))^ _1 ( l - ^ ) x {r(i' + [z'1} + \n'x)W - r(t' + [ii-l}W
+ [A_1])(^ - M)
+ [A-1] + [z-l])(z -
\)}/r2(t),
[^})^/r2{t)
Additional
Symmetries,
String
119
Equation
where t' = t—[z *] — [fi 1 ] . The expression in the braces can be transformed according to the Fay identity (see Sec. 6.4.12) (where so = 0, s\ = \~1,S2 = | i - \ s 3 = z-1). It is equal to -(/x - A)"r(i' + [z _1 ])T(t' + [A-1] + [/x"1]). In order to obtain the action of X on w(t, z) we must multiply this by exp
Z(t,z). Now, the equality e x p ^ , z ) e x p ( - ^ , A ) + ^ ( i ^ ) ) 2 : - 1 (l x (A - n)r{t - [^1])r{t
^)
+ [A"1] - [ z " 1 ] ) / ^ * )
= (A — ^i)w{t, pL)d~1w* {t,
\)w(t,z)exp£(t,z)
must be proven. Dividing by (A — n)w(t, fi) and multiplying by d we have aexp(«M) - « ( , A , ) , - . (l - * ) "
^(,
^
^
+ lA-.]Mt-^-])exp({(MHai|A))
or 3r(t + [A"1] - [z" 1 ]) • r(t) - r(t + [A"1] - [z" 1 ]) • 9r(t) = - ( z - X){r(t + [A"1] - [z-'Mt)
- r(t + [ A " 1 ] ) ^ - [z" 1 ])}
which is the differential Fay identity (Proposition 6.4.13), i.e. it is true. All the transformations are convertible. • 7.3
String Equation
7.3.1. In the string theory of modern physics the equation [Q,P] = 1 is known where Q and P are differential operators of orders, say, h and p. It is called the string equation. The problem is to find such pairs of operators. This resembles an older problem: to find pairs of commuting differential operators, [Q, P] = 0 which is closely related to the theory of the GD hierarchies of equations and will be discussed later. Now we are dealing with the string equation. In the works by HoSeong La [La] and, virtually, Goeree [Go] (though he does not say this explicitly) the problem is connected with the additional symmetries of the GD hierarchies discussed above. Both authors consider the case of lowest orders, h = 2,3. The string equation
120
Soliton Equations
and Hamiltonian
Systems
is equivalent to the requirement that the operator does not depend on the parameter of an additional symmetry. In [Di93] operators of any order are studied. There were also works (Fukuma, Kawai and Nakayama [FKN], Kac and Schwarz [KS] and Schwarz [Schw]) where the string equation is related to the Sato Grassmannian (see below). We mention here only a few articles devoted to this popular subject. 7.3.2. The equation [M,L] = - 1 implies [M,Ln\ = -nL71-1. [ML-n+1,Ln] The additional symmetry dtn+11Ln as dln+1ALn
=
Therefore,
-n.
= - [ ( M L ~ " + 1 ) _ , L n ] can be written
= {(ML-n+1)+,
Ln]+n.
(7.3.3)
The operator P = n~1(ML~n+1)+ is of an infinite order. If we wish to have an operator of a finite order, say p, then we must take in T only finite number of variables U distinct from zero, namely, i < p + n + 1. The flows di with larger i do not preserve the order of the operator P. Suppose that Ln is a purely differential operator, i.e. belongs to the nth GD reduction of KP. The string equation is a condition of the independence of this operator of the additional variable t*_n+ll: d'Ln+11Ln = 0. According to Eq. (7.3.3), this is [Ln, P} = 1.
(7.3.4)
Now we give explicit formulas for the action of d*0 and c^*+1 x on w(t, z). 7.3.5. Proposition. The following equations hold: f -zlw(t,z), if/<0 dt0w(t,z) = { , (7.3.6) 1,0 \(di-zl)w(t,z), if Z > 0 (we do not identify tf0 with U, see Remark 7.1.8, namely, d^0 does not act on £(t, x)) and 9t+i,Mt,
z) = ( ~z1+ldz \
~ lt-i +
J2
ktkdk+l J w(t, z)
k=-l+l
)
(7.3.7)
for I < 0 and df+ltlw(t,
z) = (~z1+ldz
+£
ktkdk+l + (
for I >0;
Additional Symmetries, String Equation
121
Proof. This is a simple calculation. dj*0 = —{(f>dl4>)-<j) which is — <j>dl if / < 0 and di
dt+ltl4> = -{MLl+1)-4> = - Uf^ktkdk-1dl+1r1)
= -[iYjktkdk+lr1) V
fe=l
4>-U Y khd^r1)
V k=~l+l
-l-\
I_
oo
= - Y, ktk
kt
A+l
k=-l+l
Since [d, x] = — [z,9 z ], we have an anti-isomorphism between functions of x, d and dz,z. The commutator [<j>, x] corresponds to [dz,w(t, z)\ = wz(t, z) which yields (7.3.7) if the operator is applied to exp£(i, z). Let I > 0. Then oo
fc=i oo
= Y ktkdk+lcj> - [, x]d1+l *;=i
+ ([, x}d1+l
Applying this to exp£(i,z) we obtain (7.3.8).
D
The expression for non-negative / is much more complicated than for negative ones. Let us write three first formulas, for / = 0,1 and 2. dlAw(t, z) = I -zdz + Y
d
kt
kdk I w(t, z),
2,iw(t, z) = I ~z2dz - wi + Y ktkdk+i \ fc=i
w(t, z),
and dlxw(t, z) — I -z3dz
- w\d - 2w2 + w\ + Y, ktkdk+2 fe=i
J w(t, z).
This page is intentionally left blank
Chapter 8
Grassmannian. Algebraic-Geometrical Krichever Solutions
8.1
Infinite-Dimensional Grassmannian
8.1.1. Revisiting the soliton solutions of Sec. 5.3. We want to generalize the soliton construction of Sec. 5.3.2. For that, we try to formulate in the possibly most general terms the feature of this construction which works in the proof that the constructed pseudodifferential operator is a solution to the KP hierarchy. The Baker function is w(t,z) = (f>exp^(t,z) = w(t,z)exp^(t,z) where <(> = $d~N. The function w(t,z) has the determinant expression
w(t,z) =
2/1
VN
1/i
VN
N
z
Z-N+1
(8.1.2)
(N-l) N
(N-l) 2/i (N)
VN
2/i
1
z-
1
The following property of the Baker function corresponds to the property $2/fc =0,fc = l , . . . , i V o f the dressing operator: w(t,ak)+akw(t,pk)
= 0,
k = l,...,N,
ak = ak(pk/ak)N.
(8.1.3)
A remarkable and, actually, decisive circumstance is the fact that the relations (8.1.3) do not depend on the variables tk. The Baker function is a series w
v zi E ^ i=-N 123
(8.1.4)
124
Soliton Equations
and Hamiltonian
Systems
satisfying (8.1.3). All the series of the form (8.1.4) satisfying (8.1.3) make a linear subspace W of the space H of all series. The Baker function is a special element of W that is singled out by its property: (w(t, z) exp(—£(£, z)))+ = 1. The conditions (8.1.3) give N equations for AT coefficients v-i,... ,V-N enabling one to find uniquely these coefficients if the non-negative part of the series is given. In other words, if a non-negative "tail" is given, one can attach to it, and uniquely, a negative "head" so that the resulting series belongs to W. In a generic case, i.e. for almost all values of the variables tk, the subspace Wexp(—£(i, z)) has the same property. If one takes the non-negative function 1 and attaches the negative head to it so that the sum belongs to Wexp(—£(t,z)) then he obtains w. As it will be shown, these general properties of the subspace W suffice for the constructed function w(t, z) to be a Baker function of the KP hierarchy. Thus, the solutions correspond to subspaces such that their projection to the subspace of non-negative series is a bijection. The manifold of such subspaces is called the (infinite-dimensional) Grassmannian. (This is not the general definition, see below, this is the so-called big cell of the Grassmannian.) We have not specified what kind of series we are talking about: one- or two-way infinite, or convergent etc. There are variants. Sato, who suggested this theory, considered formal series. In order to avoid multiplication of two-way infinite series, one must take some finiteness assumptions. Segal and Wilson [SW] suggested more analytical theory involving L2-convergent series on a circle. 8.1.5. Now we go to more precise definitions. We are close to the SegalWilson variant but in a less general form. More detail can be found in their article. Let H be a Hilbert space represented as a sum of two infinitedimensional subspaces, H = H+ © H~. The Grassmannian Gi H is a set of all subspaces W C H which differ not much from H+ in the following sense: if p+ and p- are orthogonal projectors on H+ and H- then the restriction of p+ to W,p+|,,, : W —t H+, is a Fredholm opertor, i.e. it has a finite-dimensional kernel and cokernel, and p-\w is a compact operator. Most interesting is the case where the index of the operator p+\w is zero, i.e. the kernel and the cokernel have the same dimension. The subspace W is said to be transversal to H-, or simply transversal, if p+\w '• W —> H+ is just a bijection.
Grassmannian.
Algebraic-Geometrical
Krichever
125
Solutions
The space H is always taken as £ 2 ( 5 1 ) , where S 1 = {z e C, |z| — 1}. The subspaces H+ and H- are spanned on the bases {zk}, fc > 0 and {zk}, k < 0 respectively. Now we consider some transformations of the Grassmannian. Let g(z) be a function holomorphic and non-vanishing in the circle \z\ < 1, and g(0) = 1. Then g(z) = exp^^°tfc,z fe = exp£(£, z), where {tfc} are some numbers; we assume that they are real. We consider the mapping
H^g-'H which can be written in the block form
- - ( : : ) according to the decomposition H = H+ © H-, a : H+ —• H+, b : i/_ —> H+, c : H- —> H-. We shall assume that for almost every set {tk} the subspace g~xW is transversal. 8.1.6. Definition. A function of z and g (i.e. on {tk})'- ww(g,z), which depends on W as a parameter is called a Baker function if for almost all {tk} -l
(i)
ww{g,z)eW,
(ii)
g~1ww(g,z)
= 1+ ^
a^ .
2 = —OO
The above condition of transversality implies that for a given W a unique Baker function exists: if g~lW is transversal then there is a unique element / e g~lW for which p+f = 1, i.e. / = 1 + £ l L aiZ\ Now we shall show that the Baker function in the new sense, ww(g, z), is the Baker function of the KP-hierarchy already familiar to us (see Chap. 6). 8.1.7. Proposition. If w = w\y is a Baker function, 8.1.6, then for every r > 2 a unique operator Br = dr + Br2dr~2 + • • • + Brr exists satisfying the equation drw = Brw, where dr — d/dtr, the coefficients {Bri} being differential polynomials in {a^}. Proof. It follows from w = g(l + S - o o aizl) drw = g(z)(zr + alZr-1
that
+
0(zr~2)),
dqw = g(z)(z" + axzq-1 +
0{z"-2)).
126
Soliton Equations
and Hamiltonian
Systems
Therefore Br can be chosen in such a way that g(z)0(z~1).
drw — Brw =
To the left stands an element v € W. We have g~xv = 0{z~1). vanish due to the fact that g'1W is transversal.
This must •
8.1.8. Corollary. A Baker function % , 8.1.6 is the Baker function of the KP hierarchy in the sense of 6.2.1, and Br = Lr_. Proof. This follows from the universal property of the KP hierarchy, 6.2.8. D Now we give a definition to the r-function. Let W be transversal. Then a mapping (p+\w)~x '• H+ -> W exists. We consider a mapping (p+\w)~l
H+
. . . g~\
_ i „ . P+v TJ
> W —> g
9 , TT
W —> H+ —> H+ .
8.1.9. Definition, r-function Tw(g) = Tw(ti,t2, • • •) (we consider it as a function of ti,<2, • • • depending on a parameter W) is the determinant of the above mapping: rw{g) = detgp + g~ 1 {p+\ w y l • We do not study the existence of this determinant in the general form (the mapping is close to the identical one). If necessary, it will be simply calculated. 8.1.10. Proposition. Let W be transversal and A : H+ -» H- be a mapping given by the graph W, i.e. A = p-{p+\w)~l, then rw(g)
= det(l +
a~lbA).
Proof. Represent the elements of the space H as pairs (x,y), yeH— Then gp+g~1(p+\w)~1x
= gp+g~l{x,Ax)
x € H+,
= gp+(ax + bAx,cAx)
= g(ax + bAx, 0) — a'1 (ax + bAx) = (1 + a~1bA)x e H+.
O
For the existence of the determinant it is sufficient that a~xbA has the trace.
Grassmannian. Algebraic-Geometrical Krichever Solutions
127
8.1.11. Lemma. The identity Tw(ggi) =
TW(g)Tg-iw(9i)
holds. Proof. To the right stands detgip+g^ detigip+g^ip+lg-iw^p+g^ip+lw^g). {p+\w)~1gx € VF and g~l(p+\w)~1gx p+ cancel out and we obtain det(g1p+g~1g-1(p+\w)-1g)
1
(p+|g-ny)_1-detgp+5'_1(p+|w)_1 = Further, if x € H+ then _1 e g W . Hence (p+\g-\w)~x and
= d e t ^ g i p + ^ i ) " 1 ^ ^ ) - 1 ) = rw(ggi)
• •
8.1.12. Proposition. The following connection between the T-function and the Baker function g-lww(g,
z) = TW(ti - 1/z, t2 - l/2z2, t3 - l / 3 z 3 , . . .)/TW{tx, t2,.. •)
holds (which is (6.3.9)). Proof. Let us apply Lemma 8.1.11 to g\{z) = 1 — z/(, where C is a fixed complex number, |C| > 1- We have gi(z) = explog(l — z/Q = ex P ( - J2zk/kCk), whence g(z)gi(z) = exp^i°(*fc ~~ l/kQk)zk and Tw(ggi)/TW{g)
= TW{h
- i/C,*2 -
i/2C2,...)/TW{t1,t2,...).
It remains to prove that Tg-\w{g{) = g~1ww(g,C)- According to Definition 8.1.6 this means: (i) as a function of £, Tg-iw(gi) belongs to g-xW; (ii) Tg-iwfa) = 1 + /o(C), where f0 e # _ . Let us represent the action of g^1 : H —> H in the above block form. A is the operator H+ —> i/_ corresponding to the graph g~lW. The action of b on the basis {z~k} of H- is
bz~k = (sr^-fc)+ = ((i + zK + *2/C2 + • • -)z~k)+ = CkgT\z). Further, a~xbz~k = g\{z)C,~kg^x{z) — (~k. Therefore, for any function fi(z) € H- we have a~1bfi(z) = /i(C). Thus, the operator a~lb sends all the functions of z to constants (£ is here a parameter), i.e. this is an operator of the rank 1. Then, so is the operator a~xbA, and det(l + a~xbA) = 1 + tr a~xbA. The trace will be calculated if we apply a~lbA to f(z) = l , a - 1 6 A ( l ) . We have A(l) = f0(z) € H-, and 1 + f0{z) e g~xW. Now det(l +a" 1 6A) = 1 + a,-xbf0(z) = 1 + / o « ) . This proves both (i) and (ii) at once. •
128
Soliton Equations
and Hamiltonian
Systems
8.1.13. Proposition. Let an element of the Grassmannian W € Grif satisfy the condition znW C W for some n (the submanifold of such elements we denote as GT^'H). Then Bnw = znw, dna,i = ^ n ^ i = n n d3nai = --- = 0, Bn=L {L _=Q). Proof. We have Bnw — znw € W and -l
Bnw - znw = dnw - znw = Y^(dnai)zi
• g.
(8.1.14)
— oo
As above, the fact that g~1W is transversal implies that YH®nO,i)zl = 0 and Bnw — znw = 0, i.e. L™ w = znw. On the other hand, Lnw = Ln
= [L™, Ln), i.e. 3mLn = [ ( L " ) ^ / n , Ln].
D
8.1.15. Example. Return to the example of the subspace W given by Eqs. (8.1.4) and (8.1.3). Let fa = ee*fc where en = 1. It is clear that if w e\V then znw e W. Thus, this W corresponds to the nth GD. Take the simplest case, n = 2, N = 1. The space W consists here of the functions having in z — 0 a pole of no more than the first order, regular in other points of the circle \z\ < 1, and satisfying the condition f(—p) = A/(p) for some A ^ 0 and p, 0 < \p\ < 1. The Baker function has the form w = exp£(£, z)(l+az~1) and satisfies the condition exp £(t, —p)(l—ap~1) = Aexp£(i,p)(l + ap^1). Denoting 6{t) = tip + t3p3 + t5p5 + ••• and A = exp(2a) we obtain a = — ptanh(9(t) + a). Now L = d2 + u, where u = —2a' = 2p2/ cosh 2 (0 + a). We have obtained the one-soliton solution of the GD hierarchy in the narrow sense, i.e. where n = 2: u = 2p2 cosh~2 (pt + p3t3 + p5t5 + • • • + a) (see (5.3.10)). The r-function can also be written. For the AT-soliton solution it has the form of a determinant similar to that obtained earlier (6.3.10), (see also [SW]). Thus, all the soliton solutions can be obtained. 8.2
Modified Definition of the Grassmannian r-Function
8.2.1. Sometimes it will be convenient to use a different definition of r-functions related to elements of the Grassmannian (see [Di93(a)]).
Grassmannian.
Algebraic-Geometrical
Krichever
Solutions
129
8.2.2. Definition. Let W be an element of the Grassmannian such that p+\w is a bijection. Let A : H+ —> if_ be a mapping given by the graph W, i.e. A = p _ ( p + | t y ) - 1 . Let lw '• H —• H- denote projection parallel to W, i.e. lw = P- — Ap+. Consider the mapping lw ° 9 '• H- -> H-- Then TW{t) = Tw(ff) = det(/ w o
fl).
(8.2.3)
8.2.4. Remark. If we consider the same mapping on i/_ extended by constants: fl_ = { S - o o ^ 2 1 } ^ e n *^ e Baker function ww can be denned as an element of the (one-dimensional) kernel of the mapping lw°9 '• H- —> H- (this element must be normalized). Also notice that while in the Segal-Wilson definition 8.1.9 the determinant of a mapping H+ —> H+ was involved, in Definition 8.2.2 this is the determinant of a mapping H- —> H-. 8.2.5. Proposition. Both definitions of the r-function are equivalent. Proof. The main trouble with this proof is in the fact that we must compare two mappings in different spaces, H+ in the first case and H- in the second. Let g~l{t,z) = X^o°Qs^ s , Qo = 1 be the expansion in z (the coefficients Qs are the so-called Schur polynomials in —t). We start with the second mapping. It sends the base elements to the following: z- fe ^ (gz-k)_
- A(gz-k)+
, fc = l , 2 , . . . .
Addition of a linear combination of the previous lines to any line does not affect the determinant. Therefore, the following mapping must have the same determinant:
The first term vanishes, so fc-i
z
k
H->
z
k
-Alg^QsZ-^*) 0
•
(8-2.6)
130
Soliton Equations
and Hamiltonian
Systems
The first definition 8.1.7 is connected with the mapping i € f f + ^ g((x + Ax) • g~1)+ =x + g(Ax • g~~1)+ e H+ .
(8.2.7)
The mapping is a sum of an identical mapping and a contracting one (more precisely, of the trace class): Tx = g{Ax • g~1)+. If Ax is expanded in the basis of H-, i.e. in z~k, then the image of T will be contained in the span of
A = g(Afk • g~l)+ + fk . If Afk € H- is expanded in basis: Afk = Y^jLi akjZ~i then the above mapping is fk*-*fk + Yl'jLi akjfj> *-e- * n e determinant we are looking for is det(ay -f Sij). Now, the following mapping has the same determinant: H. -> # _ : z~k M. z-k + Afk = z~h + A » ( z - V 1 ) + (notice this place: here we passed to mappings in i?_). It remains to transform: z~k +Ag(z-kg-1)+
= z~k + Ag L'1
= z~k + A\l
=
~Y,Qszs)
~Y,QsZsg\
z~k
z~k
z-k-A^2(Qsz'k+sg), s=0
which coincides with (8.2.6).
•
As an example of how easy it is to work with this new definition let us prove 8.2.8. Proposition. The Baker function ww(t,z) connected by Eq. (6.3.9).
and the r-function are
Proof. The plan of the proof will be the same as in Sec. 8.1: first we prove
131
Grassmannian. Algebraic-Geometrical Krichever Solutions
8.2.9. Lemma. For two different transformations g and g\ we have Tw{gg\) =
Tw(g)TWg-i(9i)
and then apply this formula for gi(t, z) = 1 — z/( where £ is a parameter. Proof of the Lemma. Tw(g) will symbolize the restriction of the operator lw°g to H-, hence Tw{g) = det Tw{g)- Let y e H-. We can decompose yg\ (in accordance with if = H-®g~xW) as j/gi = wg~1+yi where w e W and j/i G H_ are some elements. This means that j/i = 7V g -i(<7i)j/. Multiply the obtained equality by g: we have ygig = w + y\g. Again we decompose: V\g = w\ + 1/2 where iui e W and j / 2 £ # - , i-e. yi = Tw(g)yi- Now yg\g = w + u>i+Tw(s)7V g -i(0i)2/, i.e. Tw{g\g)y = Tw{g)TWg-x{g{)y and TW(ffi3) =
Tw{g)TWg-i{gl).
Thus, we obtained the required formula not only for determinants but even for mappings themselves which implies its correctness also for determinants. And this formula follows directly from definitions. • Nowlet f f l = 1-z/C, Kl > 1- Then 5 3 i = e x p X ; f t s z 5 e x p l n ( l - ; z / 0 = e x p ^ ^ ° ( i s — l/s(s)zs, i.e. multiplication of g by g\ corresponds to the translation of arguments: (ti,t2, •..) i-> (*i — l/£, * 2 _ 1/2C2, • • •)• It remains to prove that Tjyg-i(<7i) is equal to u>w(t,C)To this end we calculate the mapping TWg-\ (gi) on the basis. For z~k, k > 2 we have that z~k(l - zjQ e ff_. Thus TWg-i{gi)z-k
= z~k
z-k+l
— ,
fc>2.
For k = 1 we have z _ 1 ( l — z/£) = z"1 — 1/C- In order to obtain the projection to H- parallel to Wg~1 recall that ibw(t, z) € Wg~l and ww — 1 € H-. We write: 2 _ 1 9i = z~x - -u>w(z) + -(ww(z) and T W g -i(9i)z~ 1 = z~l + {^-/QYLT expansion of the Baker function w.
WiZ l
~
wnere
- 1) v>i are coefficients of
132
Soliton Equations and Hamiltonian
Systems
The matrix of this mapping is
/1+u/i/C
WC
WC
-1/C
1
0
0
-1/C
1
--A ...
V
..."
/
The determinant can easily be calculated and it is equal to 1 + W\/C, + W2/C2 + w3/C3 H = ww(t, C) as required. • 8.3
Algebraic-Geometrical Solutions of Krichever
8.3.1. Algebraic-geometrical constructions of solutions of the KdV equation were suggested by Matveev, Its, Novikov, Krichever, Dubrovin and others (see, e.g., [DMN], [Its76], [Kri77], [Dub77]). Segal and Wilson [SW] have shown how naturally the Krichever method can be incorporated into the Grassmannian concept. In this section, it is supposed that the reader wields some basic principles of the Riemann surfaces theory. Some definitions and formulations of the necessary theorems are listed in the Appendix of this chapter. See also: Springer [Spr], Forster [For] (there is no 0-function in this book), and Dubrovin [Dub86] (excellent lectures not translated into English). Let X be a compact Riemann surface of the genus g. A divisor is a formal linear combination with integer coefficients of a finite number of points of the surface, D =
X I m"i^i>
Wj G Z .
We define the sheaf OD (see [For], Sec. 16.4). Let U C X be an open set. Then OD{U) consists of all meromorphic functions f in U submitted to the divisor —D. The latter means that, at the points of the divisor D, if rrii > 0 the function / must either be regular or have poles of no more than mjth order. If m, < 0 the function must have zeros of no less than - m , t h order. The set of all OD(U) is called the sheaf Op (with respect to the operation of restriction of functions to subsets). Recall also the definition of a cohomology group with coefficients in a sheaf (in this case in OD). Let {C/J = U be an open covering of X. A fc-cochain is a function CJofc)..ifc = C(fe)(Uio, Uh,..., Uik) which attaches to each ordered set of neighbourhoods Uio, Uix,..., Uik having a non-empty
Grassmannian.
Algebraic-Geometrical
Krichever
133
Solutions
intersection a function of Oo(Uio n Uix n • • • nU i k ). This C^...^ is assumed to be skew-symmetrical with respect to io,..., ik- The set of all cochains forms an abelian group (even linear space) and is denoted as Ck(U,Ou). The boundary operator 6 which increases the dimension of a cochain by 1 is defined as fc+i (=0
(for example, (iCC>) W l = c f - C$, {SC^)^ = c\% - c\H + C\H etc.). It is easy to check that S2 = 0. A cocycle is a cochain such that 5c = 0. The group of the cocycles is denoted as Zk; Zk = ker(Cfe -> Ck+1). A cochain C^ is called a coboundary if there is a f^k~^ € Ck~1 such that 5/
+ deg(£>),
where deg(D) is the degree of the divisor, J ^ m j . The theorem guarantees also that dimif 0 and dim if 1 are finite. It is easy to grasp what is H°(X, OD)- An element of this group is a set of functions c\ ' € 0{Ui,D) such that q - c | ' = 0 on the intersections Ui n Uj. This means that a global function c^ € 0(X, D) exists such that c\ ' are restrictions of c^ on Ui. There are no coboundaries of the zero dimension, hence H^(X,OD) 3* OD[X). 8.3.2. We fix a point Poo of the Riemann surface. Let z _ 1 be a local parameter in a neighborhood Ux of this point not containing the points of D. The point P^ itself corresponds to z~l = 0. Without loss of generality it can be assumed that the closed domain X ^ : \z\ > 1 is contained in Uoo. Let S1 be the curve \z\ = 1 and X0 = (X \ Xoo)- The local parameter z~x enables us to identity XQO with the closed domain \z\ > 1 of the Riemann sphere C P 1 .
134
Soliton Equations and Hamiltonian
Systems
Now we define an element W of the Grassmannian. This will be the set of functions on S1, i.e. of elements of the Hilbert space H, which are boundary values of functions of OJJ(XQ) on the Riemann surface. 8.3.3. Proposition. The operator p+\w : W —> H+ is a Fredholm operator. Its index is ind(W -> H+) = dim H0(X,OD)
- dimff 1 (A',0D) - 1 •
Proof. First let us study the projection W —> zH+. The kernel of this mapping, ker(W —> zH+), comprises the functions on S1 which, on the one hand, belong to W, i.e. they can be extended to Xo as elements of OD(XQ), and which, on the other hand, can be extended to Xoo as holomorphic functions. Thus, they belong to OJJ(X), i.e. they represent the elements of H0(X,OD). The cokernel of this mapping, coker (W -»• zH+) = zH+/lm(W -> zH+), consists of functions / € zH+. Two of such functions must be identified if there exist functions zH+) comprises all the functions on S1, identified if / i — /b = (v?o — foo^s1 > fo S OD(X0), ifoo e O D ( X O O ) . This coincides with the definition oiHl{X,0D). Thus, ker(W -» zHoo) = H°(X, OD) ,
cokei(W -» zH^)
= H^X,
OD) .
Therefore ind(W -> zH+) = dimtf 0 - d i m i / 1 . Now we consider the mapping W —»• H+. We have l e W since constants belong to OD(X0). Then 1 e ker(W ->• ,z# + ) but 1 i ker(W -» H+). We have dimker(iy —> H+) = dimker(W -> zi/+) — 1, and dimcoker(W —> # + ) = dimcoker(W -» ztf + ). Hence ind(W -> H+) = md(W -> * # + ) - 1 as required. At the same time we have obtained that the kernel and the cokernel are finite-dimensional since the corresponding cohomology groups are finite-dimensional. • 8.3.4. Corollary. The index of the operator p+\w '• W —¥ H+ is zero if and only if deg(D) = g. Proof. See the Riemann-Roch theorem.
•
Solely this case will be of interest to us. More than that, we shall assume that the divisor D consists of g distinct points, and all m, = 1.
Grassmannian.
Algebraic-Geometrical
Krichever
Solutions
135
8.3.5. Remark. We do not prove that the projector p-\w : W —> H- is compact since this is not important. We shall consider further the transformation W •->• exp£(x, z)W of the Grassmannian and the corresponding Baker function. However, the Baker function can be constructed explicitly in terms of the ^-function (about ^-functions see the Appendix of this chapter). 8.3.6. Baker-Akhiezer-Krichever written in the form
w{t,P)=explf^tklJ
X
lemma. The Baker function can be
nJS-hfco
0(AP) + E f *fcUfc ~ AD) - K)fl(-4(Poc) - A(D) - K) 6(A(P) - A(D) - K)e(^(Poo) + £ ~ = 1 tkUk - A(D) - K) '
Here tip^ is the Abel differential of the second kind, see 8.A.5, Jp tip = zk + YIT bkrz~r where z~l is a local parameter in Px,, 27riUfc are the vectors of /3-periods of the differentials tip^ : 2mUkj = L. ^ P ^ I a n d K the Riemann vector (8.A.4), A(P) is the Abel mapping of the point P, A(D) is the Abel mapping of the divisor D. Proof. The following facts must be verified, (i) That the function w(x, P) determined by this formula is single-valued on the Riemann surface, (ii) That it belongs to O(X0) in X0. (iii) That in XQQ it can be written as exp(^2xkzk)(l + Y^T' diZ~l). All this easily follows from properties of the ^-function. When passing round the a-contours w(x, P) evidently does not change. If it is the f3j contour, the exponential is multiplied by exp(2iri^2xkUkj). It is also clear that the quotient of 6-functions is multiplied by exp(—2m ^ XkUkj), i.e. w does not change either. Thus, (i) is proved. This formula defines w on the whole Riemann surface except at P = PQQ, the only singularities being the points of the divisor D, which are simple poles (see 8.A.4). This proves (ii). Finally, if P —> P ^ the quotient of the ^-functions tends to 1 and the exponential can be represented as e x p ^ ^ ° xjtzfc(l + Y^T aiz~l), a s required. (In the KdV-equation theory, expressions like this with quotients of ^-functions first appeared in an article by Its [Its76].)
136
Soliton Equations and Hamiltonian
Systems
8.3.7. Proposition. The r-function has the form Tw(h,t2, •••) = exp (y~] XiU + HijUtjJ x 6 aiPoo) + J2tkVk
- A(D) - K I ,
fe=i
where {A,} and {/%} are constants. Proof. Equation (6.3.9) has to be verified. Using 8.3.6 we can write exp(— J3 tkzk)w in Xoo as exp I ^2 bkrz~rtk
X
1 1 1 + "Y^cnz"1 j
e{A{P) + Y°?tkVk-A{D)-K) e(A(Poo) + ET tkUk - A(D) - K) •
The first two multipliers can easily be represented as exp ( ^ A ^ i - l / i ^ ) + 'Y^VijiU - 1/izi)(tJ
~ l/jzj))/exp
{^2XiU + ^Mi}*i*jj
if {A,} and {fiij} are properly chosen. Then we transform the ^-function in the numerator: e (U(Poo) + A(P) - A(Poo) + £
tkUk - A(D) - K ) .
According to 8.A.6 we have z_1
p
(A(P) - AiP^j
= f
Uj=
Jpx
Vi{z-X)dz-X
f Jo
oo
= £ *-{fc+1Vjfc)(0)/(* + 1)! fc=0 oo
.
z k 1
= -J2 ~ ~
nJ£ 1} /2«(* + i)
J
k=0 oo
& oo
1
= - E *-"- Uk+i,j/(k + 1) = - E *-"Ukj/k • fc=0
*;=!
Grassmannian.
Algebraic-Geometrical
Krichever
Solutions
137
This implies
9 (A(P) + £ tkUk - A(D) - K j
= 0 (A^)
+ £((t fc - l/kzk)\Jk - A(D) - K J .
The rest is clear.
•
8A. A p p e n d i x . Abel M a p p i n g a n d t h e 0-Function 8.A.I. If the genus of a Riemann surface is g this surface is homeomorphic to a sphere with g handles. A basic system of closed contours a.\,... ,ag, (3\,..., f3g can be chosen such that the only intersections among them are those of a, and ft with the same numbers i. Let the Riemann surface be covered with charts (t/j,z,), where z, are local parameters in open domains [7$, the transition from Zi to Zj in intersections UiDUj being holomorphic. If in any Ui a differential pi(zi)dzi with meromorphic ipi(zi) is given and in the common parts t/, n Uj, fi(zi)dzi = (fij(zj)dzj, then we say that there is an Abel differential il on the whole surface with restrictions Cl\ui = ipi(zi)dzi. The Abel differential is of the first kind if all the (fii(zi) are holomorphic. There are exactly g linearly independent differentials of the first kind u\,..., vg. They are normed if / u)j = Sij, which condition determines them uniquely. We shall always assume them normed. The numbers L ujj = JB^ are called /S-periods. The matrix B — (By) has the following properties: 1) Bij = Bjit 2) r = IraB is a positive definite matrix. We consider a g-dimensional vector A(P) — {fP u)j}, where Po is a fixed point of the Riemann surface and P is an arbitrary point. This vector is not uniquely determined, but depends on the path of integration. If the latter is changed, then a linear combination of a and /3-periods with integer coefficients can be added: (A(P))j ^ (A(P))j + Ylini^ij + 52?wii-By, i.e. A(P) >-> A(P) + Y^ni&i + 2 m » B i , where <5, is the vector with coordinates <5y, Bj is the vector with coordinates B^. Thus A{P) determines a mapping of the Riemann surface on the torus A = C9 /T, where T is the lattice generated by 2g vectors {<5i,Bi} (which are linearly independent over R). This mapping is called the Abel mapping, and the torus A is the Jacobi manifold J or the Jacobian of the Riemann surface.
138
Soliton Equations
and Hamiltonian
Systems
The Abel mapping extends by linearity to the divisors: A(YlnkPk)
=
EnkA(Pk). 8. A.2. Abel Theorem. The kernel of the Abel mapping consists of principal divisors. The latter means that they are divisors of zeros and poles of meromorphic functions on the surface. (If P*. is a zero of the function, then rifc > 0 and nk is the degree of this zero. If P\. is a pole, then nk < 0 and \nk\ is the degree of this pole.) Of special interest is the case of divisors of degree g with all nk = 1, i.e. of non-ordered sets of g points P\,..., Pg of the Riemann surface. All the sets of such kind form the symmetrical gth power of the Riemann surface. The Abel mapping has the form
A{P1,...,Pg)=\J2jJ"uA , j = l,...,g. The symmetrical gth power is a complex manifold of the complex dimension g and it is mapped on the Jacobian which is a manifold of the same dimension. A problem of conversion arises (the Jacobi inverse problem). It can be solved with the help of the ^-function. For an arbitrary p € C 9 , let 0(p) = J2 exp{7ri(Sk, k) + 27ri(p, k)} , kezs (5k, k) = ^
Bijkikj ,
(p, k) = ^pih
.
The series converges by virtue of the properties of the matrix B. The 6function has the properties
e(-p)=6(p);9(p
+ 5k)=9(p); (8.A.3)
0(p + Bfc) = 0(p) exp{-7rt(Bfcfe + 2pk)} . Note that the ^-function is not defined on the Jacobian because of the latter property. 8.A.4. Riemann Theorem. There are constants K = {&,}, i = l,...,g (Riemann constants) determined by the Riemann surface such that the set of points P i , . . . , Pg is a solution of the system of equations 9
[Pi
T2 i=l
Uj = lj, Jp
°
l=
(h,...,lg)eJ
Grassmannian.
Algebraic-Geometrical
Krichever
139
Solutions
if and only if Pi,..., Pg are the zeros of the function 6{P) = 0(A(P)-l-K) (which has exactly g zeros). Note that while the function B(P) is not uniquely determined on the Riemann surface (it is multivalued), its zeros are, since branches of 9(P) differ from each other by exponential factors. 8.A.5. We define now Abel differentials of the second and third kinds. The Abel differential of the second kind, £lP , k — 1,2,... has the only singularity at the point P which is a pole of the order k +1. The differential can be represented at this point as dz~k + (holomorphic differential), z being the local parameter at this point. Such a differential is uniquely determined if it is normalized: Vi f Q,\,' = 0. The Abel differential of the third kind CIPQ has the only singularities which are simple poles at the points P and Q with the residues +1 and — 1, respectively. It is uniquely determined by the same condition. 8.A.6. Proposition. If z is a local parameter in a neighborhood of the point P (the point P corresponds to z = 0) and u>i =
o(fc )
1
dk~l
2/z=o,
and p uii,
(see e.g. [Dub86] (6.11) and (6.12))
i =
l,...,g
i=l,...,g
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Chapter 9
Matrix First-Order Operator, AKNS-D Hierarchy
9.1
Hierarchy of Equations Generated by a First-Order Matrix Differential Operator
9.1.1. An essentially new hierarchy can be constructed based on a firstorder matrix differential operator l = d + U + zA, where A = d i a g ( a i , . . . , a „ ) = const.,
U = (uap),
a,/? = 1 , . . . , n ,
uaa=0,
with distinct aa. The first term is, in fact, Id where I is the matrix unity. The elements of the matrix U are generators of the differential algebra A. Any linear differential nth order equation can be reduced to a first-order system. This enables one to consider the GD hierarchies as reductions of the new hierarchy (Sec. 9.4). In this sense the new hierarchy is more general. The special case of the hierarchy when n = 2 is called AKNS (Ablowitz, Kaup, Newell, and Segur [AKNS73]). The general matrix case was studied by Dubrovin [Dub77] who significantly extended the theory (e.g., one of the first, he considered not separate equations but collections of them; the term "hierarchy" was coined later). This is why we call the hierarchy AKNS-D. In the next chapters we shall see that the AKNS-D hierarchy can be embedded, in its turn, in a vast ZS hierarchy, and even this is not the end. Thus, a "hierarchy of hierarchies" emerges. Resolvents are series R = J^°° RjZ~i that commute with 1, i.e. [l,i?]=0,
i.e. R' + [U + zA,R} = 0.
As it will be shown, their elements belong to A141
(9.1.2)
142
Soliton Equations and Hamiltonian
Systems
9.1.3. Proposition. The set of all resolvents is an algebra over the field of formal series c(z) = ^ ° ° CjZ -1 with constant scalar coefficients. The proof is obvious. 9.1.4. How can the resolvents be constructed? Without loss of generality one can take iQ = 0, this can always be achieved by multiplication by a suitable power of z. Equation (9.1.2) is the same as the recurrence relation R[ + [U,Ri] = [Ri+i,A],
i = -1,0,1,2,
The first of these equations, for i = — 1, implies that RQ is diagonal, Ro = B = diag(6i,... ,bn). The diagonal part of the second equation R'0 + [U,Ro] = [Ri,A] yields B' = 0 and R0 = B = const. The nondiagonal part of the same equation enables us to determine the non-diagonal elements. They belong to A. We continue. Suppose all matrices up to Ri are already found. From the ith equation we can express non-diagonal elements of the matrix i2»+i as differential polynomials of elements of U and Rj, j < i- The diagonal part of the (i + l)th equation gives derivatives (diag Ri+i)'. We shall prove that the integration can be performed in A. 9.1.5. Exercise. Find Ro,Ri and R%. Answer. (Ro)jk =bj5jk,
(Ri)jk = —
ujk
(j 7^ k),
(Ri)u = 0
dj — ak
(ttj - ak)2
J
£^k
a.j - ak \aj - ap
ap-akJ
9.1.6. Lemma. If i? is a resolvent, then tri? = const. Proof. It suffices to take the trace of Eq. (9.1.2).
•
9.1.7. Proposition. All the elements of Rk belong to A. Proof. As it was already stated, the recurrence procedure requires integration, and it is not known beforehand whether this integration can be performed within the algebra A- To have a formal right to do this we must properly extend the algebra. In the Appendix of this chapter we prove that there is an extension A\ of the algebra A, A C A\ enjoying the property: for any / G A\ there is an element g € A\ such that g' = f. The elements
Matrix First-Order
Operator, AKNS-D
Hierarchy
143
of all Rk belong to A\. We have to prove that they, in fact, belong to A. We use induction. For k = 0,1,2 we already know this. Suppose that for some k it is already proven that the non-diagonal elements of matrices Rj for j < k + 1 and the diagonal elements for j < k all belong to A. Moreover, suppose that all of them, except diagonal elements of RQ belong to .Ao, i-e- do not contain constants. According to 9.1.3, Rl for any I is a resolvent. Thus, tr Rl = ci(z) = const (Lemma 9.1.6). The coefficient in z~k yields tr R^Rk + Flk(Rj) = (ci)k where Fik(Rj) £ Ao- In other words, Sfl&iT^-^fcW = (ci)k ~ Fik(Rj)- At first we can assume that all the elements {bp} are distinct. Then this system of equations determines all the diagonal elements {Rk)pp as elements of A. These elements are determined by integration, i.e. up to constants. Therefore, by choice of constants, we can make them belong to AQ. The obtained matrices depend on the first of them, RQ, linearly. Therefore, the restriction that bp are supposed to be distinct is irrelevant. • 9.1.8. Proposition. Resolvents are uniquely determined by constants in the diagonal elements of all the matrices {Rk}Proof. It suffices to show that a resolvent vanishes if there are no constants in the diagonal parts of all terms {Rk}- We have already seen in 9.1.4 that the first non-vanishing term is always a constant diagonal matrix. • 9.1.9. Proposition. The algebra of resolvents (see 9.1.3) is n-dimensional over the ring of constant diagonal series. As a basis of the algebra, the resolvents Ra can be chosen such that RQ is the matrix
and the other i?£ do not contain constants. Thus, any resolvent can be represented in the form n
R=J2ca(z)Ra. a=l
The basic resolvents Ra have the properties: RaRf} = 5apRa and Y^l Ra = I, i.e. they make a spectral decomposition of the unity. Proof. The product and the sum of resolvents are resolvents. In order to prove the equality of two resolvents, it sufices to make sure that they have the same constants in their diagonal parts. It is clear that linear combinations R = X^a=i ca(z)Ra can have any set of constants in all terms
Soliton Equations
144
and Hamiltonian
Systems
Rk, i.e. all resolvents can be obtained. To prove the claimed properties of Ra, just compare the terms with z° on the left and on the right and show that they are equal. • 9.1.10. Proposition. All resolvents commute. Proof. RWRW
and # ( 2 ) fl ( 1 ) have the same constants.
•
9.1.11. Definition of the AKNS-D hierarchy. Let fc
i=0
where a = l,...,n,k = 0 , 1 , 2 , . . . , and let tka be some variables. The AKNS-D hierarchy is the collection of equations dkal = [Vka,l], dka = d/dtka
.
(9.1.12)
These equations can be written in various forms. For example, dka{U + zA) + 8Vka = [Vfca, U + zA],
(9.1.13)
which is the zero curvature form, or, taking into account the recurrence formula (9.1.5), dkaU=[A,R^+1]
or
dkaU + dRl = {Rt,U}.
(9.1.14)
From the first of these equations it is obvious that the diagonal elements of the matrix U are conserved by the equations. This is why the hierarchy could be restricted to the submanifold of matrices U with zero diagonal. 9.1.15. Proposition. By virtue of Eqs. (9.1.12), dkaR0
=
[Vka,R0\.
Proof. We have [1, R13] = 0 which yields
0 = dka\l,R^ = [[VfcQ,l],^] + frdkcRP] = -URP, Vka], 1] + [1, dkaR?] = [1, dkaRP - [Vka,R'3}}. Hence, dkaR@ — [Vka,R^] is a resolvent. As it is easily seen, this resolvent does not contain constants hence it must vanish. That completes the proof.
•
Matrix First-Order
Operator, AKNS-D
145
Hierarchy
9.1.16. Proposition. The equation dkaVip - dlffVka = [Vka, Vip] holds. Proof. dka(zlR?)+
- dw(zkRa)+
= (zl[Vka,R^})+
[Vka,Vw]
- (zk[Vl0,Ra})+
= \{zkRa)+,zlRP]+ = [(zkRa)+,
-
-
[Vka,Vl0]
- [(zlRf})+,zkRa}+
(zlRe)+]+ + [(zkRa)+,
-
[Vka,Vip]
{zlR?)-]+
-[{zlRt})+,zkRa]+-[Vka,Vw] = [Vka,Vl0] + [zkRa,zlR?\
- [V^, Vl0\ = 0.
•
9.1.17. Proposition. Vector fields {dka} commute. Proof. The action of [dka, dip] on generators of the differential algebra A is: (dkadi(} - diPdka)l = dka[Vip, 1] - dif}[Vka, 1] = [dkaVl0 - dl0Vka, 1] + [Vip, [Vka, 1]] - [Vka, [Vt0,1]] = [[Vka, Viff],l) + [Vip, [Vka,\\] - [Vka, [Vw, 1]] = 0 (using the Jacobi identity).
•
9.1.18. Exercise. Prove the identity - 2_, aadia = d. a
Recall that there was the identity d\ = d for KP. Now there are many variables t\a. The identity means that t\a and x are always involved in combinations (t\a — aax). 9.1.19. Exercise. Prove that flows defined by vector fields doa describe similarity transformations of the matrix U. More precisely, U{t0a,tla,t2a,...)
= e x p l ^ £ a i o a ) U(0,tla,t2a,...)exp
(-^-Ba<0a) •
Soliton Equations and Hamiltonian
146
Systems
9.1.20. The equations of hierarchy can be restricted to matrix submanifolds, namely, Lie subalgebras, for example, to SU(n, C). Take, as a basis, matrices 5l =
Ko-l)'
^
=
^3 = 2 (" o) •
2 (-1 o) '
The rules of commutation are [Si, S2] = S3, [S2, S3] = Si and [53, Si] = £2Let A = Si. Resolvents are also supposed to be in SU(n, C). This leaves only one resolvent that starts with RQ = Si, i.e. in the old notations R = (i/2)(R1 -R2). Correspondingly, we are writing dk = (i/2)(dki - ^ 2 ) . Let R = r^Si
+ rWS2 + r^S3
and
U = u2S2 + u3S3 .
(Strictly speaking, we should write U = U1S1 + U2S2 + U3S3, but it is easy to show that m is conserved, we can just restrict the solution to m = 0.) Recurrence relation from Sec. 9.1.4 takes the form r
( l ) '
+ M 2 r
( 3 ) _
U 3 r
( 2 )
J2)' + y,rW rt + u3ri (3)'
= 0 )
- r(3) —ri+1,
(1)
(2)
and the equation of the hierarchy (9.1.14) is dku2 + 0r£ 2) + u3rkl) = 0 , dkU3 + dr[3' - u2rk1' = 0. 9.1.21. Exercise. Write these equations for k — 2 and k = 3. Answer. If k = 2, then denoting t2 = t, dtf = f,
v>3 + u'2' + ~u2(ul + u\) = 0 , or, letting u = 112 + 1113, -iii + u" + ^u\u\2 = 0. it
This equation is called the nonlinear Schrodinger equation.
(9.1.22)
Matrix First-Order
Operator, AKNS-D
147
Hierarchy
If k — 3, then (denoting £3 = t) u + u'" + § | u | V = 0.
(9.1.23)
This equation admits the restriction to the real u: 3 u + u'" + -u2u'
=0.
(9.1.24)
This is none other than the modified KdV equation. 9.1.25. Proposition. The quantities J;/? = f tr ARfdx the equations of the hierarchy.
are first integrals of
Proof. dkaJiff=
[trA[(zkRa)+,R%dx
ftiA[(zkRa)-,R%dx.
= -
Below, in 9.3.12, it will be proven that all / tr A[R^, RPk]dx = 0. 9.2
•
Hamiltonian Structure
9.2.1. We do not assume now that diag U = 0. Later we discuss the reduction to the submanifold tr U = 0. The Lie algebra V consists of vector fields d
° = Yl aip/du<jk = E t r a{i)d/dU{i),
(d/dU<%k = dlduki .
These derivations commute with d and can be taken to A, the space of functionals / = / fdx where / e A. We have daf = / ^cijkdf/Sujkdx
= / traSf/SUdx,
(Sf/5U)jk
=
5f/5ukj.
The module Ct° is, as usual, A, and the dual space CI1 consists of matrices X, Xjk e A. The pairing is (da,X)
= I tr
aXdx.
The Hamiltonian mapping is X -> H(X) = dA{x),
A(X) = [1, X} = X' + [U, X] + z[A, X].
148
Soliton Equations and Hamiltonian Systems
9.2.2. Proposition. [dA(X),dA(Y)}
= dA([XiY]+dMX)Y-dMY)X)
•
Proof. We have \dA(X),dA(Y))
= 9dMX)A(Y)-8MY)A(X)
,
where dA(x)A{Y)
-
dA(Y)A{X)
= d[lx][l,Y] - (X o Y) = \[l,X},Y} + [l,d[hX]Y] = \X',Y] + {[U + zA,X],Y} + [1,d[hX]Y]
-(X<*Y)
-(X<*Y)
= [X, Y]' + [U + zA, [X, Y}} + [1, d[lx]Y - d[lY]X] = [1, [X, Y] + dA(x)Y
- dA(Y)X\ = A([X, Y] + dA(x)Y
-
dA(Y)X).
a Thus, Im H is a Lie subalgebra of V. We define the form w as usual: "{dA(X),dA(Y)) = (dMX),Y) = JtvA{X)Ydx. 9.2.4. Proposition. The form UJ is closed.
(9.2.3)
Proof. In the next calculation we use the following: for any three matrices a, b and c: tr [a, b]c = tr [b, c]a, da\ = a, and dc / tr A(a)bdx = / tr <9C([1, a]b)dx = / tr([c, a]b + A(a)dcb - A{b)dca)dx since the operator A is skew-symmetric. Now, d^(9A(Xi), dA(X2), dA{X3)) = dA(Xl)u(dA{X2),dA{X3)) - dA(xx) j"tr
- u)([dA{Xl),dA{X2)},dA{X3))
+ c.p.
A{X2)X3dx
+ J tr A(X3)([X1,X2] + dA(Xl)X2 - dA{X2)Xi)dx = ftr[A(X1),X2]X3dx
+ J ti
A(X2)dA{Xl)X3dx
+ c.p.
Matrix First-Order
-
Operator, AKNS-D
f ti A(X3)dA{Xi)X2dx
+
+ J tr A{Xz){dA{Xl)X2
ftrA(X3)[X1,X2]dx
- dA{Xa)Xx)dx
= 2 ftTA(X3)[Xi,X2}dx
+ c.p.
+ c.p. = 2 fti(X'3X1X2
+ [U+zA,X3][XltX2])dx + 2 ftT({X3X1X2y
149
Hierarchy
-
+ c.p. = 2 fti(U + zA)[X3,
- {X3X2X1)')dx
+ c.p. = 0 .
X3X2Xi [XuX2]]dx D
A vector field corresponds to each functional: / ^ dA(Sf/su) = d[d+u+zA, sf/su] •
(9.2.5)
The Poisson bracket is {/, 9}=
I tr A(Sf/SU)6g/5Udx
= f ti[l, 5f/6U]5g/SUdx.
(9.2.6)
Limiting values when z —> oo and z = 0 are {/, ff}(oo) = / tr[A, {/,ff} (0) = Jtr[d
Sf/5U]5g/SUdx,
+ U,
Sf/SU}Sg/5Udx.
9.2.7. Both structures can be restricted to the manifold tr U = 0. We need tr(5f/5U)' = 0 or, simpler, Xr(5f /5U) = 0. The variational derivatives 5f/Suij cannot be defined independently since ^2 uu = 0. We write Sf = J2 AijSuij. The result will not change if the same constant A is added to all diagonal elements An since ^ Sua = 0. This constant is irrelevant in the Poisson bracket, but we can normalize it by a requirement that ^ An = 0. Now let 5f/8uij = Aij. Notice that the first structure can be restricted even to the submanifold of matrices U with the zero diagonal. This is why the hierarchy was restricted to this submanifold. 9.2.8. The above constructions admit a group theoretical interpretation in the spirit of the theory of coadjoint representation of a Lie algebra, i.e. Poisson-Lie-Berezin-Kirillov-Kostant bracket (Sec. 2.4). This can easily
Soliton Equations and Hamiltonian
150
Systems
be done for H^°°\ The dual space SI1 has a Lie algebra structure with respect to the commutator [X, Y] = XY — YX. The Lie algebra V is considered now as a dual space to SI1 (its own commutator is ignored). The pairing remains the same: J tr aXdx. We have a d ( X ) y = [X, Y] and it is easy to see that ad*(X)a = [a,X]. The elements ad*(X)a are tangential to the orbit at the point a. Take a = A. The symplectic form is u([A,X],[A,Y])
= J ti [A, X]Yda
which is exactly the form (*/°°) defined above. Interpretation of the form u/°) is more difficult. We need the notion of the central extension of Lie algebras with the help of cocycles. Recall the definition of the cohomologies of Lie algebras. A function F(X, Y) where X and Y are elements of the algebra is called a cocycle if F([X, Y], Z)+c.p. = 0. In our case, we use the cocycle F(X, Y) = J tr X'Ydx. (Check its cocycle property!) The central extension of the Lie algebra SI1 is the Lie algebra SI1 whose elements are pairs (f,X) where / e A and X € SI1. The commutator is [(f,X),(g,Y)}=^JtrX'Ydx,[X,Y]>j
.
The Jacobi identity is equivalent to the cocycle property. The dual space comprises pairs (A, a) where a € V and A G R or C. The pairing is ((A, a), (/, X)) = Xf + J tr aXdx e A. We have ((X,a),[(f,X),(~g,Y)})
= ((\,a),
Q
tr X'Ydx,
[X, Y]
= fti(X'Y-X
+ a[X,Y])dx
= ((0,XX' +
[a,X]),(g,Y))
whence ad*(/,X)(A, a) = (0,XX'+[a,X}). we have ad*(f,X)(l,U)
In particular, at the point
= {0,X' + [U,X]).
(l,U),
Matrix First-Order
Operator, AKNS-D
151
Hierarchy
The symplectic form is W (ad*(/,X)(l,C/),ad*(5,y)(l,t/))
= {ad*(f,X)(l,U),(g,Y)) = f ti(X'
= ((0,X' + [U,X}),(~g,Y))
+[U,X])Ydx
which coincides with o/°) above. Note that in our context the group theoretical interpretation is not obligatory. This point of view is especially emphasized in the works by Gelfand and Dorfman [GDor79, 80, 81]. Many other authors prefer to consider Hamiltonian formalism as a manifestation of the group theoretical properties.
9.3
Hamiltonians of the A K N S - D Hierarchy
9.3.1. Proposition. Formal series oo
oo
j=0
i=-l
exist, where Aj are diagonal matrices, and the elements of the matrices fa and Aj belong to A, such that the equation d + U + zA = <j)(d + A)(l>-1
(9.3.2)
holds. 4> is determined up to a multiplication on the right by series J2™ CiZ~% where C; are diagonal and constant, Co = / . The choice of {Ci} can be fixed by the requirement that {<j))u = 0, i > 0, j = 1,..., n. Proof. Equation (9.3.2) is equivalent to (j)(d + A) = (d + U + zA)(f> =
cj>'k + U(j>k + A<j>k+1 = ^2 4>k-iK , i=-l
k =-1,0,1,2,...,
(0_i = 0).
152
Soliton Equations and Hamiltonian
Systems
We have A-l = A
-V) = ^diag,
[l,A] = £/„diag ,
where t/diag symbolizes a diagonal matrix with the same diagonal as U and f^idiag = U - f/diag- Further, A* = ( t ^ f c ) d i a g . fc-1
[A,0 fc+ i] = -0' fc - (U4>k)ndias + ^2
We put (fc+i)diag = 0 and determine in succession A^ and 4>k+\-
D
9.3.3. Proposition. The resolvent i ? B starting with RQ = B and without other constants can be represented as R
B
= <j>B<j)-1.
(9.3.4)
Proof. It is clear that this expression can be expanded into a series starting from RQ = B and having no more constants. Further, \d + U + zA,RB]
= (j)[d + \,B}<j)-1 = 0.
•
Equations (9.3.2) and (9.3.4) give the representation of the operator 1 and the resolvent RB by dressing. Sometimes another dressing is used: 1 =
(b) - V' + il>(U + zA) = Atp.
(9.3.5)
9.3.6. Proposition. The variational relation 6 tr AR = tr RZSU + d tT(6(/)Bipz - 4>zBSip) holds (the subscript z denotes a derivative with respect to z). Proof. Apply the operator 5 to Eqs. (9.3.5): 5(f)' + (U + zA)54> + 6U-<j> = 5
(9.3.7a) (9.3.7b)
Matrix First-Order
Operator, AKNS-D
153
Hierarchy
and take the derivative of the same equations with respect to z: 4>'z + (U + zA)4>z + A<j> = 4>ZK + <j>Az , -ip'z +ipz(U + zA) + tpA = Arpz + Az ip.
(9.3.8a) (9.3.8b)
Now, we multiply (9.3.7a) by Btpz on the right, add (9.3.8b) multiplied by (j>zB on the left and take the trace. Transform some of the terms: tT(6
+ tr
$ZB^
-(j>zBSip)
- tr{6cl>B(ipz(U + zA) + ipA- A-02 -
Azip)
+ ( ( [ / + zA)<j)z + A4>- <j>zA -
•
9.3.9. Corollary. The equation dti AR/6U
= RZ
holds. The same can be presented as StiARk+1/SU
=
-kRk.
If tr U = 0 then these equalities hold for the non-diagonal elements. Let
hi = ^ J tr AR^+1dx
(9.3.10)
be taken as Hamiltonians. 9.3.11. Proposition. Equation (9.1.14) is of Hamiltonian type, it relates to the Hamiltonian — hk+i in the ij(°°) structure and to hk in the H^ structure.
154
Soliton Equations
and Hamiltonian
Systems
Proof. H^(5hk/6U)
= -[d + U,Rk] =
H^(Shk+1/5U)
[A,Rk+1],
= -[A,Rk+1].
D
9.3.12. Proposition. The Hamiltonians (9.3.10) are in involution with respect to both structures, H^ and H(°°\ Proof. We have
{ C £ } ( o o ) = Jtr[A,Rg]R^dx = -{h^h?}™
.
In more detail: amk = tr[A,R»}RC
= -tr(d<_x +
[U,Rg^})R^
= -d tr R^R%
+ tr(dR° + [U,
= -dtr
-trlA-R&.il^-i
RZ^RC
= tr[A, R^]R%+1
R^\)R^
- d tr i ? * . ^ = a m -i.*+i - 0 tr C+i-Rfc7 •
We continue until the first index becomes negative and amk vanishes. Then tT
[A, Rm+i]Rk+i
•= -9(RmRk+i
+ Rm-iRk+2
H
1- Ro
which proves our assertion.
R
m+k+i) (9.3.13) •
In fact, Eq. (9.3.13) is richer than the statement of Proposition 9.3.12. The right-hand side of Eq. (9.3.13) vanishes by integration and does not play any role in 9.3.12, but is very important when we consider stationary equations in Chap. 17. 9.3.14. Corollary. All the vector fields d[A,Rk] commute. This was already proven in 9.1.17. 9.3.15. Corollary. All the Hamiltonians are first integrals in involution of the equations of the hierarchy. 9.4
G D Hierarchies as Reductions of the Matrix Hierarchies (Drinfeld—Sokolov Reduction)
9.4.1. The idea of this construction is based on the simple observation that a scalar nth order differential equation is equivalent to a system of n
Matrix First-Order
Operator, AKNS-D
Hierarchy
155
first-order equations. Indeed, if Lf = (dn + un^dn-x
+ ••• + uo)/ = 0
is an nth order differential equation then choosing new variables f\ = f'ifi — / i > - - - ) / n - i = fn-2 w e S e t a system, or a matrix equation, l j / / = (d — J + U)f = 0 where J is a matrix with only nonzero elements J%,i+i = l , i = 0 , . . . , n—2, U is a matrix with only one nonzero row, namely, the last one, the ( n - l ) t h , which is (u 0 , u i , . . . , u „ - i ) , / = (/, / i , • • •, / « - i ) T (the superscript T indicates, as usual, a vector-column). Notice that in this subsection operators \u do not depend on z, in contrast to the definition in 9.1.1. When later we introduce the dependence on z, the corresponding operators will be denoted as 1[/. A more general system which can be reduced to a scalar equations is lqf = (d — J + q)f = 0 where q is an arbitrary lower triangular matrix. Coefficients of the operator L are differential polynomials in entries of the matrix q. It is also clear that different q can give the same L. 9.4.2. Proposition. Let AT be a group of matrices I + u where v is a strictly lower triangular matrix, let \qi = g~1\q2g where g € N. Then equations l g i / = 0 and lg2f = 0 determine the same scalar equation Lf = 0. Conversely, if two operators l 9l and lQ2 reduce to the same scalar operator L, then they are connected by a similarity transformation with some
geM. Proof. Let L\ and L^ be scalar operators related to l qi and 1Q2. Let / be an arbitrary solution to l q i / = 0. Then h = gf is an arbitrary solution to lq2h = 0, and the zero component of this vector, h, is an arbitrary solution to L2h = 0 while / is an arbitrary solution to L\f = 0. Evidently, h — / , due to the structure of the matrix g. Two monic differential operators with the same set of solutions coincide. Conversely, let l qi and \q2 yield the same L, let / and h be solutions to l g i / = 0 and \g2h = 0. We can consider pairs / and h such that their zero components coincide, f = h, since both / and h are solutions of the same equation Lf = 0. Writing the equations in coordinates, it is easy to find a matrix g € Af such that h = gf, g being independent of the choice of the solution f = h. Then l 9l and g~x\q2g have the same set of solutions and the same leading term d. They must coincide. •
156
Soliton Equations and Hamiltonian
Systems
The similarity transformations of differential operators lg are called "gauge transformations". The freedom of the choice of a gauge determines the flexibility of the method. 9.4.3. Corollary. On every orbit of the action of the gauge group TV there is one and only one operator of a form lu = d — J + U where U has only one nonzero row, the bottom one. Elements of this row, Ui, are differential polynomials of elements of q if d — J + q belongs to this orbit. Proof. Indeed, take the operator L related to lg = d — J + q and the operator lu = d — J + U where Ui are coefficients of this L, then apply Proposition 9.4.2. The submanifold of operators lu in the manifold of all lq is transversal to the orbits of the gauge transformations, the elements of lu parametrize the orbits. Another possible gauge is to take q = V = diagonal(u n ,..., v\). This yields L = (d + vn) • • • (d + vi). The transition from the "£/-gauge" to the "V-gauge" is the Miura transformation. 9.4.4. Remark. The following formula for L with a given lg can be used. L = det(<9 -J
+ q),
where the non-commuting elements in the expansion of the determinant det aki must stand in the order: aki to the left of ay if k > i. This can be proven by induction. Besides, u n - i = trg. 9.4.5. Thus, we have a manifold of operators lu- The infinitesimal change of such an operator, i.e. an element of the tangent space is a matrix having one nonzero row, the bottom one ( a o , . . . , a n - i ) . Denote this matrix Sa. It corresponds to an infinitesimal change of the operator I i - > i + a where a = a„_!d"-1 + a„_2dn"2 +
ha0.
Designate this tangent space as S+. The dual, i.e. cotangent space consists of matrices Q with the pairing tr(5 0 Q). It is clear that only the last column of Q is relevant, the rest of elements are indifferent. Thus, the dual space S- is the space of all matrices over the subspace of matrices with the zero last column. If the last column is X = (Xo,... , X n _ i ) T , we denote this element of S- as Qx9.4.6. Theorem. For any X, there is a unique matrix Qx £ Qx, i-e. a matrix with the last column X such that [lt/,<3x] € S+, i.e. has zeros
Matrix First-Order
Operator, AKNS-D
Hierarchy
157
everywhere except the bottom line. This commutator is [lu,Qx] = S-AW(X) . where AW(X)
= -{LX)+L
- L{XL)-
= L(XL)+ -
(LX)+L
and
x = d-1x0 + --- + d-nxn-1. A^(X) is the Adler mapping of X. (We denoted the last column of the matrix, (Xo,. •. ,Xn-i)T and the 1 n corresponding operator d~ X0-\ \-d~ Xn-i by the same letter X. It will not lead to confusion. As we shall do later, we identify columns of matrices with the elements of R-/d~nR(see 3.1.1), and rows with differential operators of order < n—1, like a = a n _ i d n _ 1 + a „ _ 2 C ? n - 2 - | hao above.) The proof of the theorem will be in the next subsection. 9.4.7. As it was mentioned, it is very convenient to associate a differential operator of an order < n — 1 with every row of the matrix, and an integral operator, more precisely, an element of R-/d~nR(see 3.1.1) with every column. Namely, n—1 j=0
n —1 i=0
One can, for example, imagine a matrix Q acting not in an abstract linear space but in R-/d~nR-, then Y = QX = ^ o " 1 3 ~ i - 1 Tes(QtX). A product of two matrices will be (QR)ij = res(QiR^) and so on. Designate the last, (n—l)th, column of the matrix Q as X, X = Qn~l = d~1Xo + • • • + d~nX„-i. In addition to the statement of the theorem, we shall prove two formulas: Qi = di(XL)+-(diX)+L,
t = 0,...,n-l
(9.4.8)
and Qj = (X{Ld-j-1)+ Let P=[lu,Q]. P
- (XL)_9_J'-1)_ .
Then
ij = Q'ij + Qi,j-1 ~ Qi+l,j - Qi,n-lUj,
i = 0, . . . , n - 2
(9.4.9)
158
Soliton Equations and Hamiltonian
Systems
and n-l •*•n—l,j
=
u
Vn-lj + Qn—l,j-l ~ Qn-l,n-l j + / J uaQa,j a=0
•
In terms of operators this is Pi = d o Qi - Qi+\ — Qi,n-iL,
i = 0,..., n - 2
and n-l
-Pn-1 = 9 o Q„_! - <5 n _ 1]n _jL + ^ U Q Q a . a=0
The first n — 1 rows vanish which gives recursion relations
Qi+i=doQi-Qin_1L,
i=
0,...,n-2.
Multiplying on the left by 9 ~ ' _ 1 and summing over i = 0 , . . . ,n — 2 we obtain d~n+1Qn-i = Q0 — (X — d~nQn-i>n-i)L. Taking positive part of this equality, we get Q0 = (XL)+. This coincides with the formula (9.4.8) for i = 0. After that, the recursion formula permits to perform induction and prove the formula for Qi for all i's; the formula for Pn-\ permits to identify it as the Adler mapping. Finally, Q J is an element of R-/d~nRsuch that res Qid~i~l = resd'Q-?. We have for any A and B res A+B = res AB_, therefore, vesQid-i-1 = lesd^iX^+d^-1 - XiLd^-1)^) = resa i ((XL)+a--»'- 1 XiLd-i-1)-)and Q> = ( ( X L ) ^ ' " 1 - X{Ld~^1)^ • We can replace UQ with UQ — zn, L with L = L — zn and \u with \u = d — J + U — znA where the only nonzero element of the matrix A is a„_i,o = 1. In this case we write Qx instead of Qx- In particular, the statement of the theorem will be [If/, Qx] = S(-A(X))
where A(X) = -{LK)-L
+ L(XL)_ .
(9.4.10)
9.4.11. Proposition. The matrix Qx can be written as the operator
Y e R_/d-nR-
.-> Qx(Y) = (X(LY)+ - (XL)-Y)-
e R-/d~nR_
.
(Recall that this expression is already familiar to us: [X, Y}i = Qy(X) — Qx(Y), see (3.1.3).)
159
Matrix First-Order Operator, AKNS-D Hierarchy
Proof. Using (9.4.8) we obtain n-l
QX(Y)
= Y, 9-'-1 res (QtY) = £ fl"*"1 r e s ^ (XL)+Y) i=0
0
n-l
Y^d-'-1 resdd'X)+LY) = ((XL)+y)_ j=0
n-l a_i_1
- £
res(^X(Ly)_)
= ((XL)+y - X ( L y ) _ ) _ .
D
i=0
9.4.12. Proposition. A matrix Q is a resolvent, i.e. commutes with ly, if and only if Q = Q x , where X is a resolvent of the operator L : A(X) = 0. Proof. This directly follows from (9.4.10).
•
9.4.13. As usual, the reduction un-\ = 0 is provided by the condition res [L, X] = 0 which allows to express X n _ i in terms of the preceding Xi's. 9.4.14. Exercise. For the operator L = d2 + u, write the matrix Qx and Gnd[lu,Qx]. Answer. X0
2 °
Qx =
•Ixa.
U0X0
2X'°J
0
Wu,Qx] = 2 °
(uXo)' - uX'0
0
9.4.15. We have a one-to-one mapping of the manifold of operators L onto the manifold of orbits of operators lq with respect to the gauge transformations lq i-> g\qg~l. We want to construct a symplectic form on the manifold of orbits which will determine a Poisson structure on this manifold and on the manifold of L. First of all, what are the vector fields on the manifold of orbits? The vector fields on the manifold of all triangular matrices preserving orbits must be taken, i.e. the vector fields which transform functions constant on orbits into functions with the same property. Then they must be factorized with respect to vector fields tangential to orbits, i.e. sending
160
Soliton Equations and Hamiltonian
Systems
the functions constant on the orbits to zero. One representative of each class can be obtained if a field tangential to S+ is taken and transferred to the whole orbit with the help of the gauge transformations. If a vector field tangential to S+ at a point \u has the form d[\UtQ], then at a point lg = <7U/<7_1 of the orbit it has the form dg[\UtQ]g-i = d^ gQg-\\. The gauge transformation is glqg*1, therefore a vector tangential to an orbit is cfy „], where u is a lower strictly triangular matrix. Any vector at a point \q can be repesented as a sum of a vector of the form ^q,gQg~x\ and a vector 9.4.16. Proposition. A symplectic form which is defined on vectors cfy ^ by the formula (9.2.3) ;
(d[l,,Q1]><9[i„Q2]) = / tr[lg,<5i]<52rfa;
degenerates on vectors tangential to the orbits of gauge transformations and therefore can be transferred to the manifold of orbits. Proof. If v is a lower strictly triangular matrix, then
since [1Q) ffQp-1] = <7[l[/,<3]<7~1 is a triangular matrix.
•
9.4.17. Proposition. The symplectic form (9.2.3) reduced to the manifold of orbits parametrized by S+ is ^[VsQxff-M-^.sQvs-1]) = u(d[\v,Qxh
d
[lv,Qy]) = /
= - / tr S^A(x))Qvdx
tr
=-
Ptf.
res
Qx]Qvdx
A(X)Ydx
which is the AGD symplectic form on the manifold of operators L (the minus sign is not important). The proof is obvious. • 9.4.18. Remark. This reduction can serve as a proof of the closedness of the AGD symplectic form, since for the matrix hierarchy the proof is considerably simpler, see 9.2.4.
161
Matrix First-Order Operator, AKNS-D Hierarchy
The Poisson bracket in matrix terms is: {/,
(9.4.19)
The Hamilton equation with a Hamiltonian h is: dtlu = [QSh/5LM.
(9-4.20)
It can also be written as an equation of zero curvature: dt(U -J)+
dQSh/SL
= [QSh/sL, U-J]
(9.4.21)
which is the compatibility condition for the system (d + U-J)f
= 0,
(dt-QSh/SL)f
= 0.
(9.4.22)
9.4.23. Exercise. Let L = d4 + uzd2 + u\d + uo- Take the Hamiltonian h = res!* 1 / 2 , find \JJ, Qsh/5L a n d write the Hamiltonian equation (9.4.20). Answer. / d -1 0 0
h=
Qsh/SL
=
0
o\
9-1 0 0 d -1
\W0
Ui
(
u2/2
U2
)
d) 0
1
0
«' 2 /2
U2/2
0
1
u'2/2 — uo
u'2 — ui
-u2/2
0
u'2/2 — tti
-U2/2)
\u'i'/2-u'0
3«J/2
- u[ -•
u0
\
The coefficient uo can be replaced by (UQ — zn), this will give a family of Hamilton equations with the same Hamiltonian and different Poisson brackets. 9.4.24. Now we show how we could arrive at the formula (9.4.8) for Qx starting from the equations of the nth GD hierarchy. We have d\.L = [L + , L], or [dk — L+,L] = 0. This is the compatibility condition for the system (dk-Lyn)frk/n
= 0,
Lf = 0.
162
Soliton Equations
and Hamiltonian
Systems
(The function / is not the Baker function, it is not supposed to depend on a parameter z.) Let / = ( / , / ' , . . . , / ( " _ 1 ) ) T , \u = d- J + U, as before. So, If// = 0. Let us try to find a matrix Qx such that dkf — Qx]• We have dkf{i) = Fdkf
= S'L*/n/ = ^ ( L ( f c - n ) / " L ) + / =
&{L^~n),nL)+f
(since (L{*-n)/nL)+f = L{*~n)/nLf = 0). Hence 0 f c / « - d\XL)+f, is 5hm-n/5L, see 3.6.1). Further, since L F = 0,
(X
n-l
Sfe/W = (3 4 (XL)+ - ( a ' X ) + L ) / = Q i / = J ] Q y / W . j=o
Thus, (dk - Qx)f = 0 where the matrix Qx is given by Eq. (9.4.8), Qt = <9l(AX)+ — (0 8 X) + L. The compability condition of the system lyf = 0, (dk-Qx)f = 0is d*\u = [QxAu]9A.
Appendix. Extension of the Algebra A. to an Algebra Closed with Respect to the Indefinite Integration
Not every element of the algebra A is an exact derivative. However, a differential algebra can be embedded into an algebra where all elements are derivatives. This can be made in the following way [GD78(b)]. Let Ai be the set of all formal power series oo
x
f( )
u
= z2 T\ak
a
'
k£A,
k=0
where all {a/j} except for a finite set of them have the property ak+i = a'k. Addition and multiplication are as usual. Action of the operator d is 00
A:—1
°°
k
w*) = £ (jrri)r = E T\ak+1' k=0
v
'
0fc
fc=0
the sequence of the coefficients shifts to the left: (ao, a\,...) t-> (ai, a2,...). It is easy to see that this differentiation has the necessary property d(fg) = f'g + f'g. Therefore A\ is a differential algebra. Any element of A\ has an indefinite integral J f(x)dx = Y^f ^jafc-ii where a_i E A is arbitrary and the coefficients shift to the right.
Matrix First-Order
Operator, AKNS-D
Hierarchy
163
The embedding of A into A\ is given by the formula / 6 ^ 4 / ( i ) = € S o ° T\f^ ^ i - ^ ^s e a s y t o c n e c k ^ a t this is, indeed, a mapping of differential algebras.
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Chapter 10
Generalization of t h e AKNS-D Hierarchy: Single-Pole and Multi-Pole Matrix Hierarchies 10.1
Single-Pole Matrix Hierarchy
10.1.1. This chapter is based on [Di94]. There are many possibilities to generalize the AKNS-D hierarchy. First of all, the operator L was a polynomial of first degree in z. We can take, instead, l = d + U = d + Uo + UlZ + --- + Umzm + Azm+1,
d = d/dx
(10.1.2)
where A is a constant diagonal matrix with distinct and nonzero diagonal elements. (Notice the change of notations: former U is now Uo, and what became now U, was U + Az.) The whole theory of Chap. 9 can be reproduced for this case. The differential algebra A is generated by elements of all the matrices Ui. A resolvent is a formal series R = ^ £ ° RkZ~k with matrix coefficients commuting with 1: [1, R] = 0 or R' + [U, R] = 0 .
(10.1.3)
Resolvents form a ring, and even a commutative one. Elements of Rk belong to A. Without loss of generality we can assume that fco = 0 (dividing the resolvent by z~ko). First we prove the existence of resolvents. 10.1.4. Proposition. There exists a formal series <\> = ^ ^ ° 4>kZ~k belonging to A (i.e. all elements of all matrices € A), being
(10.1.5)
166
Soliton Equations and Hamiltonian Systems
Proof. Rewrite the required equality as (d + £/) = 4>{d + A) or as a recursion system:
Uifa+i
j=0
ra =
22 fa+i^-i i=-k+l
+ 0oAfc .
(10.1.6)
Let A_ m _i = A, 4>Q = I and the diagonal elements of all fa be zero. Then the recursion relation allows to determine non-diagonal elements of
c z_1
( ) = Yl Ckz~k• k=ka
Thus, Ra are basic resolvents. Proof. It is not difficult to show that a resolvent without constants in the diagonal elements is zero, i.e. resolvents are uniquely determined by constants in the diagonal elements. On the other hand, series ^2aRaca(z~1) can have any set of constants in the diagonal elements. • 10.1.8. Definition of the hierarchy. Let the elements of matrices Uk depend on some variable symbolized by tia and let dlal = [(zlRa)+, 1],
dla = d/dtla
(10.1.9)
be a system of differential equations for the elements of Uk- This system is meaningful because the right-hand side can be represented as — [(z'i? a )_, 1] and is, therefore, a polynomial in z of a degree < m. It is easy to prove that two equations of this kind, for different I, a, commute, thus all of them can be integrated simultaneously. After that, the operator 1 will depend on all variables tia. The totality of all Eqs. (10.1.9) is called the (m + l)-hierarchy.
Single-Pole
and Multi-Pole Matrix
167
Hierarchies
We can consider linear combinations of (10.1.9) also as hierarchy equations, dtcl = Y/cia[(zlRa)+,l},
(10.1.10)
where tc is some variable and dt0 = Y^ cla<^ia • Proposition 9.3.6 can be generalized for this case: 10.1.11. Proposition. 5 tr UZR = tr{5U • R)z + d(6<j>R0ipz - 4>zR0Stp).
(10.1.12)
(In 9.3.6 we had Uz = A and tt(5U • R)z '= tr SU • Rz.) Prom this equality one can obtain, using i(da)5f = daf, the following: if da is a vector field, then da tr UZR = tr(daU • R)z + 9 ( S a ^ 0 ^ -
(10.1.13)
10.1.14. Proposition. There are the following first integrals of the hierarchy (10.1.9): >kj3
I
tvresUzRl3zkdx.
Proof. dlaJk/3 =
= -
ftrres(diaU-R0)zzkdx
ftriesdiaU-R0kzk-1dx
= - [trres[{Razl)+,l}
•
R0kzk~ldx
= f tr r e s ^ , /] • {Razl)+kzk-1dx
= 0.
D
It is also possible to build two Hamiltonian structures for this hierarchy. We skip this. 10.1.15. The rest of Sec. 10.1 will be devoted to a motivation for the next section. The idea is to get rid of a distinguished position of the variable x and the operator 1 making them just one of the variables of hierarchy and, correspondingly, one of the operators, equal in rights with all the others.
168
Soliton Equations
and Hamiltonian
Systems
Why, considering the GD hierarchy, took we the second coefficient, un-i, being zero (or constant)? Since the flows of the hierarchy preserve this coefficient, taking it variable would prevent the distinguished variable x to be one of the "time" variables t\, £2, • • • or their linear combination. By the same reason, considering the AKNS-D hierarchy, we required that the diagonal elements of the matrix U were zero. Now, for the hierarchy based on the operator (10.1.2) we must also have a restriction on the matrices Ui that guarantees that x is among the variables tc (see 10.1.11) of the hierarchy. Let = symbolize the equality of m first terms of two asymptotic expansions in z~x. 10.1.16. Lemma. As a consequence of Eq. (10.1.9), <9Zatrt/pm=20
(10.1.17)
for any p > 1. Proof. We have <9jQlp = — {(zlRa)-,V]. Let us take in this equality only the terms with Z P(™+I), Z P ( ™ + I ) - 1 ; . . . ; 2 ( P -i)(m+i) F o r t h e s e t e r m s y> c a n be replaced by U*> and dlaV by dlaUp. Then dia tr Up m=2 0.
•
10.1.18. Example. For m = 0 we have tr £/p = tr A p z p +ptrvl p - 1 t/o.z p ~ 1 = const and diag UQ = const. Now we have one distinguished variable x and the corresponding derivative d, and variables of the hierarchy tia, or, more generally, tc from (10.1.10) and corresponding derivatives dia and dc. We want x to be one of tc, i.e. to belong to the hierarchy. We shall find the conditions for that. The lemma implies that a necessary condition is that for every p e Z+ there must be dtrUpm=20
(10.1.19)
since for all the dia's this is true. 10.1.20. Proposition. Equation (10.1.19) is a necessary and sufficient condition for d to be one of dtc 's, for some c, more precisely, m+l
n
d = 22 z2 Cladla ' 1=0 a=l
Cm+i,a being equal to — aa.
Single-Pole and Multi-Pole Matrix
Hierarchies
169
Proof. It remains to prove the sufficiency. The first dressing formula implies IP = (d + Uy =
= const, i = —m —
A + = const. Let us take a resolvent:
R =
Oiz-1).
Then /
m-fl
n
\
U = R+ = U J2 5Z A-i.a^Sa^"1 i=0 a = l
The equation corresponding to the resolvent R is c?tcl = [R+, 1]. On the one hand, this is m+1
ft J
n
0 J ] E ^-i.aZ'Ea^-1} ,1 i=0 a=\
/ 4.
m+1 n — / _, / _, A - j , a d j a l •
J
t=0 a = l
On the other hand,
dtci = [u,i] = [u,d + u} = -du. Therefore, m+1
n
d=-Y/Y,X~i>°'di<*-
(10.1.21)
i=0 a = l
(When m = 0, this formula involves c?ia and do a while the corresponding formula in Sec. 9.1.18 contained only d\a. The reason is that in Chap. 9 we had a condition diag U = 0 which implied Ao = 0.) • 10.1.22. The second dressing formula, the Baker function. Let 00
n
£. = 2_, 2J *(aZ ^a • i=0 a = l
170
Soliton Equations and Hamiltonian
Systems
Letting <j> = ipe t and taking into account (10.1.21), we obtain m+l
n
\
(a + A-^^A.^% i=0 a=l
r1
/
= V>(<9 + A_W> -1 . Finally, denoting w = -0exp(/A-dx) = (/>exp(JA_efo;)exp£ = u)exp£, i.e. u) =
(10.1.23)
The entries of the coefficients, wit of the new dressing series, w = YlZo wiz~t; do not belong to A, in contrast to fa. Equation (10.1.23) expresses entries of the matrices Ui in terms of differential polynomials in {wi)ap. The differential algebra of such polynomials is denoted by Aw. Thus, we have an embedding A C Aw. The function w(t,z) = ti)exp£ is called the Baker function. Equation (10.1.23) implies l(w) = 0 where l(w) symbolizes the action of the differential operator 1 on w while \w is a product of operators of the first and zeroth order. For a proof, it suffices to apply both sides of the equality \w = wd to the function / = 1. The hierarchy equation (10.1.9) is, in terms of w,
dlaww-1-[(zlRa)+,l}=0. Therefore, the equation dlaw = (zlRa)+w
(10.1.24)
implies Eq. (10.1.9), and it is an extension of Eq. (10.1.9) that was given on A to the whole differential algebra Aw. It is not difficult to show that the extensions of the flows 9j Q to Aw commute as well (see below). The advantage of the dressing formula (10.1.23) is that it allows us to consider it; as a universal dressing series, not depending on /. Taking various constant diagonal matrix polynomials A + we will get different I, all of them are equal in rights. There is no distinguished variable x either. In the next section we are presenting the theory of the single-pole hierarchy in this sense from the very beginning. In the section following the next section we extend the theory to the multi-pole hierarchy.
171
Single-Pole and Multi-Pole Matrix Hierarchies
10.2
Single-Pole Hierarchy. Presentation not Depending on a Distinguished Operator 1
10.2.1. Now we are starting again from the other end, forgetting about the operator 1. Let w = ^ ° ° WiZ"1 be a formal series, WQ is a constant and diagonal, and the entries of matrices m* for i > 0 are considered as independent generators of a differential algebra Aw10.2.2. Definition. equations
The single-pole hierarchy is the totality of all the
diaw = -(zlRa)-w
Ra = wEaW'1.
where
(10.2.3)
This formula gives the action of an operator dia on the generators of the differential algebra Aw; we can extend it as a derivation to the whole algebra. Letting w = w exp£(£, z), where £ is as in Sec. 10.1, we get an equivalent form of the equations of the hierarchy dlaw = Blaw,
Bla = (zlRa)+ .
(10.2.4)
The same equation can also be expressed as wdla • w'1 = w(dla - zlEa)w'1
= dla - Bla .
(10.2.5)
Thus, dressing of dia yields a first-order differential operator (10.2.5) that is an Zth degree polynomial in z. The expression w is called a formal Baker function. 10.2.6. Proposition. Operators dia commute. Proof. (diadm/}~dml3dla)w
= -dia(Rpzm).w
- ((l,a) «• (m,/?))
= -dia{{wEpzmw-1)-w) =
- ((J,a) «• (m,(3))
((Razl)-wEpzmw-l)-w-(wEf3zmw-1(Razl)-)-w + {wEf3Zmw-l)-{Razl)-W
= {Razl)-{Rl3zm)^w - (R0zm)^{Razl)-W
- ((I,a) & (m,/?)) (Razl(R'3zm)+)-w
+ -
((RPzm)+Razl)-w
172
Soliton Equations and Hamiltonian
Systems
+ (R)3zm)_(Razl)-w
-
{{R0zm)^Razl)-w
+ {Rxz\Rpzm)J)-w
-
(Razl)-(Rf}zm)-w
= {RazlR<3zm
- R/3zmRaz1)-
=0
since Ra and R@ commute.
•
10.2.7. Proposition. Operators dia — Bia commute, i.e. diaBm0 - dmpBia - [Bia, Bmp] — 0.
(10.2.8)
Proof. This follows from Proposition 10.2.6 and Eq. (10.2.5).
•
Let Xi, I = 0 , . . . , m + 1 be a sequence of constant diagonal matrices, A; = diag(A (a ) and d = - Y^1 E L i h<*
l=-^2Y,Xia(d,a-Bia)
= d + U,
(10.2.9)
( = 0 <*=1
where U = YZLV E a = i ^i<*Bla- Then l = wdw~1 =d + U.
(10.2.10)
We came to the formula (10.1.23). However, now 1 is but one of the operators of the hierarchy. All of them have absolutely equal rights. 10.2.11. Proposition. The hierarchy equations (10.2.3) imply dm.pl =
[Bmp,l].
Proof. This follows almost automatically from (10.2.4) and (10.2.10).
• If another linear combination is taken: dt = — P+l
YM=O
E a = i ^la^ia and
n
1=0 a = l
then [d + U,dt + V}= 0, or dtU - dV = [U, V].
(10.2.12)
Single-Pole
and Multi-Pole
Matrix
Hierarchies
173
This is the Zakharov-Shabat equation for U and V having a single pole at infinity (i.e. polynomials). This explains the name single-pole hierarchy. In the next section we consider the multi-pole case. What about first integrals and the Hamiltonian formalism? In a theory where all variables are equal in rights, and there is no distinguished x and distinguished time variable, it is more appropriate to use the field formalism that will be discussed in the last two chapters. 10.2.13. Remark. If w is replaced by w^ = w • c(z), where c(z) = Y^ CiZ~% and c* are constant diagonal matrices, then w\ satisfies (10.2.3) as well, and operators (10.2.5) do not change at all, neither does Eq. (10.2.12). We say that w and w^ are equivalent. The corresponding Baker functions w and i//1) are also equivalent. A Baker function, if needed, can always be replaced by equivalent one. In particular, one can make u>o = I. 10.3
Multi-Pole (General Zakharov-Shabat) Hierarchy
10.3.1. Definition of the multi-pole hierarchy. Let afc, k = 1 , . . . , m be a set of complex numbers. Let, for every k, oo
Wk = y~]wki(z ~ aicY , i=0
be a formal series, the entries of matrices u>ki being taken as generators of a differential algebra. Then this algebra is extended by elements (det w^o) - 1 and the obtained algebra is called Aw. One also can consider an additional point at infinity and the corresponding series oo i=0
In principle, infinity can be sent to a finite point by a fractional linear transformation, and this point will not differ from the others. However, more convenient for applications is to preserve this point at infinity to get formulas including the single-pole hierarchy of the preceding section. Practically, this means that expansions in z — Ofc at finite points must be replaced by those in z'1 at infinity, sub- or superscripts — by + and vice versa. Often we are not writing explicitly the term at infinity and even not mentioning it, just keeping in mind that one of the points can be at infinity and necessary corrections should be done.
174
Soliton Equations and Hamiltonian Systems
The formal series Wk can be inverted within the algebra Aw. Let = WkEaW^1 ,
Rka
Rkal
= Rka(z ~ ak)~l ,
where Ea has the same meaning as before. (According to the above said, ™oo = E o °
w
kZ~k,
R
ooa = WooEaW^1 and Rooal = RooaZ1.)
We consider two kinds of objects. Such quantities as ibk and Rkal are formal series, or jets, at the points a*,. The algebra of all such jets will be called Jk and J = ® Jfc. If jk € Jk is a jet, then j£ symbolizes its principal part, i.e. a sum of negative powers of z — ak, and j£ the rest of the series (now we are writing ± as superscripts, there will be too many subscripts). If the principal part contains finite number of terms (and we tacitly assume this unless the opposite is said or is evident from a context) it can be considered as a global meromorphic function; the algebra of global meromorphic functions is G. Global functions are objects of the second kind. A global meromorphic function gives rise to a jet at every ak- In particular, j ^ can be considered as a jet at a point a^ different from ak, more precisely, as an element of j£. Its expansion in z — a^ will be denoted
10.3.2. Definition. A hierarchy corresponding to a fixed set {ak} is defined by the equations dkaiwkl = < ^ _ k a l / ' { -ftfccJfci^fci '
' , otherwise
dkai = d/dtkai •
(10.3.3)
tkai are some variables. (If k = oo, the superscripts "+" and "—" should be swapped.) 10.3.4. Remark. A hierarchy given on a subset of points can be in an obvious way embedded into a hierarchy on a whole set; two hierarchies can be embedded into one. One can consider the inductive limit. Further in this section, for simplicity of writing, we shall unite indices a and I into one subscript a = (a, I) and write dka and Rka f° r dkai and Rkal10.3.5. Lemma. Equalities UkalRkiaili
hold.
=
__
I lRkai'Rki<*ih].
otherwise
Single-Pole and Multi-Pole
Matrix Hierarchies
175
Proof. It can easily be obtained from the definition of Rkal •
D
10.3.6. Proposition. All operators dkai commute. Proof. One has to prove [dkiai, dk2a2]wk3a3 = 0 in three cases: (i) all of fcj coincide, (ii) only two of them coincide, (iii) all are distinct, (i) We have (dkaidka2
- dka2dkai)wk
= -dkaiRla2wk
- (1 •» 2)
= [Rta^Rka2]
+
Wk + RlAwk
" (1 «* 2)
+ [Rka1,Rka2} + Wk
= RL1Rta2^
+
-[Rta2>Rkai] +
= [Rkai,Rka2]
Wk-RtaiRta2Wk
Wk = 0
since Rkai and Rka2 commute, (ii) First we consider {dkaxdkcL2 - dka2dkai)wkl
= dkaiRka2wkl
-
dka2Rkaiwkl
= -[flfaM.flkaJfcWk! + R^Rla^k,
~ (1 <* 2)
-{Ria1Rka2)k^k1+{Rka2Rkai)kWki-{l^2)
= = — [•Rfcai)-Rfeo2]l"fei
which is zero since Rkai and Rko,2 with the same k commute. A notation A^ means the principal part of an expansion of A in powers of z — ak. A+k=Ak-A~k. Then we take (dkadkiai
- dkiaidka)wk
= dkaRkiaiwk
+
dkiaiR^awk
= [Rka'RkiaAk^k
-
RkiaiRkaWk
+ [Rkiai> Rka}kWk + = [flfca.flkiailfc^k "
+ RkaRk^k
+
+ iRkiai> RtJt% =
(l-^fca'-^fciaiJfei ~
+
RtaRkiaiWk [Rkiai>Rka}™k
RkaR^aVk
+ lRkiai> R'kM™k [Rka'Rknnl
l(Rka)k,Rkiai}k)Wk.
176
Soliton Equations and Hamiltonian Systems
In the parentheses there is a function [R^a, R^ a ] minus its principal parts at both poles, a,k and a ^ . Thus, this is a constant. This expression approaches zero when z —> oo which implies that the constant is zero, (iii) \dkiai,dk^Wk3
= dkiaxRk2a2Wk3
-
= lRkiai^Rk2a2}k2^3
dk2aiRkiaiwk3 + Rk2a2Rkiai™k3
~ (l-^fciai'^aalfci + \.Rkiai>
-(1^2)
R
k2a2\k2
-lRkiai>Rk2a2])™k3The expression in the parentheses vanish by the same reason as in the previous case. • The following proposition can readily be proven by a simple straightforward computation: 10.3.7. Proposition. A dressing formula Wfci (dkal - Ea(z-
ak)~lSkk!)w^
= dkai -
Bkai, (10.3.8)
Bkal = Rkal
holds, as a consequence of Eq. (10.3.3). The operator dkai ~ Bkai is assumed to act in J^ • However, it does not depend on k\ at all and can be considered as a global function of z with the only pole of the Zth order at a^. Let wk = wk exp£ fc ,
oo n where 6 = ^ Z X I l^alEa{z 1=0 a = l
- ak)~l •
10.3.9. Definition. The collection w = {wk} is the formal Baker function of the hierarchy. Equation (10.3.8) can be rewritten in terms of the Baker function as WkldkalW^
= dkal ~ Bkal •
(10.3.10)
10.3.11. Proposition. All the operators dkai — Bkai commute. Proof. This is a corollary of the theorem 10.3.6 and Eq. (10.3.10).
•
Single-Pole
and Multi-Pole
Matrix
Hierarchies
177
One can consider arbitrary linear combinations of the above constructed operators, X
1 = Y,
kd(dkai - Bkal) =d + U,
(10.3.12)
k,a,l
where d = Y,k,a,i ^kaidkai and U = - Ylk,a,i ^kaiBkai- Two such operators commute which yields equations of Zakharov-Shabat type dU1-d1U
= [U1,U\-
(10.3.13)
Functions U and U\ are rational functions of the parameter z.
10.4
Example: Principal Chiral Field Equation
10.4.1. Consider the simplest case of the Zakharov-Shabat equation (10.3.13) when each of the rational matrix-functions U and U\ has one simple pole, and these poles are distinct. We change the notations: [di + U,dv + V} = 0,
U = U0 + zA,
V = z~1U^1.
(10.4.2)
So, U has a pole at z = oo and V at z = 0. The equation is equivalent to the system A , = 0,
UQt1)=[A,U-1\,
tf_u
= [£/_!,%].
(10.4.3)
Very often, Eq. (10.4.2) is represented in a different form using the gauge transformation: % + U = fc(a{ + U)h~l = «% + h(U0 +zAdv + V = h(dv + V)^1
=dn + hiz^U-i
h^h^h-1, -
h^h^h-1
in order to make each pair of parentheses on the right-hand sides a product of a matrix by a scalar polynomial of z and z _ 1 : h-1^
= U0 + A,
h-1^
= £/_!.
(10.4.4)
The compatibility condition is satisfied by virtue of the system (10.4.3). Denoting M = 2hAh~1,
N = 2hU-1h~1
(10.4.5)
178
Soliton Equations and Hamiltonian
Systems
we have [d^ + U, dv + V] = 0 where % + U = % - | ( 1 - z)M,
dn + V = dv-
1(1 -
z~l)N,
and the zero curvature equation is M„ = ±[N,M],
N( = ±[M,N].
(10.4.6)
Conversely, any solution of (10.4.6) can be obtained in this way from Eq. (10.4.2). Suppose, M and TV satisfy Eq. (10.4.6). Reduce M to a diagonal form (we consider a generic case when all eigenvalues of M are distinct): M = 2hAh~1. The matrix h is determined up to a diagonal right factor, h >-» hfi. Find Z7_i from the equality N = 2hU-\h~l. Thus, Eq. (10.4.5) is satisfied. Now, we prove that choosing the factor /i one can satisfy the second of Eqs. (10.4.4). Indeed, substituting M and N from (10.4.5) into the first of Eqs. (10.4.6), we get [h'1^ - U-ly 2A] + 2An = 0. If h is replaced by h[i then this equations modifies: [h-1^
+ ^n'1
- U-X,2A] + 2Ar, =0.
Taking the diagonal part of this equality, we have Av = 0. The nondiagonal part gives nondiag(/i_1/i,7? — f/_i) = 0. Then // can be found such that h~1hv + ^vfJ'~1 — U-\ = 0 for all terms, diagonal and non-diagonal. Finally, the first of Eqs. (10.4.4) serves as a definition of UQ. Now, it is easy to check all the equations (10.4.3). The system (10.4.3) or its equivalent (10.4.7) is the principal chiral field equation, or the sigma model. Usually, it is restricted to the Lie algebra u{n) of skew-hermitian matrices. If one prefers to deal with the poles in the finite part of the complex plane, this can easily be achieved by a fractional linear transformation z = (A - 1)/(A + 1). Then U = -Af/(A + 1),
V = N/(X - 1),
[«% + U, dn + V] = 0.
(10.4.7)
We are discussing this equation again, in Chap. 20, concentrating on its Lagrangian and Hamiltonian structure as well as on its first integrals. 10.5
Grassmannian
10.5.1. We use the following notations: Ck are disjoint circles around the fixed points afe, k = 1 , . . . ,ra, and f2 is the part of the Riemann sphere
Single-Pole and Multi-Pole Matrix Hierarchies
179
outside all the circles; Hy. are Hilbert spaces of vector functions ffc(z) € C n on the circles, subspaces H£ consist of functions on Cfc which can be expanded in non-negative powers of z — ak, and H^ contain expansions in negative powers. Now, H = 0 f c # f c , and H+ consists of {ffc} such that ffc G H£ under an additional constraint ^fc ffe(afc) = 0. Finally, let H* consist of f = {ffc} € H such that ffc are boundaryvalues on the circles Cfc of a holomorphic vector function in the domain 17; this function will be denoted by the same letter f. It is easy to see that H = H* © H+. Indeed, let f = {ft,} € if be an arbitrary element. Then each ffc can be decomposed into ffc = f^ + f^. Elements f^~ are holomorphic outside the corresponding circles Cfc, and elements f^ are holomorphic inside the corresponding circles. Now,
ffe= ( l X + c ) + t e f c - c ) ' where gfc = f+ - £ ,^fc f" and c = m _ 1 £ f c gfe(afc). We have, {£< fr + c } 6 H* and {gfc - c} € H+. The decomposition is unique. Let P* be the projector P* :H^H*. One of the points ak can be at infinity. Then the corresponding H£ consists of series in non-negative powers of the local parameter z"1. It will be tacitly assumed henceforth that if one of ak is infinity then the local parameter z — ak is replaced by z _ 1 . 10.5.2. Definition. An element of the Grassmannian, W € Gr, is a subspace of H with the following properties: (i) the projection P* : H —> H* restricted to W is a bijection, and (ii) (z — a\)~1W = (z — a^)~lW = • • • =
{z-am)-lWcW. For example, a trivial element of the Grassmannian is W = H*. We think about vectors as vector-rows. A matrix is said to belong to W if so do all its rows. One can consider the following transformation of the Grassmannian. If f = {ffc} e W, then fexp£ = {f fc exp&} where & = E S o E L tkaiEa(z — a,k)~l, £ = {£fc}- The set of all f exp£ is called Wexp£. For almost all tkai the subspace Wexp£ is a new element of the Grassmannian. 10.5.3. Definition. A Grassmannian pre-Baker function, corresponding to an element of the Grassmannian W, is a matrix function w S W such that
180
Soliton Equations
P*wexp(-£)
and Hamiltonian
Systems
= c,
where c is a constant in z (however, it can depend on variables t). An element W € Gr is invariant with respect to multiplication on the left by a matrix which is constant in z, since this leads to linear combinations of rows. The projector P* commutes with this multiplications. Therefore, if w is a pre-Baker function then so is a gauge-equivalent function ew where e is any matrix independent of z. All pre-Baker functions can be expressed in this way in terms of one of them, e.g., corresponding to c = I (the normalized pre-Baker function). Let w = {wk} be a pre-Baker function and Wk — Wkexp(—£*). This means that u>k has a form c + Wkto + u>k,i(z — ak) + • • • and Ylwk,o — 0For a normalized function c = I. Thus, the normalized pre-Baker function has expansions wk = (I + wk,o +wk,i(z-ak) fc = l , . . . , m ;
+ ---)expffe g W,
7 ^ Wk,o — 0. k
These equalities are equivalent to the definition of the normalized pre-Baker function. Let t u b e a pre-Baker function. We have dkaiWi - Rk~aiwi = {dkaiWi + 5kiWiEa(z - a t ) - ' - R^aiWi) exp& = I ( {dkaiWi + Rtal™i)
(10.5.4) eX
P&'
i = k.
10.5.5. Definition. A pre-Baker function is called a Baker function if for every (kal) a relation {dkaiWi} € (z — a,j)~1W holds (in fact, this subspace does not depend on j , see the definition of the Grassmannian). 10.5.6. Proposition. A Grassmannian Baker function is a formal Baker function of the hierarchy, in the sense of Sec. 10.3.9. Thus, a solution of the hierarchy equations is related to any Grassmannian Baker function. Proof. The left-hand side of (10.5.4), {dkaiWi — RkaiWi}> bekmgs to (z — 0'k)~1W. Therefore, the expression in parentheses on the right-hand side, let it be gt, is in (z — Ofc)_1W^exp(—£). Then {(z — dj)gi(z)} 6 Wexp(—£) for every j . On the other hand, this is an element of H+ plus, maybe, a constant. Let 7, be arbitrary matrices. Then {^jlj{z — aj)gi(z)} € Wexp(—£). It is easy to see that choosing matrices 7^ one can achieve that
Single-Pole and Multi-Pole
Matrix
181
Hierarchies
E i E j 7 j ( o i - Oj)5i(°i) = °- T h e n {LjlAz ~ aj)9i(z)} is in H+ and, + therefore, in Wexp(-£) n i / . This implies that { ^ • 7^(2 - aj)gi(z)} = 0 and ^ = 0. The latter means that wt satisfy the equations of the hierarchy. D 10.5.7. An example: soliton-type solution. One starts with a specification of an element W G Gr. Consider a linear space H*(D) of meromorphic vector functions f in the domain fl with a fixed divisor D of simple poles bj, where j = 1,...,7V. Collections of boundary values of these functions, {ffc}, on the circles Cfc will be denoted by the same letter f, and the linear space of them by the same symbol H*(D). This will not lead to any ambiguity. Now, let W C H*(D) be a subset of meromorphic functions in H*{D) satisfying Nn conditions v(^j) - 7 ^ = 0 where i = 1 , . . . ,Nn, Hi G f2 are arbitrary points, and T/, are given vector-columns. Collections of their boundary values are symbolized by the same letter W, and this, generically, is an element of the Grassmannian. The property (ii) of the definition of the Grassmannian is self-evident. Notice that (z — a,j)~1W consists of those elements which are boundary values of meromorphic functions vanishing at infinity. Let us prove that the property (i) is also satisfied. We have to prove that if there is an element f = {ffc} G H*, then a unique element g = {g^} G W can be found such that h = f — g = {ffc — g^} = {hfc} is in H+, i.e. its analytical prolongation inside every circle Ck exists, being Efchfc(afc) = 0. Given ffc are boundary values of a holomorphic in Cl function f. Let g = f + b 0 + 5Zi=i ^j(z ~ k j ) - 1 where hj are vectors that have to be found, in all (N + l)n unknown components. First of all we require that g G W which is equivalent to Nn scalar equations g(//») • r)i = 0. Then we impose one more constraint EfcLi(bo + E ? = i kj(afc — ^ ) _ 1 ) = 0 which gives n more equations for the unknown coefficients, in all (N + l)n equations. Generically, this system can be solved uniquely. After that, boundary values of b 0 + Ylj b j ( z - fy)-1> c a ii them hfe, belong to H+, and gfc = ffc + hfc that implies P*g = f, as required. We are looking for a Baker function in the form w = $(z) exp Y^£fc = 11 + Y, Bj{z - bj)'1 fc
\
J= l
exp ^ /
&. k
Here Bj are matrices. Then w^ = (^ + E , = i ^ 3 ( 2 ~ ' , j ) - 1 ) e x P E a i t ^ ' W e have the following equations for elements of the matrices Bj
182
Soliton Equations
$ ( / x i ) e x p ^ ^ f e ( / i i ) -rji = 0, fc
and Hamiltonian
Systems
z = l,...,JVn
or, in coordinates, n
^2$ap{Hi)yi3i
= 0,
a = l,...,n,
i=
(10.5.8)
l,...,Nn,
0=1
where y0i = exp I ^2 ^2 twiVi \*;=1 1=1
- afc) M J70 • J
Matrices Bj are determined by this equation uniquely. We have chosen a gauge in which wt are boundary values of a function constant at infinity, therefore dkai^i are boundary values of a function vanishing at infinity and {dkdWi} € (z — a,j)~lW, i.e. this is a Baker function. 10.5.9. Proposition. The solution to the system (10.5.8) is given by the formula $ a/3
A-1 5a£
ya,Nn
Val
(z - h) 1810
x
yu
(UNn - bi)
X
ynl
(VNn - bi)
1
y11
(fJ'Nn - &iv)
1
5ni3
|
(fix - bi)
1
(z - bN)
1
| (fix - bN)
1
(z - bN)
1
Snl3 | (m - bN)
1
(z - bi)
1
| (^ - h)
S1p
ynl
(fJ-Nn - &Af)
yi,Nn
yn,Nn
Vl,Nn
yn,Nn
Here A is the cofactor of the element Sa0. (The structure of the determinant is the following. It has Nn + 1 rows and columns. All the rows except the first one can be parted into N groups, n rows in each of them. The rows, except the first one, can be labeled by j , •y where j = 1 , . . . , N and 7 = 1 , . . . , n. The columns, except the first one are labeled by % = 1 , . . . , Nn.
Single-Pole and Multi-Pole
Matrix
183
Hierarchies
The nonzero entries of the first column are on the (j, (3) places, i.e. on the /?th place in each group, and also the upper left element if a — f3.) Proof. Left-hand side of Eq. (10.5.8) is represented by a determinant where the first column coincides with the ith, hence it vanishes. Taking into account the division by A, we see that $ has the desired form I + 1
EjLi^*-^)- -
n
10.5.10. Expression of the Baker function in terms of r-functions for solitons. The next very natural topic in this context would be the rfunction. According to the common definition, we could expect a relation between the Baker and the r functions something like Wk,ap =
^bp
-
— exp 2 ^ Zi(z) •
(10.5.11)
We do not give here a general definition of a T-function and restrict ourselves to the soliton-type solutions. We show that for them the formula (10.5.11) can be written, indeed. The general case is discussed in [Di96] though not in terms of the Grassmannian. The summary of these results will be given in Chap. 12. In order to obtain u>k, one has to multiply $ by exp£] f c i jfc£fci(z) and expand it in powers of z — afc. There are two cases: (i) a = (3 and (ii) a ^(3. (i) Let us transform the determinant adding the first row multiplied by — (z - bj)_1 to all j(3th rows, j = 1,...,N, i.e. annul all elements of the first column except the first one. Expanding along the first column, we get that 3>/3/3 = f l ^ i O 2 — &j) -1 multiplied by an N x N determinant with the entries: (z — Hi)(ni — bj)~1ypi on the (fij, i) place and (/Xj — bj)~1yli on the (lh *) place for 7 ^ / 3 . An obvious identity oo
Ofc
Hi - z = (fii - afe) exp i=i ' V^*
—
flfc
implies N
oo 1
Wk,p0 = ]J(z - b^A' j'=i
det(f*£f)exp £ fci^fc
$ > f c l j 8 i ( * " Ofci)" i=i
where ( T ^ f ) with fixed fc and /3 is an Nn x Nn matrix, "/j is a number of
Soliton Equations
184
and Hamiltonian
Systems
a row and i that of a column,
jife/3/3
1 V-yi i
otherwise.
bj
We denoted ~fc/3 Viy •
V-yi = e x p fci=i1=1 ,N
The factor n^LiC 2 — &j) -1 does not play any role in the dressing formula (10.3.8), it just cancels out. Thus, except for the factor e x p ^ f c S S i Infill2 ~ aki)~l the whole dependence on z is in modified time variables tkl72 i-» £/tl7( — Sk1,k^-y0l~1{z — flfc)'- This is just what we need in order to obtain (10.5.11). Thus, a r function for a matrix element Wk,pp is Tk,00 = det Tk(3P where
rpk/3/3
fJ-i -
bj
ym,
1 y~(i , in — bj
if 7 = P otherwise.
(ii) Now the element $ a) g with a ^ /?. The (1/9) row multiplied by (z — b\)(z — bj)~x must be subtracted from the (j(3) row, for all j = 1 , . . . , N. We have N
wk,a0 = U(z
- b^A-1
det(f *?f )exp £
j=l
^*M'(Z ~ kx^kl=l
where 2/o
if j = l,
1 ~y~fi : Hi - b.
otherwise.
7 = J9
rpk(x{5 7J'.»
ak
^~
Single-Pole
and Multi-Pole Matrix
The determinant detTka/3
185
Hierarchies
is a r-function for Wk,a/3, i-e. Tk,ap where
Vai, Tkoc/3
if 3 = 1, 1 = P
(Mi - o,k){bj -
h)
(jii - bj)(jii
bi)
1 ~y~fi i
-
ypi,
if j > 1, 7 = /3 otherwise.
In this particular example we obtained the following fact. There are matrix-functions Tfc(t) such that Eq. (10.5.11) holds, being r(t) = A.
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Chapter 11
Isomonodromic Deformations and the Most General Matrix Hierarchy
11.1
Isomonodromic Deformations
11.1.1. The problem of deformations of ordinary linear differential equations with rational coefficients on the complex plain which preserve the group of monodromy was posed by Riemann. It was solved in simplest cases by Schlesinger, Gamier and others at the beginning of the 20th century and then was practically forgotten. Interest to a concrete mathematical formula revived in the second half of the century. Riemann's problem happened to be closely connected with the fast developing theory of integrable systems and attracted attention of people working in that area. In the '70 and at the beginning of '80 the problem was completely solved by efforts of several mathematicians (the most advanced work was [JMU], that article also had detailed historical remarks and bibliography). From more recent works let us mention [HI]. There is a formal analogy between the theories of isomonodromic deformations and integrable systems. Lax has shown that integrable systems appear as deformations of linear differential operators preserving the spectrum, and here we have deformations of differential operators preserving the monodromy group. Beyond the formal analogy, there is a far closer relationship. One can construct a really huge hierarchy of integrable system that is, so to say, the "mother of all integrable systems". It contains isomonodromic deformations and the multi-pole hierarchy of the preceding chapter as its restrictions. It is a very natural union of both theories as if they were created for each other. We closely follow in this chapter the presentation in [DiOO].
187
188
Soliton Equations and Hamiltonian
Systems
11.1.2. Given the equation (d/dz-M(z))w = 0,
M(z) = M+(z) + J2M-(z),
(11.1.3)
k
where Tk
Too
JLT
r=l
*•
k>
r=0
All M's are matrices, superscripts "+" and "—" are just notations that will be convenient later. Let ZQ be a fixed point and WQ a fundamental matrix of solutions in a neighborhood of this point. If the solution is analytically continued along a closed loop 7 not passing through singular points ak, it comes back as a fundamental matrix wy not necessarily coinciding with wo- In fact, w-y depends not on 7 but on its homotopy (even homology, see [Ahl]) class. There is a constant matrix Ny such that wy = woNy in a neighborhood of ZQ. All the matrices iV7 form a group, a matrix representation of the fundamental group. If the point zo is replaced by another one or if the fundamental matrix of solutions WQ is replaced by another one, all the matrices JV7 will undergo a similarity transformation TNyT~x with the same T. The group of matrices Ny, up to the choice of a basis, is the monodromy group. Isomonodromic deformations of an equation (d/dz — M(z))w = 0 are the deformations such that the monodromy group stays the same. Consider a gauge transformation: w^P(z)w,
M ^ PMP-1 + {dP/dz)-P~1,
(11.1.4)
where P(z) is a meromorphic matrix function. It is easy to see that the matrices of monodromy are not changed. Indeed, u?7 = woNy implies that P(z0)w1 = P(zo)woNy, i.e. the solution Pw has the same matrix of monodromy. We are interested in one-parameter families of deformations. Let parameter of deformation be a and matrices P in (11.1.4) be P(a) being P(0) = / . We have w(a) = P(a)w(0) and M(a) = P ( a ) M ( 0 ) P - 1 ( a ) + (dP{a)/dz)-P-l{a). If Q = DaP-P~l where Da = d/da, then Daw = Qw and DaM = dzQ + [Q,M].
(11.1.5)
Conversely, if M — M(a) and there is another matrix function Q = Q(a) such that (11.1.5) holds, then the solution of (11.1.3) can be chosen so that
Isomonodromic
Deformations
and the Most General Matrix
189
Hierarchy
it depends on a as Daw = Qw. Indeed, 0 = Da{dz - M)w = (dz - M){Daw
- Qw)
+ (dz - M)(Qw) - (dzQ + [Q, M))w = (dz - M)(Daw
- Qw) + Q(dzw - Mw) = {dz - M)(Daw
-
Qw).
Thus, Daw — Qw is a solution of (11.1.3), if it is zero at an initial point (which can always be achieved by the choice of initial conditions), then it is identically zero. Now, solving DaP = QP, find P. Equation (11.1.4) is recovered which provides an isomonodromic deformation. It is not difficult to show that this is a general form of a family of isomonodromic deformations. Indeed, let w(a) and M(a) be a family of isomonodromic deformations. When a point z returns to the point ZQ, the matrix wo changes to wy = WQN^, and u>o(a) to w1T~1(a)N7T(a), by definition of an isomonodromic deformation. For the solution w(a) = w(a)T~1(a) we have w-y(a) = w0(ct)N7 and daw1{a) = daw0(a)N7. Eliminating iV7 = w0~1w1, we get daw~, • w~x = dawo • WQ1 , i.e. Q(a) = daw(a) • w(a)_1 is a single-valued function. Now, taking a derivative of dzw — Mw = 0 with respect to a, we easily obtain DaM = dzQ + [Q,M]. This does not mean that Q can be arbitrary. It is naturally to require that in a process of deformation new poles do not appear, though the existing poles can move, and that the order of poles does not change, i.e. the structure of Eq. (11.1.3) remains the same. For that, the right-hand side of (11.1.5) has to have singularities only at the points where M(z) has them and of less than or equal order (cf. [HSS]). The order can increase by 1 if afc depend on a. For example, Q cannot have poles distinct from a^ and oo. Further, such deformations will be constructed. 11.1.6. Exercise. A parallel translation aj >-> a, + a of Eq. (11.1.3) is an isomonodromic deformation. What is the matrix Q in this case? Answer. Q = M. 11.1.7. Our first problem is one where a pole, say Oj, is taken as a parameter of deformation. Let £)j = d/dai. We use the following notations. If R(z) is a rational function, then Rk means its Laurent expansion at a^ and .Rj" is the principal (i.e. the singular) part of the expansion, and R^ = Rf. — R^ . In particular, M\. is the Laurent expansion of M at ajt. Evidently, (Mfc)~ = M^ which explains the notation introduced before, and (M)£ = 53 .-fe M~ + M+,|fc.
190
Soliton Equations
and Hamiltonian
Systems
11.1.8. Proposition. Equations*
{
-\M~,Mr}-
when i 4- k
,
, ,
, A M + = - [l M - , M + ]Jo0 + fc' °° ' °° (11.1.9) can be written as a closed system of ordinary differential equations on coefficients Mkr- (The expression [Mr, Mk]k is the principal part of the Laurent expansion of [M~,Mj7] at the point <Xfc, similarly must be understood [-A/i~,M+>]+) where M^~ is a series in z - 1 . ) The system (11.1.9) is equivalent to DtM = -dzMr -[M~,M], (11.1.10) i.e. it has a form (11.1.5) with Q = —M^, and therefore defines a family of isomonodromic deformations. -a z M f c - + [M,M fc -]^
wheni =
Proof. The matrices Mk depend on a* in two ways: explicitly, through the factors z—at when k = i, and implicitly since the matrices Mkr are supposed to depend on the parameter of deformation a*. The derivative can be represented, correspondingly, as Z>j = d/ddi + D*. We have dM^ /da,i = 0 when i ^ k and dM~ /den = —dzM^~. Equation (11.1.8) takes the form
{
-\M,~,Mr]~
when i ^ k
[M, Mk ]k
when i = k
(11.1.9') The orders of poles on the right-hand side do not exceed those of Mk , therefore, the equation reduces to a system for the coefficients Mkr- It is easy to see that the system (11.1.9) means the equality of principal parts of the left- and the right-hand side of Eq. (11.1.9) at all the poles hence (11.1.9) and (11.1.10) are equivalent. • 11.1.11. In the preceding section one parameter a, was shifted. Is it possible to shift several parameters at the same time? In other words, are Eqs. (11.1.9) with different i compatible? We need to prove that DjDiM = DiDjM. DjDiM = £>i(-dzM7 = dz[M7,M-}7
-
[Mr,M]) + \[M-,M-]-,M]
- [M-, -dzMj
-
\MJ,M\\.
a W h e n all the poles are simple and the matrices are of second order, this is the Sch'lesinger system (see, e.g. [IKSY]).
Isomonodromic
Deformations
and the Most General Matrix
191
Hierarchy
Transform the terms with dz. Using the fact that a meromorphic function is a sum of all its principal parts, we get dz[M7,M-}7
+
[M-,dzM-}
= dz\M~,M-\
- dz[M-,M-)~
= [Mr,dzM-}
+
+
[M-,dzM-\
dz[Mr,M-}j,
i.e. this expression is symmetric with respect to i and j . The rest of the terms: llM-,M-}-,M]
+
lM-,[M7,M}]
= [[M-, M-\~, M}} + [[Mr ,M-),M] = [[Mr, Mr]j,
+ [M-, [Mr, M]\
M\ + [Mr, [Mr, M}\,
also shows that this part is symmetric. Hence, DiDjM respect to i and j .
is symmetric with
11.1.12. Besides shifting the poles, there are other possibilities to get isomonodromic deformations. We want to have independent parameters of deformation. Therefore, new parameters of deformation must stay invariant under the deformations of the first kind. We assume that the eigenvalues of matrices Mk,rk are distinct, and in the case r^ = 1 we assume more: that their differences are not integers. All the assumptions have a generic character. 11.1.13. Lemma. There is a formal matrix series uik = Who + wki {z-ak)-\ and a formal series of diagonal matrices rp __
lk,rk
-Tfc,rfc-1 r
{z - ak) *
{z - afcf*-
Tki 1
_,
_,
,
,
z - ak
such that dz-Mk
=
wk{dz-Tk)wr\
Mk = w^wr1
+ dzwk • wr1.
i.e. (11.1.14)
Soliton Equations and Hamiltonian
192
Systems
The representation is not unique, it has some gauge freedom, since u>k can be multiplied on the right by a diagonal matrix series dk with a corresponding modification of Tk'- Tk >-» Tk + dzdk • d^1. Proof. Equating terms with (z — ak)~Tk in the equality MkWk = WkTk + dzu>k, we have Mk,rkwko = WkoTk,rk, i-e. Mk,rk = WkoTk,rku>k~o hence Tk,rk is a diagonal form of the matrix Mk,rk and Who is a matrix which reduces it to the diagonal form. Let Vk = w^Wk and Nk = w^M^WkoThen NkVk = VkTk + dzVk- Notice that Nk,rk = Tk,Tk- Equating terms with (z - ak)~rk+1, we have Nk,rk-i + Nk,rkVki = Tk,rk-i + VkiTk,rk or Tk,rk-1
+ [vkl,Tk,rk]
= •'Vfc.r,,-!
when rk ^ 1- In the case r^ = 1 the left-hand side has an additional term, +Vki • The right-hand side is known. Taking the diagonal part of the equality, and assuming that the diagonal part of Vki is zero we get Tk,rk-i while the off-diagonal part of the equality gives the off-diagonal part of Vki- (Here it is important that in the case r^ = 1 the difference of two diagonal elements of Tkjrk is not integer, by assumption.) It is easy to see that this procedure can be prolonged, and this will provide all the needed coefficients. Diagonal parts of all Vkr with r > 0 are taken as zero. • It is not difficult to compute (this will be done in Sec. 11.2) that if one requires DiWk
=
-dzWk + M^Wk
when i = k
-M~ibk
otherwise,
and
{
—dzTk
when i = k
0
otherwise
then the equations of isomonodromy deformations (11.1.10) will be satisfied. In both cases i = k and i ^ k we have D{Fki = 0. This is why diagonal elements of these diagonal matrices can be taken as independent parameters of the second family of isomonodromic deformations. Notice that the gauge can be chosen freely only for initial conditions, after that it is uniquely defined. The commutativity of Di in their action on Wk does not automatically follow from the proven commutativity in their action on M, due to the freedom of gauge. Nevertheless, this is correct which will also be shown in Sec. 11.2.
Isomonodromic Deformations and the Most General Matrix Hierarchy
193
Instead of Tk, we introduce a diagonal matrix series r)k such that Tk = dzVk^k
=
5 Z 5 Z *fc«r(z _ a-k)~rEa + ^2 *ka log(z r=l a=l t
Tkr(z - ak)r+1
^
ak)Ea
a
_^
+
r=0
where Ea is a matrix (Sia5ja); tkar and Afea are diagonal elements of the matrices —Tk,r+i/r and Tkt\. The logarithmic term is included in rj^. For a pole at infinity, one has to replace (z — ak)~r by zr, the logarithmic term is absent. In view of further importance of v^, we give it a special designation, £&: nt-l
& = Yl
n
Y2tkcr(z
- ak)'rEa
+ Y^^ka\og(z
r=l a=l
- ak)Ea .
(11.1.15)
a
11.1.16. Corollary of the lemma. Let oo
wk = wk exp T)l = ^2
w
kr(z -ak)r
.
r=0
Then Mfc = wkdz£kwk~l
+ dzwk -wl1
and
dz - Mk = wk(dz -
(d^k))^1. (11.1.17)
Indeed, 9zWfe • w^"1 = dzwk • w^1 — wkT£w~^1. It is possible to shorten this formula by letting wk = wk exp £fc . Then Mfc = dzwk • w^1,
i.e. dzwk = Mkwk .
(11.1.18)
This formula is local, at the point ak. Thus, wk is a local formal solution of equation (dz — M(z))w = 0. From 11.1.16 it also follows that Mk~ = (wk(dz^k)w^\
= (wk(dz^k)w^\
.
Soliton Equations
194
and Hamiltonian
Systems
The equations of isomonodromic deformations in terms of wk stay the same as those for u>k (since (Dt + dz)r)f = 0):
{
—dzWk + Mtwk
when i = k
—Mi Wk
otherwise.
(11.1.19)
Now we introduce a new object dual, in a sense, to M^ that will playin what follows a role not less important than M^ itself. This is (dkai = d/dtkai): Rkal
= Wk(dkaltik)wk~
,
dkalik
= Ea{z
- dk)~
,
dooalZoo =
EaZ
and M (compare R^al = (wk(dkai^k)wk% h = (wk(dz^k)w^\). The quantities tkai, I < rk will be chosen as new parameters of deformation.
11.1.20. Proposition. Equations „ ,._ j-[Mk,R-Jk wheni^fc ,11101x diaiMk = { _ (11.1.21) {dzRkai-lM'RkJk wheni = fc can be written as a closed system of ordinary differential equations on coefficients MkT. It is equivalent to dialM
= dzR~al + [RTal,M]
(11.1.22)
which coincides with (11.1.5) for Q = R^al and defines an isomonodromic deformation. (One of aj and ak or both can be at infinity, then the corresponding superscripts change to "+"•) Proof. One has to prove that the multiplicities of poles on the right-hand side do not exceed those on the left-hand side. This is evident when i ^ k. When i = k, terms with (z — a,i)~r, r > r-j are the same as in \dz - Mi, Riai} = wi[dz - {dz£,i),dial£,i}w~l
= -Iw^z
-
ai)~l~1Eaw^1
= —IRiaJ-f-i , and in the latter expression there are no such terms since / < rj. The proof can be completed in the same manner as in Proposition 11.1.8. • The commutativity of all above constructed deformations will be proven in the next section in a more general setting. It will also be proven that
Isomonodromic
Deformations
and the Most General Matrix
Hierarchy
195
the second set of isomonodromic deformations in terms of lik has the form
{ 11.2
-R£ai™k
when i = k
R-iaitik
otherwise,
dial = 9/dtiai .
(11.1.23)
General Matrix Hierarchy
11.2.1. What is the telegraph? Imagine a gigantic cat. They jerk its tail in Rome, and it meows in London. What is the wireless telegraph? Just the same but without the cat. An old joke. All what we studied in Sec. 3.1 was Eq. (11.1.3) and its deformations which preserve the group of monodromy. Now we are going to deal with just the same, but without Eq. (11.1.3), leaving only its deformations. Then, deformations of what? It happens that many auxiliary structures were already constructed so that the deformations can start their own life among those constructions. The system of differential equations we discussed in the first section happens to be merely a restriction of a larger system to a submanifold determined by Eq. (11.1.3). The larger system is, in fact, a huge integrable system containing most if not all two-dimensional integrable systems as its restrictions and reductions. There is a list of all needed structures. 1°. There are N points ak € C and formal series oo
wk = y^wkr(z
- ak)r
r=0
and also
Woo =
oo y^^WpprZ' r—0
The coefficients wkr where k = 1 , . . . , N, oo are nxn matrices whose entries are dependent variables of the problem.
196
Soliton
Equations
and Hamiltonian
Systems
2°. rfc —1 n
n
r=l a=l
a=l
and r-oo + l Soo —
/
y
n /
_, *ooarZ
-^a j
r=0 a = l
where Ea are matrices (<$ia<Jja), {*itar} are independent variables, so are {ak}', rk are finite but not specified natural numbers. 3°. wk =u}fcexp£fc,
k =
l,...,N,oo.
They are called formal Baker functions. 4°. Rkar
= U>k(dkar£k)Wk
'
dkar^k
= - E a ^ ~ a*:)_r ,
^ocar-Coo = £'o-2 : r .
5°. Mfc = c?zu;fc • u;^ 1 ,
i.e. 3zu;fe = MkWk
or, equivalently, Mfc = dzwk -Wk1 +
tikidztk^k1
or dz - Mk = WkdzW^1. 6°. Now we are writing the equations of hierarchy. The first set of equations is
{
-Rtal'Wk
when i = k
diai=d/dtial.
(11.2.2)
R-ialWk otherwise, The left-hand side and the right-hand side are formal developments in powers of z — ak, R~al is the singular part of the expansion in powers of z — a^, this is a rational function, it can be expanded in powers of z — ak- We are also using the notation Rfal = Riai — R~al. In terms of the Baker functions, the equations are diociwk = R~alwk •
(11.2.3)
Isomonodromic
Deformations
and the Most General Matrix
197
Hierarchy
Indeed, dkaiwk = -Rkalwkeik
+
wk{dkai^k)eik
= -• R fcai™fc e5 ' t + RkadWkeik
= RkalWk
.
The case i ^ k is obvious. The second set of equations is
{
—dzwk + MuWk ' \
when i = k ,. .
D^d/dai.
(11.2.4)
—Mt wk otherwise, The operator Di is the "total partial derivative" with respect to aj, taking into account both the explicit and implicit dependence on a,. In terms of the Baker functions, the equations are DiWk = -M-wk. Indeed, Dk£,k =
(11.2.5)
-dz£k.
Dkwk = (Dkwk)eik + (dzwk)e^k
- wk{dze^k)
= (~dzwk + M£wk)eik
= M~£wk - Mkwk = -M^wk
-
dz(wke^k)
.
The case i ^= k is obvious. Notice that both operators, diai, as well as Di, commute with dz. 11.2.6. Remark. Among all the structures listed here, we do not see one which played an important role in the first section: M. The local series Mfc we deal with now are not necessarily Laurent expansions of the same meromorphic function M at the points ak, they are independent. This is why the hierarchy constructed here is considerably wider than the isomonodromy system. The latter is its restriction (see Proposition 11.2.13 below). 11.2.7. Lemma. The following equations follow from the equations of hierarchy (11.2.3) and (11.2.5) (or (11.2.2) and (11.2.4)) and definitions of Rk0m and Mk: (11.2.8)
dialRkj3m = [RiaV Rk/3m} , dialMk = dzR-al + Uitikfim
.{-[M-,RkPm]
—\
{-ozRkf3m = ~[Mr,Rk0m]
(11.2.9)
[R-al,Mk], when i ^ k ,
otherwise
+ [M^, Rkf3 + 5ik(-dzRk0m
+ [Mk,Rk0m\)
(11.2.10)
198
Soliton Equations and Hamiltonian
Systems
and DiMk = -dzMr
- [Mf,Mk].
(11.2.11)
P r o o f . We have, dialRkpm
- WkidkPmik)^
• w^1)
diaiMk = diai(dzwk = dzR-al DiRkpm
=
Rial"Wk(dk0m£k)wk~
RialWkW^1
= \R^ai,Rk0m}
= dial(Wk(dk0m£k)"Wk
=
+
)
= dz(R7a[Wk)
•
• w^1 - dzwk
• w^1RTal
[R-al,Mk].
Di{wk{dkpm(,k)Wk~l) 6ik(wkdz(dk/3m^k)w^1)
= -[Mr,Rk0m]
-
= ~[M~ ,Rk0m\
- Sik(dz(wk(dk0m^k)w^1)
-
[Mk,Rk0m])
since Dk{dk0mik)
— -dz(dk/3m£k)
•
Finally, DiMk = Di(dzwk
• w^1) • w-1 + dzwk
= -dz(Mt-wk) 1 1 . 2 . 1 2 . Proposition.
• w^Mr
= -dzM~
- [M,r,Mk].
D
All vector fields (11.2.3) a n d (11.2.5) commute.
P r o o f . Compute: [diahdkfimlWj
= dialRl0mWj + Rk0mRi«lWJ
- dk0mR7alWj
=
[R~al,Rk0m}kwj ~ R7alRk0mWi
~ iRkPm'KiafcVj
•
lii^k, [dtcd,dkf}m]Wj =
[R7al>Rkf3m}kWi + Rk0mRicclW3
=
lRial>Rk0m\
~ [Rk0rmRU7Wi * \RiaVRk0m\
= °
KalRk~0mWJ
Isomonodromic Deformations and the Most General Matrix Hierarchy
199
and if i = k then [dicd,dipm]Wj = \R7anR7(lm)7WJ
[R7al,Rtpm\7Wi
+
+ Rif)mRialWi
- [R7f3m>Ri<*l]7wi -
= [Ri*l,Rtf3m}7WJ = [Rial,RiPm\i
-
R
7alR7pmWJ
[R70m'Ri^]7WJ
Wj =0 ,
Rial and Ripm commute since they are obtained by the same dressing from two diagonal matrices. Now, let i ^= k, then \DuDk}wj
= -Di[M^Wj)
+
= [M-,Mk}-Wj
Dk{M~Wj)
- [M^,Mi]rWj
= [Mr,M^Wj
-
- [Mk-,M-}7Wj
= {M-,Mk-]Wj
- [M-,Mk]Wj
[M-,Mk]wj -
[Mf,Mk-\Wj
= 0.
Let i ^ k, [diai,Dk\wj = -diai(MkWj)
-
= -lR7avMk}k~wJ
-
= -\R7avMk\k~wJ
-MkR7am
= -[RZ.i>MkH
Dk{RTalWj)
M R
k 7*iVj
+ \Mk,R7J7™i + [Mk>R7J7™j
+ [RZd'M^wj
+
R
+
7aiMk~Wj R
7aiMkwJ
= o.
Finally, let i = k, [diai,Di\wj
= -dial(M~Wj)
-
= -(9zR7ai + [R7ai^i]-)wj
Di(R7alWj)
-
M-R-alWj
- (-d*R7«i + [Mjt,Ri«i]T)wj + R7aiMiwJ = - ( [ ^ > M 4 7 H ' - M7R-alWj = -lR7*i, M~};WJ + [R^M^Wj
- ([M+.-RrJi'M +
R
7aiM-wj
= 0.
One point can be at infinity, the proof will not change.
•
200
Soliton Equations and Hamiltonian
Systems
11.2.13. Proposition. The equations of hierarchy are compatible with the restriction
M+ = J2Mi>
* = 1,2,...
(11.2.14)
(one of aj or at may be oo in which case the superscripts + and — must be swapped in the corresponding term). Restricted equations coincide with the isomonodromic Eqs. (11.1.9) and (11.1.21). Proof. It is easy to check from the definitions that
(Y,Di + dz j wk = f - ^ M f + M+ J wk hence Mk = Ei^k M~ ls equivalent to (%2tDi + dz)wk = 0. The latter manifold is tangent to the flows of hierarchy since the operator (%2i Di+dz) commutes with o>ia; and Dt. Being restricted to this manifold, Eq. (11.2.11) yields \-[M-,M~}l DtMk- = { ( -dzMk~ - [Mr, Ej M-]~k
wheni^k (11.2.15) when i = k
and Eq. (11.2.9) yields o w \{R7avMk\k diaiMk~ = { [ 9zR^al + [Rka,, Ej Mr]~
when i / A : (11.2.16) when i = k
which coincide, correspondingly, with isomonodromic deformations (11.1.9) and (11.1.21). • 11.2.17. Return to the non-restricted system (11.2.5)-(11.2.8) but concentrate only on Eq. (11.2.8): dialRkffm
=
[RialjRk0m\
or, in terms of Baker functions, dialWk
= R~aiWk •
This is nothing but the multi-pole Zakharov-Shabat hierarchy of Sec. 10.3.
Isomonodromic Deformations and the Most General Matrix Hierarchy
201
The extension of the ZS hierarchy by joining new variables afe and rational functions Mk is reminiscent of additional symmetries by Orlov and Shulman (Chap. 7), but it is very important that variables of the additional symmetries have been identified with the poles Ofc. That was prompted by the theory of isomonodromic deformations.
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Chapter 12
Tau Functions of Matrix Hierarchies
12.1
Segal-Wilson's T-Function for A K N S - D
12.1.1. In Sec. 10.5.10 we expressed the Baker function of the soliton solution of the matrix multi-pole hierarchy in terms of the tau function. However, for a general solution it was not done. Here we do this in the simplest case of the AKNS-D hierarchy on the base of the Grassmannian theory. Besides, we give a short summary of results about the tau function for the multi-pole hierarchy without using the Grassmannian. The Grassmann theory of the tau function for this hierarchy stays open (though the definition of the Grassmannian was given in Sec. 10.5). We follow Ref. [Di93(a), 96]. Represent the operator 1 = d + U + zA in the "second dressing form" 10.1.22,m = 0: l = wdw~1,
Ra =
wEaw~l
where
(
oo
n
\
^2^2zkEatka)
k=0a=l
oo
' ™(z) = ^ W j Z ^ , J
W0 = I.
j=0
The equations of the hierarchy are equivalent to dkaw = (zkRa)+w
or
dkaw =
-(zkRa)_w.
The function w is called the wave Baker function. 12.1.2. Definition of the Grassmannian. Let H be L 2 (S, C n ) , a space of series v(z) = X^oo^fc 2 '' where vk € C n , \z\ — 1, let H+ and H- be spaces of truncated series: H+ = {2o°}) H- = {X^lL}! H = H+ ® H— Let p± 203
204
Soliton Equations and Hamiltonian
Systems
be natural projections of H onto these subspaces. We shall think of vectors as vector-rows. The Grassmannian, Gr, is the set of all subspaces W C H enjoying properties (i) p+\w is a one-to-one correspondence (this is not the general definition but a generic case), (ii) zW C W. We say that a matrix function belongs to W if all its rows do. Let g(t,z) = exp£ = e x p £ ] £ l 0 ^J^=lzkEatkaWe consider a transformation g~* of the space H: g~x : H -> H, « 4
vg~x.
If W e Gr, then for almost all t the subspace Wg~x s Gr. 12.1.3. Definition of the Grassmannian Baker function. ww{t,z) is a Baker function corresponding to a W e Gr if (i) for any t = (t\,t2,- • •), ww € W as a function of z, (ii) p+(wwg~1) = 1. Together the conditions mean that ww (t, z) is the only element of W of the form ww(t,z)=
I/ + ^
Wi{t)z~l J g(t, z) = ww(t, z)g(t, z).
12.1.4. Proposition. The Segal-Wilson (S-W) Baker function ww{t, z) is a Baker function in the sense of Sec. 12.1.1 (a formal wave function). Proof. Let d = — Yl™ aadia- It must be proven that (i) l(w) = 0 for some 1 = d + U + zA, where w = w\y(t,z) and (ii) dkaw = (zkRa)+w, a 1 where R = wEaw~ . From the definition of ww it follows that w'w~1 = -Az — U + (w'w'1)where U = [wi,A]. Let (w'w*1)- = Q. One must prove that Q = 0. We have w' + (U + zA)w = Qw . The left-hand side is an element of W. Then (w' + (U + zA)w)g~1 = Qwg^1 is an element of Wg"1. The right-hand side is 0(z~1). This means that p+lQwg"1) = 0. Recall that p+ is one-to-one mapping of Wg~x onto H+. Therefore w' + (U + zA)w = 0, and (i) is proven. Further, for Ra defined as above we have (dka - (zkRa)+)w
= (dkaw + (zkRa)_w)
-g,
(w =
wg'1).
Tau Functions
of Matrix
205
Hierarchies
The left-hand side belongs to W. Hence (dkaw + (zkRa)-w) is an element of Wg~x whose p+-projection vanishes (since this is 0 ( z - 1 ) ) . Thus dkaw+(zkRa)^w
=0
which is equivalent to (ii).
•
12.1.5. An example of elements of the Grassmannian. Let im, where i = l,...,Nn, be some points inside the unit circle, \m,i\ < 1. Here N is a natural number ("the soliton number"). Let r/j be Nn vector-columns in Cn which span this space (this is not a basis, of course, they are too numerous). Let W be the set of all elements of H (vector-rows) of the form v(z) = YI^N vkzk satisfying the relations v(m,i)-r]i — 0. This W is for almost all sets {rrij, 77*} an element of the Grassmannian. The corresponding solutions are solitons. 12.1.6. An explicit expression for the Baker function will be given. One has to find
(
\
N
I+ J2Mt)z-s 1
00
n
e««-*>, t(t,z) = J2Y,tk«zkE<* )
k=0a=l
such that 1 1 + f^ wsmTs j e«*.m<> • m = 0,
i=
l,...,nN.
The crth component of this vector equality is N
n
+ e««"' m '>», te = 0 ,
Y, E ^'^mr'M^^THp
00
&(i, m,) = £
8=10=1
tkpmk
.
fc=0
Multiply this by m f and notice that m f _ i e ^ ( * ' m ' ) Letting y0i = e x p ^ ^ m , ) ) ^ we have N
n
Y2 H ws,a0d^fsy0i + d?ayai = 0 . s=l/3=l
From here, ws>a/g must be found and then N Wa/3{z) = ^
WStapZ
S
+ Sa0 .
=
d^e^'"1^.
206
Soliton Equations
and Hamiltonian
Systems
It is not difficult to check that the result is the following: The (a/?)th element of the matrix w{z) is given by the determinant of the (Nn + l)th order: |
0
1
1
Vn.Nn
|
0
9liyi,Nn
|
0
1
z
\
0
2/n
Vl,Nn
Vnl
#111/11
-N
dlnVn,Nn
dlnVnl
Wa/3 =
11
0
Vl,Nn
ZN-1
aft-'i/m
d
ln
0
Vn,Nn
dlLVa.Nn
ZN5a/3
We have here N blocks of rows containing n rows each, and one separate (the last) row. Nonzero elements of the last column are on the /?th place in each block, and also the last element (when a ^ (3). A is the minor of the first Nn rows and columns. 12.1.7. The T-function for this example. We proceed in the same way as for the KP hierarchy, i.e. making zeros in the last column. If a = /3, then we can annul all elements of this column except the last one. If a ^ (3, then the last element of the last column is zero, and it is possible to annul all elements of the last column except that of the order zN_1. As it is easy to calculate, l
dipypi - -di^ypi
.
.
1
8=1
sz°
= dip e x P Yl ( ts P
Z
ml • rap.
Therefore, for diagonal elements we obtain: waa{t,z)
=
A(...,tsa-l/szs,...)
A(t)
(12.1.8)
Tau Functions
of Matrix
207
Hierarchies
the term l/szs being subtracted only from the time variables with the second subscript a. For non-diagonal elements we have wap(t,z)
=z
^
-^-
,
(12.1.9)
where Aap is the cofactor of the element z " - 1 of the last column. This can also be described as the minor A where the /3th row of the last block is replaced by the a t h row of the next block (which is not involved in A), and the sign must be changed. Now it is quite natural to consider as the T-function the following matrix:
12.1.10. hierarchy. of the KP Let W projection
(Aa0(t),
ifa^/J
\A(t),
i f a = /3.
The general definition of the T-function for the AKNS-D The definition is similar to the modified Segal-Wilson definition T-function in Sec. 8.2. be an element of the Grassmannian. Let lw be the operator of H -> if_ parallel to W. Then
TW(t) = rw(g) = det Tw(g),
where Tw(g) = lw ° 9 • H- -> E- . (12.1.11)
Let Ra/3 : H- —> H be an operator given by its action on basic elements: _k Ra/3Z
f-eQ, ep = <
^C
e
p
iik = l,p = P otherwise.
Here {ep} is a basis in C n . Put Twap{t) = TWaj3{g) = detTWap(g),
(12.1.12)
where Twap(g) = lw °9° Ra/3 : H- -> H-. 12.1.13. Proposition. Elements of the Baker function can be expressed in terms of the r-function as n wWaa(t,z)
\
rw(...,tsa-l/szs,...) = ^ p—i TW{t)
'-
/ioii/i\ (12.1.14)
208
Soliton Equations and Hamiltonian
Systems
and A, WWa0{t,Z)
\
• ,ts0 - 1/SZS,...) tL 1-:
-lTWa0(-• = Z
/ioiic\ (12.1.15)
, <* T P •
Tw\t) Proof. First we prove a lemma which follows directly from definitions: Lemma. Tw\g)Tw(g9l)
= TWg-i(9l).
(12.1.16)
(The proof does not differ from that in the case of KP, see 8.2.9.) Now, let gi = d i a g ( l , . . . , 1 — z/C, 1 , . . . , 1) where 1 - z/C, is on the ath place. Then gg\ is the same as g where tsa are replaced by tsa — l/sC s a n d tsp with /? 7^ a remain unchanged. If we take the determinant of both sides of Eq. (12.1.16), then the left-hand side will coincide with the right-hand side of Eq. (12.1.14). The mapping on the right-hand side of Eq. (12.1.16) preserves all basic elements z~kep with p ^ a, sends elements z~kea with k > 2 to z~kea — z~k+1 /Cea £ H-, and it remains to find its action on z~1ea. We have -i
giz
-i
ea = z
1
-i
ea - -ea = z
1/
ea-
.
N
1-
- ( e a - wWa) ~ jWWa ,
where % a is the a t h row of the Baker function. The operator l\y annihilates the last term belonging to W and does not change the rest of them belonging to H-. Now, the determinant of the mapping TWg~i(gi) can be calculated in the space i/_ modulo the subspace spanned on the basic elements z_fce/g with (3 y£ a and is equal to 1 + W(Waa)l/C
-l/C o
w
{Waa)2/(
W(Waa)zK
i
o
-l/c
i
''' = WWaa(t,Q
•
which proves Eq. (12.1.14). In order to calculate non-diagonal elements, let us multiply Eq. (12.1.16) to right by the operator Rap- The determinant of the left-hand side is equal to the right-hand side of Eq. (12.1.15). The mapping on the right-hand side preserves all basic elements z~kep with p ^ /?, sends z~kep with k > 2 to z~ke@ — (z~k+1 /Qep € H- and it remains to find its action on 2 _ 1 e / 3.
Tau Functions of Matrix
209
Hierarchies
The operator Rap sends this element to —e a , the operator g\ does not do anything. We have - e a = ( - e a + u>wa) — u>wa- The projector lw annihilates the last term belonging to W and does not change other terms. The proof is completed with the calculation of the determinant: W{Wa/3)2
W(Wa0)3
-i/C
1
0
o
-i/C
1
W(Wa0)l
•• •
WWap(t,0
-C
D
as required.
12.2
Tau Functions for More General Matrix Hierarchies
12.2.1. In this section we briefly summarize some results of the article [Di96] concerning the definition of the T-function for the single-pole and the multi-pole Zakharov-Shabat hierarchies. Contents of Sec. 10.2 can be completed by the following results similar to those for the KP hierarchy. 12.2.2. Proposition (Universal property). Let w be a series w = £ ^ WiZ~%, WQ = I and w = tuexp£. All the functions depend on variables thalf w satisfies an equation of the form dkaw — Bkaw where BkaW is a polynomial in z then this is an equation of the hierarchy, i.e. Bka = (zkRa)+12.2.3. Proposition (Bilinear identity). Let w be a series w = YLV WiZ~l' u>o = I and w = u)exp£. All the functions depend on variables tkali w satisfies the hierarchy equations (10.2.4) then the following bilinear identity reszzldkiai
• • • dksasw • u T 1 = 0
holds for arbitrary sets of indices, i > 0. Conversely, if there is another series v = Y^viz~li exp(—£)v and
(12.2.4)
VQ = I, v =
res 2 z'd fclQl • • • dksasw -v = 0 for all sets of indices, then v = w _ 1 and w i s a Baker function of the hierarchy.
210
Soliton Equations and Hamiltonian
Systems
12.2.5. Proposition. For any Baker function there are functions r(t) and Tap(t) and constant series cp(z) such that w M z ) = C0(2)
-JT-T
(12.2.6)
and wa/3{z) = z
cp(z)
7-7
,a^(3
(12.2.7)
T(t) hold. Coefficients cp(z) are insignificant if Baker functions are considered up to the equivalence. 12.2.8. The transition to the multipole hierarchy will be clearer if we consider first the not normalized single-pole hierarchy. This means that we no more require that WQ is the unit matrix. The definition of the hierarchy (10.2.4) must be adjusted to this requirement since (10.2.4) implies that wo = const. Let J4( + ) symbolize the purely positive part of an expansion in powers of z, i.e. without the constant term, and A(_) negative part with the constant term, i.e. the constant term passes from the positive part to the negative one. Equation (10.2.3) will be replaced by dlaw = -{zlRa)(-)W,
Ra = wEaw~1
(12.2.9)
and Eq. (10.2.4) by diaw = Blaw = (zlRa){+)w.
(12.2.10)
It can be proven that the operators dia commute as well. 12.2.11. (10.2.3).
Proposition. If w satisfies (12.2.9), then v = w^w
satisfies
12.2.12. Proposition. To every Baker function of the not normalized singlepole hierarchy there exist functions r and rap such that wap{z) = cp(z)
7-
,
(12.2.13)
where cp(z) are constant series. 12.2.14. Now, turn to the multi-pole hierarchy, Sec. 10.3. If the dependence only on "local variables" thai at a point ak is considered, and that on "alien variables" tkiai where k\ ^ k is ignored, then we have a single-pole
Tau Functions
of Matrix
211
Hierarchies
not normalized hierarchy (the pole is not at infinity but at z — ak, one can change variables z — ak = C - 1 )- Then it is possible to represent the Baker functions in terms of r-functions as Wk,a0{t,z) = ck0(z)—
,.:
e*k ,
(12.2.15)
where operators of translation Gkp(z) are defined by Gkf}f{t) = / ( • • • )*fei,7,! - hk!Sff-yjiz
- ak) , • • • J
and Ckp(z) are constant series in z — ak- Here the constants Ckp{z) can depend on alien variables. The following fact is true: 12.2.16. Proposition. If w = {wk} is an arbitrary Baker function, then there are functions r(t) and Tk,ap(t) and constant series Ckp{z) such that «*,«*(*,*) = ck0{z)Gkp{z)^{t)e^
.
(12.2.17)
Notice that the last factor is exp ^ £; and not just exp £fc, therefore the expression in front of it is not Wkap, , ,Gkp(z)Tk,af3(t) M
Wka0 = Ckp(z)
T[t)
'
y-,..<.
' ' e2^***
.
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Chapter 13
KP, Modified KP, Constrained K P , Discrete KP, and q-KP
13.1
Modified G D (Cont'd)
13.1.1. This chapter is based on the paper [Di99]. We are already familiar with the modified GD hierarchy (Sec. 4.3). Now we add some more information about the Baker functions related to the modified GD. In the next section, the definition of the modified KP hierarchy will be given. The latter has very interesting connections to the discrete hierarchies which will also be described in this chapter. The most general of them is the Toda lattice hierarchy (Ueno and Takasaki [UT]). We shall study a restricted variant of this hierarchy, the so-called 1-Toda lattice hierarchy where only a half of all time-variables involved in the general Toda lattice hierarchy is kept. This 1-Toda lattice hierarchy was recently studied by Adler and van Moerbeke [AvM99], they called it the discrete KP. In the discrete hierarchies the continuous variable x is replaced by a discrete index i, and the derivative d by a shift operator i i-» i + 1, infinite matrices are analogues of pseudodifferential operators, e.g., the matrix A = {6iti-i} is an analogue of d. Upper and lower triangular matrices are similar to differential and purely integral operators. All usual constructions of KP: dressing operators, Baker functions, tau functions have counterparts in the discrete theory. (For example, the role of the dressing operators play here the lower triangular matrices with unities on the diagonal.) The "time" variables are equal in rights, no one is distinguished. All this would be good but not surprising if not one truly miraculous circumstance. If the first of the time variables is distinguished as x, d\ = d, and if the elements of a row of a matrix are interpreted as coefficients of a pseudodifferential operator (the diagonal element being a coefficient in <9°), similarly to that what we did study the Drinfeld-Sokolov reduction in Chap. 9, then each row of a dressing matrix of the discrete KP presents a pseudodifferential 213
214
Soliton Equations and Hamiltonian
Systems
operator which happens to be a dressing operator of a genuine, continuous KP. In a way, this is a manifestation of the "universal" property of the KP hierarchy (Chap. 6). In this way one obtains not only one solution of KP but a chain of interconnected solutions, exactly what we call a modified KP. This is a very beautiful theory (see also [AvM99]). Another relation between the continuous and the discrete equations is a usual relation between the finite difference and the derivative: the second is the limit of the first when the step tends to zero. The finite difference is a "deformation" of the derivative. Considered from this viewpoint, discrete hierarchy is called q-KP. 13.1.2. This section is a continuation of Sec. 4.3. Each operator L, can be represented in a dressing form Li = Widnw~1 where Wi = Y^Wi,id~l with u^o = 1- Operators Wi are determined up to multiplication on the right by constant series Ci(z) = Y^f Ci,ld~l with Ci,0 = I-
13.1.3. Lemma. equality:
Properly choosing c,(z), one can always achieve the (d + Vi)-Wi = wi+1-d.
(13.1.4)
Proof. From (4.3.3) we have •wi+idnw~^1{d + Vi) = (d + vi)widnw^1
or
dnwT^ (8 + Vi)wi = w~^{d + Vi)widn , i.e. w~+l(d + Vi)Wi commutes with d being a first order \l/DO. It has a form d ( ! + Z ) r Ci,kd~k). If we replace wi+1 by wi+i(l + Y^° ci,k9~k)~1, then 11)^(8+Vi)ibi = d, i.e. (13.1.4). Now we can start with some i and improve in succession u),+i, Wi+2, •. • ,u),+ n . We have wi+n = (d + Ui+„_i) • • • (d + Vi)wid~n = LiL~lWi = m and Wi depends on the index i periodically, like Lj.
•
It is not difficult to show that {dk} defined by (4.3.6) commute. Baker functions Wi(t, z) corresponding to the dressing operators u>i(t, d) are Wi{t, z) = ibi(t, d) exp£(£, z) where £(t, z) = J2T ^z'• Equation (13.1.4) is equivalent to (d + Vi)wi(t, z) = zwi+i(t, z).
(13.1.5)
13.1.6. Remark. It is worth mentioning that in terms of Grassmannians the relations (13.1.4) or (13.1.5) mean the following. If Vi are elements of
KP, Modified KP, Constrained KP, Discrete KP, and q-KP
215
the Grassmannian related to Wi, then zVi+\ C Vi (see [Di93(a)]). Grassmannian considerations help to build examples; for instance, let H be the space Li on the circle \z\ — 1, and
Vi = I / ( z ) = E
tek\tt«i)
= e'aifim);
l = l,...,N,en
= l\. (13.1.7)
Functions / are supposed to be prolonged into the circle and ai are distinct points, 0 < |a; | < 1, while a; are arbitrary nonzero numbers. It is easy to see that all properties are satisfied. A transition from one solution Li of KdV to the others, Lj, is a Backlund (or Darboux) transformation, see Adler [Ad81].
13.2
Modified K P and Constrained K P
13.2.1. There exist several definitions of the modified hierarchy given by various authors. The first was suggested by Kuperschmidt [Kup], see also Cheng Yi [ChYi] and Gestezy and Unterkofler [GU]. All the definitions are trying to transfer the relationship between KdV and mKdV to the KP situation and, first of all, to factorize the KP operator. And there is a big obstacle on this way, even insurmountable one. A pseudodifferential operator cannot be represented as a product of first order differential operators. Suggested palliatives like a product of a finite number of operators with one VtDO factor and the rest of them being first order differential operators have a disadvantage being not symmetric and not allowing a Backlund transformation. We abandon the very idea of the factorization and we will base our definition on a concept of a collection of KP operators connected by Eq. (4.3.3). There are too many variables here, all of them cannot be independent. The first problem is to find a complete set of independent variables. We suggest the following construction. Let LQ be a KP operator and let Vi, where i £ Z, be variables. For i > 0 let Li = (3 + «<_!) • • • (9 + u 0 )Lo(d + vo)-1 •••(d + ^ - i ) " 1 and L_i = {d + v-i)-1
•••{0 + v^)-lL0{8
+ v-!) •••{d + w_ 4 ).
216
Soliton Equations
and Hamiltonian
Systems
Evidently, Li+1(d + vi) = (d + vi)Li,
ieZ.
(13.2.2)
Thus, we take the collection of all coefficients of LQ along with the set of all Vi as independent variables. Instead, we could take coefficients of some other Li0 and the same set of Vi; it would be a different system of variables. 13.2.3. Definition. The modified KP hierarchy is a system of equations: dkL0 = [(Lk)+,L0],
dkvi = (Lki+1)+(d + vi)-(d
+ vi)(L^+.
(13.2.4)
13.2.5. Proposition. For all i e Z , the equation dkLi = {{Lki)+,Li]
(13.2.6)
holds. Proof. We use induction. Let i > 0 and let for all smaller non-negative indices the Eq. (13.2.6) be proven. We have: + Vj-i)" 1
Li = (d + Vi-i)Li-i(d and
dkLi = -(d + vt-JLi^id
+ v^y'KL^+id + Vi-!)-1
-(d + Vi-MLlj+W + {(Lk)+(d
+ v^)
- (0 + i; i _ 1 )(L*_ 1 ) + }L i _ 1 (5 + t ^ ) "
+ (8 + vi-1)[(Lk_1)+, = -Li(Lk)+
Li-^d
+
Vi-!)'1
+ (8 + Vi-!)Li-!(Lk_!)+(d
+ (Lk)+Li -(d
+ Vi-ML^J+Li-iid
+ (d + Vi-!)[(Lk_!)
+ Vi_i)
+
,Li-!}(d + Vi.!)-1
+
Vi-!)-1
+
Vi-!)-1
= [(Lk) + , Li] .
Similarly, L-i = (d + v _ i ) _ 1 L _ i + 1 ( a + v-i) and dkv-i = (Lk_i+1)+(d
+ v^) -(d
+ v-i)(Lk_i)+
.
1
KP, Modified KP, Constrained KP, Discrete KP, and q-KP
217
Then, dkL-i
= -(d + v-i)-1{{Lk_i+1)+{d - (d + v-MLij+W
+ v-i)
+ v-^L-i+^d
+ (d + v.i)-1L-i+1{(Lk_i+1)+{d
+ v-t)
+ v-i) -(d
+
v-MLlM
+ (d + v _ i ) - 1 [ ( i - i + i ) + . i - i + i ] ( 9 + v-i) = -(d + v^r\Lk_i+1)+L-i+1(d
+ «_4) + (L* 4 )+L_i
+ (0 + v.i)-1L.i+l{Lk_i+1)+{d
+ v-i) - L_,(L*,)+
13.2.7. Proposition. Derivations <9fc commute. Proof. The fact that dkdiLi = didkLi is known; this is a property of the KP hierarchy. Now,
dkdlVi - dtdkVi = dk{(Lli+1)+(d + Vi) -(d + « 0 ( L | ) + } - N O k Vi)[(L )+,L\\+
= [(L* + 1 ) + ,L{ + 1 ] + (3 + Vi) - (3 + + (Lli+1)+{(Lk+1)+(d - {(Lk+1)+(d
+ v^ - (d +
+ Vi) -(d
Vi)(Lk)+}
+ Vi){Lk)+}{Ll^+
= {(Lk+1)+Lli+1-(Lli)_Lk}+(d
- N O
+ vi)
-(d + Wi){L?(L{)_ - L[(Lk)+}
- N i )
= { i t A 1 - i ' + i i ? + i } + ( 9 + «i)
Dressing operators can be introduced: Wi(t,d)=
o J2
^t.5".
w
*o = l
(13.2.8)
a = —oo
such that Li = Widw'1,
(d + v^ • tbi = wi+1 • d,
(13.2.9)
218
Soliton Equations and Hamiltonian
Systems
Baker functions Wi(t,z) -Wi(t,d)exp^(t,z),
LiWi(t,z) =
zwi(t,z),
(d + Vi)wi(t,z) = zwi+1(t,z)
(13.2.10)
and adjoint Baker functions: w*(t,z) = (u; i (t,a)- 1 )*exp(-^(< ; z)), (d-Vi)w*+1(t,z)
L*w*{t,z) =
zw*{t,z),
= -zw*(t,z).
(13.2.11)
Here the asterisk "*" in w* just belongs to the notation while in w* and L* it means formal adjoint of operators. 13.2.12. Proposition. There is an automorphism of the mKP: tk •-»• ik = (-l) fc-1 ifc ,
Vi^Vi
vbi >->Wi = (u>Zi)* ,
= -u-i-x,
Wi !->• u>i = w*Li,
Li n- Li = -L*_i, z i-> i = — z.
(13.2.13)
Indeed, it is easy to see that the equations defining the hierarchy tolerate this transformation. • Notice that any streak of Eqs. (13.2.4), finite of semi-infinite, 0 < i < i\ or i\ < i < 0 (in particular, one equation, i = 0) form a closed system. Especially interesting are the semi-infinite cases, 0 < i < oo or - c o < i < 0. These are one-sided mKP's: m K P + and mKP_. The automorphism (13.2.13) interchanges them.. Also notice that the mKdV is a restriction of mKP to the case when vi+n = Vi and LQ = (d + v„_i) • • • (d + v0)There is another interesting special case when one of the operators L", for example, LQ is a "constrained KP" operator of the form PQ~l or Q~XP where P and Q are differential operators. Then it can be included in a chain of operators L™ of a similar form with ord Pi — ord Qi = const. It looks especially elegant when we start with a purely differential operator, say LQ . Then all L™ and Lnii are ratios of differential operators of orders n + i and i: Ul = PiQ-1,
L% = Qz\P-i
(and Ln0 = Qr'Pi = P-iQZ\),
* > 0,
where Pt = (d + Vi-i) •••(d + v0)L% ,
Qi = {8 + Vi-!)
•••(d + v0)
KP, Modified KP, Constrained
219
KP, Discrete KP, and q-KP
and P_i = L%{d + « _ 0 • • • (d + v-i),
Q-i = {d + w_i) • • • (0 + v-i).
It is easy to find from (13.2.4) that
dkPi = (PtQ-^Pi
PiiQuxPitln
-
and dkQi = (PQ-^Qi
- QiiQT'P)^
•
The equations for P and Q for the constrained hierarchy were suggested first in [Di95] and [Kri95]. More about the constrained hierarchies can be found in [OeS93, 96], [Ar97] and [vL]. 13.2.14. Proposition. If Wi are Baker functions of the mKP hierarchy, then the bilinear identity r e s ^ z ' - ' S ? 1 • • • d^Wi)
• w) = 0,
when i > j
(13.2.15)
holds for any (fci,..., km). Proof. Since d^w = L^_w, it suffices to consider only the case when m = \. Then, using Lemma 6.2.5, reszzi-j{dkwi)
-w* = = iesz(dk(d
xesz{dkwidi-je^t'^){w'fj)-le--^t'z) + v^)
•••{& +
vj)wjexz){(w;)-1e-x*)
= resg dk(d + v,_i) • • • (d + Vj)ilij • w = resddk{d + vi-1)---(d
1
+ vj) = 0.
D
The converse is also true. 13.2.16. Proposition. Let oo
Wi = Yl ™i<*z~aei{Uz), a=0
w* = Yl * 0 ~ a e ~ * ( M ) a=0
be formal expansions where Wia and w*a are functions of variables ifc, and Wio = w*0 = 1. Let res 2 zi'j (dkl • • • 8%" Wi) • w* = 0,
when i = j,j + l
220
Soliton Equations and Hamiltonian
Systems
hold for any multi-index. Then Wj and w* are the Baker and the adjoint Baker functions of the mKP hierarchy. Proof. When i = j , this is Eq. (6.2.6). When i = j + 1, we have 0 = res 2 zdkWj+iiv*
= ieszdkWj+ide^(w*)~le~^
which yields that (ibj+idwj1)differential operator: Wj+\dWj
=
— 0 and Wj+\dwJl = d + v^. Then
(d + Vi)wj =
is a first-order monic
Wj+id,
and this proves the proposition.
13.3
resddkWj+idw~1
D
Discrete K P
13.3.1. We deal with a linear space H of *DO: {a = ]C-oo a »^'} where only finite number of a^ with positive i do not vanish. A dual space H* is {b = ^2°° d~l~1bi} with finite number of non-vanishing bi with negative i; a coupling is given by (a, b) = resa ab = X^oo Oi&tInfinite matrices M = (m^) where i, j € Z will be considered as matrices of the change of the basis: Mj = 2 ^ Q = _ o c rniada are new basis vectors, instead of dl. The same matrices can also be treated as matrices of basis change in the dual space, M-? = ^ °° d~/3~1mpj being new basis vectors instead of c?--?'-1. Product P = M N of two matrices can be found as Py- = resaMjN-?. All the matrices have the property, m^ = 0 when j > i + const., and all sums make sense. Thus, the rows are associated with operators Mj and the columns with operators M J . (In the same manner we treated the Drinfeld-Sokolov reduction in Chap. 9: differential operators Mj = J2a rnia9a were associated with the rows and integral operators M-7' = YIR d~^~1mpj with the columns of finite matrices.) Let W be a matrix with rows W» = ibid1 where xbi are dressing operators (13.2.8). Let CO
w~l = Yi d-pwPj
,
w0j = 1
/?=0
and let W be a matrix with columns W-7 = 13.3.3. Lemma. W = W _ 1 .
d~^~1w~+1.
(13.3.2)
221
KP, Modified KP, Constrained KP, Discrete KP, and q-KP
Proof. The matrix elements of W are W*j = Wij-% when i > j and 0 otherwise. The matrix elements of W are W y = Wi-jj+i if i > j and 0 otherwise. Both, W and W , are lower triangular matrices with unities on their diagonals. So is their product W W , its subdiagonal elements are ( W W ) y = res a WiW» = res a w^d-^wj^
,
i>j.
Equation (13.2.9) implies that if i > j , then Wi = (d + Vi-Jwi-id-1
vj+1)wj+1d-i+j+1
= --- = {d + Vi-i) •••{d +
whence Vj+Jwj+tf-'+^d*-*-1^
( W W ) « = res a (9 + Ui-i) • • • (0 + — res 9 (d + Vi-i) •••(d + vj+i) — 0.
Thus, W W = I, the matrix unity, and W = W _ 1 .
D
13.3.4. Definition. If M is a matrix, then M + is a matrix such that (M+)i.
{
My ,
when i > j
0,
when i < j
and M _ = M - M + . It is easy to see that (M+)i = (Mid-i)+di,
(M_)i = (Mid-1)^
.
(13.3.5)
Let A be the matrix A^- = <5i,j-i13.3.6. Lemma. If M = (My) is a matrix and Mj is an operator associated with its ith row, then an operator associated with the ith row of the matrix MA is Mjd. Proof. (MA)* = ^ ( M A ) i ^ = ^Mij-xtV 3
3
= ^ M i j " - ! ^ - 1 ^ = Mid-
a
3
The matrix A can be dressed with the help of W : L = W A W " 1 . 13.3.7. Proposition. true:
By virtue of the mKP equations, the following is 9 fc W = - ( L f c ) _ W .
(13.3.8)
222
Soliton Equations and Hamiltonian
Systems
Proof. dkWi = dkWidl = (dkW • W-%
-(L^-Wid*,
Tesd{-(Lki)_widid-i-lw-l1)
=
= -resd(Lk)_(d
+ Vi-i) •••(8 +
vj+1).
On the other hand, (L%
= (WAfcW-%- =
vesaCWA^iiW-y
= resawididkd-j-1wj^1
=
resdLkwidi-:>-1w-_?;1
= res 9 Lk(d + Uj_i) • • • (d + vj+i) = resd(Lk)_(d
+ UJ_I) • • - (d + vj+1).
•
13.3.9. Corollary. The mKP equations (13.2.4) imply dkL=[(Lk)+,L}-
(13.3.10)
Proof. dfcL = ^ ( W A W " 1 ) = - ( L ^ - W A W - 1 + W A W - 1 ( L f c ) _ W W _ 1 = -[(L f e )_,L] = [(L f c ) + ,L].
D
13.3.11. Definition. Equation (13.3.10) where L = A + (lower triangular) is the discrete KP. 13.3.12. Theorem. The discrete KP (13.3.10) is equivalent to the modified KP (13.2.4). Proof. Since this is already proven in one direction, it remains to show that each solution to (13.3.10) can be obtained from a solution to (13.2.4) in the above described way. Thus, given a matrix L = A + (lower triangular), satisfying (13.3.10). The matrix L can be represented in a form L = W A W - 1 . Lower triangular dressing matrices W with unities on the main diagonal are not uniquely determined; they can be multiplied on the right by a matrix commuting with A, i.e. constant on all diagonals. Using this freedom, it is always possible to satisfy (13.3.8) (there remains a little freedom even after this operation: W can be multiplied on the right by a constant matrix commuting with A, this can be fixed with the initial conditions). Thus, one can consider
KP, Modified KP, Constrained
KP, Discrete KP, and q-KP
223
(13.3.8) as a discrete KP equation. The variable t\ will be identified with x and d\ = d. Let W j = Wid1 be operators associated with rows of W and ( W - 1 ) ^ = d~i~1Wj+i operators associated with columns of W _ 1 . In the next three lemmas there will be assumed that W satisfies (13.3.8). 13.3.13. Lemma. There exist quantities Vi such that (d + Vi)il)i =
Wi+id.
Proof. Equation (13.3.8) for k = 1 reads d{W = - L W + L + W = - W A + (A -
V)W,
where V is a diagonal matrix. The operator associated with the ith row of the left-hand side is d{ibidl) = d • xbid1 — uiidd1. The operator associated with the ith row of the right-hand side can be found if Lemma 13.3.7 is taken into account and if one observes that (AW), = (W)i+i. Then it will be —xbid%d + Wi+id%+l — Vitbid1. Equating these two expressions, one gets (d + Vi)wi = wi+id. • 13.3.14. Lemma. Wj+i
= w7+\ .
Proof. We have ( W W _ 1 ) y = 0 when i > j , therefore 0 = resdWid^'J^Wj+i
= resa(d + Vi-i) • • • (d +
vj+i)wj+iwj+i.
This can be true for all i > j only if (wj+iii>j+i)- = 0. Then Wj+iWj+i = 1 and Wj+i = wj+iQ 13.3.15. Lemma. The operators iij satisfy KP. Proof. The (ij)th - ( W A f e W - 1 ) _ is
element of the matrix equality 9fcW • W _ 1
res 9 dh^Wi^d'3'1-!!)^
= - r e s 9 widkdl~:>~1w~^1,
=
i> j
or resd(dkWi • wT1 + Widkw^l)(d
+ v,_i) • • • (d + vj+i) = 0
for a given i and all j < i. This implies {d^Wi • wj1 + ibidkw~1)finally, dkWi • w~l = —(widk'w~1)- which is KP.
= 0 and, •
224
Soliton Equations
and Hamiltonian
Systems
This also completes the proof of the theorem since (d + Vi) = Wi+idwi and dkvi = -(L^1)_(d+vi)
13.4
+ (d+vi)(L^^(L^+1)+(d+vi)-(d+vi)(L^+.
•
g-KP
13.4.1. There exist two variants of the so-called (/-deformation of the KdV that can easily be generalized to (/-deformations of KP, one belonging to Frenkel [Fr] and another to Khesin et al. [KLR]. Following mainly the line of argumentation of [AvM99], we are trying to show that the q-KP is virtually the same as the discrete KP. We shall treat here the Frenkel type definitions for ^-hierarchies. Firstly, a simple analogy. Consider the difference equation (a): o„+i — an = 0. Its solutions are an = ao, where do remains an arbitrary constant, which is an initial condition. Now let us compare this equation with (b): u(y + h) — u(y) = 0. Is this a new equation? Formally, yes, it contains a continuous parameter. However, it is clear that this is a set of infinite number of independent equations (a), or, more precisely, a continuous direct sum of these equations. A solution depends on an arbitrary function, the initial condition: u(y + nh) = u(y), 0 < y < h. Nevertheless, Eq. (b) acquires a new quality if we try to change the step h, e.g., send it to zero, and watch what happens to a solution. A similar relationship is between the discrete KP (a) and the q-KP (b). According to the Frenkel type theory, the q-KP is the following. Let D be the operator acting on functions of a variable y, 0 < y < oo, as Df{y) = f(qy) where q > 1 is a fixed real number. Let Lq = D + u0(y) + u^D'1
+ u2(y)D-2
+ •••
(13.4.2)
be a formal g-pseudodifference operator. Equations of the q-KP hierarchy are denned in the usual way: dkLq = [(Lkq)+,Lq).
(13.4.3)
It is quite obvious that Eq. (13.4.3) involves values of all functions, uo(y), m(y),... only at points belonging to a sequence qny, n e Z with a fixed y.
KP, Modified KP, Constrained KP, Discrete KP, and q-KP
225
Now, we establish a correspondence between g-pseudodifference operators and infinite matrices we dealt with studying the discrete KP hierarchy. Let D correspond to A, a function f(y) correspond to a diagonal matrix with diagonal elements f(qny) at the nth place: D^A,
/(»)-> diag/(gny),
f(y)D ^ diag f(qny)A.
(13.4.4)
We have D o f(y) = f(qy)D -». diag f(qn+1y)A
= Adiag/(g"y),
where diag f(qn+1y) is a diagonal matrix with f(qn+1y) at the nth place. The last equality means that the above correspondence is an isomorphism. The correspondence (13.4.4) also respects the subscript + since in one case it means non-negative powers of D, in the other those of A. Now, Lq !->• L = A 4- diagu0(g™2/) + d i a g ^ ^ ^ A - 1 H
,
dkLq = [(Lkq)+,Lq\ -> 8kL = [(L fe ) + ,L] which is the discrete KP. Thus, the q-KP is a continuous direct sum of discrete KP on the interval 1 < y < q, for other y the solution is uniquely determined.
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Chapter 14
Another Chain of K P Hierarchies and Integrals Over Matrix Varieties
14.1
Introduction. More About the Modified K P
14.1.1. In the preceding chapter we studied the modified KP hierarchy which was a sequence, or a chain, of the KP operators Lm, or corresponding dressing operators (we change the notation wm by 4>m)
(14.1.2)
Variables
,
3kvm = (Lkm+1)+(d
+ vm) - (d + vm)(Lkm)+
where Lm = <j>md(j>^ , .
(14.1.3) (14.1.4)
There exist various reductions of this chain, e.g., modified GD, when
228
Soliton Equations and Hamiltonian
Systems
independent of m for m large enough. For any natural number n one can construct a stabilizing chain such that its stable limit belongs to the nth reduction of KP, i.e. to the nth GD hierarchy and, more than that, after some shift of variables satisfies the string equation of Chap. 7. We show that the stabilizing sequence of r-functions is identical with some integrals over matrix varieties known in the statistical physics and quantum field theory as the Kontsevich integrals. 14.1.5. Multiplying Eq. (14.1.4) by (d + v m ) _ 1 on the right (the latter expression is understood as a power series in 5 _ 1 ) and taking the residue we get an equivalent form of this equation: 3kvm = - r e s ( 5 + vm){Lkm)+{d + Vm)'1.
(14.1.6)
A semi-infinite solution, or a semi-infinite chain is a special solution such that (j>m = 1 and vm-\ = 0 when m < 0. Let Pm = c/)mdm. Then Po — 1, Eq. (14.1.2) implies (d + vm)Pm = Pm+i and therefore Pm is an mth order differential monic operator when m > 0. Then <j>m = 1 + Wmid-1 + • • • + Wmm.d~m'• (The matrix W of Sec. 13.3 is in this case a direct sum of two semi-infinite blocks, one is the unity and one is (W^) with i and j > 0.) We shall show that all the semi-infinite solutions are simply the well-known Wronskian (or soliton-type) solutions of KP. 14.1.7. Lemma. Let {(j)m,vm} be a solution of the semi-infinite chain of equations (14.1.2-4) with m = 0, 1, 2 , . . . and let Pm = 4>mdm. There exists a sequence of linearly independent functions yo, j / i , . . . such that Pmyi = 0 when i = 0 , . . . , m — 1 and (dk — dk)Vi = 0 for all k and i. Proof. If a function y belongs to the kernel of an operator Pm, then it also belongs to the kernels of all Pm> with m! > m since (d + vm-i)Pm-i = Pm. Suppose, yj's are already constructed for i < m — 1 (if m = 1, nothing is assumed). The kernel of the operator Pm is m-dimensional, therefore there is one more function ym-i linearly independent oi y0,..., ym-2 such that Pmym-i = 0. If a function y belongs to the kernel of Pm then so does (dk — dk)yIndeed, we have o = dk(Pmy) = (dkPm)y + Pm{dky) = -{mdk
= -<j>mdmdky + (Lkm)+Pmy
+
Pm(dky).
The middle term vanishes since Pmy = 0 and (Lkn)+ is a differential operator. Thus, Pm(dk - dk)y = 0, and (dk - dk)y is in Ker PN. Take ym-\ as
Another
Chain of KP Hierarchies and Integrals Over Matrix Varieties
229
y. We have, m—1
(dk-dk)ym-i
= Y^ AkiVi,
Au=0.
i=0
The coefficients Au% do not depend on t\ = x. Now, m— l l
k
(d - di)(d - dk)ym-i
= A fe , m _i ^2 Auyi - ^ i=0
= Ai,m-i
{diAki)yi,
i=0
m— l
(3* - dfc)(c>' - di)ym-i
m—l
m—l
Y, AkiVi - 5 Z i=0
{dkAu)vi.
i=0
Operators (dk — dk) and (dl — di) commute, hence m—l
m—l
] P (di + Aiim_i)Afci • j/i = ^ i=0
(9fe + Afe)Tn_i)i4ji • j/i
i=0
and, by virtue of the linear independence of j/j as functions of t\, (di + Aiim-i)Aki
= (dk + Ak,m-i)AH
.
(14.1.8)
This is the compatibility condition of the equations (dk + Ak,m-i)oti = Aki. We can find a*, with aOT_i = 1, and then m— l
(dk - dk)ym-i Let ym-i
= 5^o"_
= ^2(dk »=o
+ Ak,m-i)ai
• &.
a
iVi- I* i s easy to see that (dk - dk)ym-i
=
Ak,m-iym-i.
Now, if we solve the system dk4>m-\ =
(afc-afc)^_1
=
o.
•
230
Soliton Equations and Hamiltonian
Systems
14.1.9. Proposition. The general construction of solutions to the semiinfinite one-Toda hierarchy, or, equivalently, the chain (14.1.2-4) is the following: let {yi\ be a sequence of functions of variables t\ = x, *2J • • • having the property (dk — dk)yi = 0. Then
Vo
Vm-1
1
y'o
Vm-1
d
(m)
(m) Vm-1
Qrr
Vo
where Wm = W(yo, • • •, ym-i)
(14.1.10)
• d-
Wrr
is the Wronskian and
vm =
-dln(Wm+i/Wm).
Proof. One way. Let {(j)m,vm} be a solution of the semi-infinite chain of equations (14.1.2-4) with m = 0,1,2, — Let yo, y\,... be a sequence of functions given by the lemma. Then Pm = <j>mdm where
+ vm)Lkm = 0.
Applying the operator dk to (14.1.2), we have 0fct>m-0m-(0 + « m ) ( £ m ) - & n = - ( ^ + l ) - ^ m + l 9 = - ( i ^ + 1 ) - {d + Vm)(j)m
or [dkvm + (d + vm)(Lkn)+
- {Lkm+l)+{d + vm))
0.
Canceling
•
14.1.11. The property (dk - dk)yi = 0 implies Vi
z
2z2'h
3z3'
1—dj
yi(t1,t2,t3,...) (14.1.12)
Another
Chain of KP Hierarchies and Integrals Over Matrix
or yi(t — [z x]) = (1 — d/z)yi(t),
231
Varieties
using the common notation. Indeed,
/ oo
1 exp .Mr-tfcz*)- ^ u =
exPj3-(^fc)_1«fc
y%
fc=l
\fc=i
= exp In ( 1
d J 2/j = ( 1
0 J yt.
The Baker, or wave function is 4>{d) exp ^ ^ ° tk£k = w(z) expYlT where 2/o 1 U>m(z)
2/o
tkZk
2/m-l
•••
2/m-l
-m+1
w« (m)
(m)
2/o
1
Subtracting the second row divided by z from the first one, then the third divided by z from the second one etc., we obtain zeros in the last column except the last element which is 1. In the ith row there will be elements jf - z - 1 i 4 i + 1 ) = yf'it - [z- 1 ]), according to (14.1.12). Thus, wm{z)
Wm(t - [z-1]) Wm(t)
which means that Wm(t) is the r-function r m (i) corresponding to wm(z), and vm = - l n ( r m + i / T m ) . What are the functions yi(t) with the property (dk — dk)yi = 0? For example, exp^^°ifca fc , linear combinations of several such functions, and integrals / / ( a O e x P ( 2 i ° tkak)da, or the more general Stiltjes integral. Using the Schur polynomials pk(h, ti,...) defined by exp Y^f tk<xk = S o ° Pk(t)ak, we see that all the above examples have the form yi(t) — ^2cikPk(t)Conversely, any series of this form has the property (dk — dk)Vi = 0 since the Schur polynomials have it, as it is easy to see. Apparently, this is, in a sense, the most general form of such functions.
14.2
Stabilizing Chain
14.2.1. Definition. The stabilizing chain is a collection { ^ m , u m i n m + 1 } , m — 0, 1, 2 , . . . where 4>m(d) = 1 + wmld~l
+ •••+ wmmd-m
= Pm(d)d~
232
Soliton Equations and Hamiltonian Systems
and um, vm+i are "gluing" variables, subjected to the relations: (a)
(d + um) •(/>m = (d + vm+1) • <j>m+i,
(b)
dk(j>m = -(L^)-4>m,
where Lm = (pmdfa1,
(c)
dkum = -res((d + um){Lm)+(d
(d)
dkvm+1
+ um)
= -res((S + vm+i)(Lm+1)+(d
_1
^
),
+ vm+i)_1) •
Equations (14.2.2c) and (14.2.2d) can also be written in a different form: (c')
dkum = res Lm - res((d + um)Ll^(d
(d ; )
dkvm+i
+ um)_1),
=resL^+1-res((9 + vm+i)L^+1(5 + vm+i)_1),
where (0 + Um)Lkm(d
+ Um)-1
= (d + Vm+X)Lkm+1(d
+
Vm+x)-1
by virtue of (14.2.2a). It is easy to find that
= 0
by virtue of (14.2.2a). Using (c'), one obtains dk((d + Um)
+ um)-l))
.
233
Another Chain of KP Hierarchies and Integrals Over Matrix Varieties
The last term transforms as: -(d + um){Lkm)-4>m = -((d + um)L^)^4>m = -((d + um)Lkm{d + um)'\d
- res(L^)0 m
+ um))_
= ~((d + Um)Lkm{d + Urn)-1)-^
+ Um)(j)m k
- res(Z4)(/>m 4- res((d + um)L m{d + Um)'1)^
.
Then dk((d + um)
dk((d+vm+i)(f>m+i)
Now, taking into account (8 + Vm+1)Lkm+1(d
+ Vrn+i)-1
= (d + Um)Lkm{d
+ Urn)'1
,
we have dk{{d + um)4>m - (d + vm+1)(j)m) = - ( 9 + -u TO +i)L^ +1 (5 + v m + 1 ) - 1 ( ( a + um)m - (d + vm+i)(f>m). Thus, the property (14.2.2a) is preserved by the flow. Before we prove (3), the relations (14.2.2c) and (14.2.2d) will be presented in a different form. Equation (14.2.2a) implies (d + v m +i)^m+i,m+i = 0 where (d + t; m+ i)w> TO +i im+ i is understood as a result of the action of the operator (d + vm+i) on the function u> m +i ]Tn+ i (not a product!). Indeed, this is the coefficient in d~m~x of the expression on the right-hand side while the left-hand side does not contain this term. Thus, in' W
vm+i
=
77l+1,771+1 !
0 1
= -dlnu>m+iim+i.
(1 A ri
A\
(14.2.4)
^771+1,777+1
Subtracting (c') and (d1) we also have dk(um - vm+i) = res Lkm - res L ^ + 1 . These two equations are equivalent to (14.2.2c) and (14.2.2d). Proof of the commutativity of \dk\- They commute in their action on all <j)m as KP vector fields. Since the last two formulas express vm+\
Soliton Equations and Hamiltonian Systems
234
and differential polynomials in coefficients of <j)m, they also commute in their action on these gluing variables. This completes the proof of Proposition 14.2.3. • Notice also a useful formula for dkVm+iobtain lesdkfd_^\
-1
From Lemma 6.4.8 we can a(fe) a
where a is a function, and res P(0) •
(d-~
\_1 1
P(d)a a
(14.2.5)
for any polynomial P(d). Now, dkVm+i = - r e s ( ( d - i / 4 + l i m + 1 / i o m + i , m + i ) ( L ^ + 1 ) + x (d -
w'm+hm+1/wm+1,m+i)-x)
(d - w'm+1 ,7n+l/ ^ m + l , m + l
which can be written as OkVm+i = - ^
•
(14.2.6)
An alternative way to get (14.2.6) is the following. Equation (14.2.2b) implies dk
one
getg dkWm+l,m+l
= (Lm+i)+(wm+itm+l)
•
(The operator (Ljj, +1 ) + acts here on i u m + i i m + i . ) Then Eq. (14.2.6) easily follows from (14.2.4).
14.3
Solutions t o the Chain
14.3.1. Let yom,---,ym-im operator Pm = <j)mdm: Pmyim
be a basis of the kernel of the differential = 0.
Another
Chain of KP Hierarchies and Integrals Over Matrix
Varieties
235
14.3.2. Lemma. Passing if needed to linear combinations of yim with coefficients depending only on £2, *3, • • •, one can always achieve dkyim = dkyim
(14.3.3)
and a Vim = y'i,m+i,
i = 0,...,m-l.
(14.3.4)
Proof. Suppose yim, where i = 0, . . . , m — 1, are already constructed. Let 2/0,m+i, • • • ,j/ m ,m+i be a basis of the kernel of the operator Pm+i,m+i — 0. The functions yo,m+i> • • • '2/m,m+i which belong to the kernel of (d+um)Pm (since (d+um)Pmd = (d+vm+i)Pm+i) are linearly independent, otherwise there would be a linear combination of j/i, m +i belonging to the kernel of Pm+i which is constant (with respect to t\ = a:) while we know that P m + i l = u>m+i,m+i / 0 by assumption. Hence, at least one of these functions does not belong to ker Pm, let it be j/J„,m+i: Pmy'm,m+i 7^ 0. Since all Pmy'i m+i belong to the one-dimensional kernel of d + um, there must be constants a^ such that Pm(y'itm+i — «i2/m,m+i) = 0- (When we speak about constants, we mean constants with respect to t\ = x depending, maybe, on higher times.) Thus (yj, m +i — a iJ/m,m+i)' form a basis of the kernel of Pm. There exist linear combinations (yj „, + 1 )' coinciding with yim: (l/tm+i)' = Vim where t = 0 , . . . ,m - 1. This yields (dk - dk)dy^m+l =0 k c (by the assumption of induction) and (d — dk)y\ ^+1 ~ ki = const. As in Lemma 14.1.7, we can prove that (dk — dk)yim+i G Ker -Pm+i> therefore c^ = 0 since constants do not belong to the kernel. It remains to consider Vm,m+i- Since (dk - dk)ym,m+i = Y,Aiyi,m+i, * n e same reasoning as in Lemma 14.1.7 will do the rest. D 14.3.5. Proposition. All solutions to the chain (14.2.2) have the following structure. Let yim where m = 1, 2 , . . . and i = 0 , . . . , m — 1 be arbitrary functions of variables t\ = x, fo,... satisfying the relations dkyim =
dkyim
and Vim = y'i,m+i,
i = 0,...,m-l.
a T h e relation (14.3.4) is what makes this chain different from the one considered in the previous section (modified KP) where it was ytm = Vi,m+i instead.
Soliton Equations and Hamiltonian
236
Systems
Let 0o = 1 and W0 = 1. Then
VOn
2/m—l,m
-l
y'm-hm
d
•d~m,
W„ (m) Vm-l,;
(m)
Tn = l , 2 , . . . ,
(14.3.6)
&°
where W m = Wm(yom> • • • >!/m-i,m) is the Wronskian. Besides, Van -din-
Vm—lm
Vm,',
Vm~\
Vr,
771 = 0 , 1 , 2 , . . .
W„ (m)
(m)
(m) (14.3.7)
where y m , m = yL m+i by definition and
iWi =
-d\n
VOn
Vm~ l,m
ym,m
VOn
^m—l,m
y?n,m
(m) 2/Orre
,,("»)
W, m+\
m = 0,l,2,
(m)
(14.3.8) Proof. Let { 0 m • u T O ,^ m + i} be a solution of the chain (14.2.2). Find ^ m according to Lemma 14.3.2. Then the operator Pm =
= -{Lkm)+<}>mdmyim .
237
Another Chain of KP Hierarchies and Integrals Over Matrix Varieties
The operator {Lkn)-
14.4
Solutions in the Form of Series in Schur Polynomials. Stabilization
14.4.1. Recall that the Schur polynomial are defined by the equation
(
oo
\
oo
53tkZ,k I = YlPk{t)zk , pfe = 0 when k < 0 . fc=i / fc=o A grading can be introduced: the variable U has the weight i, z has the weight — 1. Then the polynomial Pk(t) is of weight k. It is easy to verify that the Schur polynomials have properties diPk = Pk-i = d% , Pk{t)-CXdpk{t) = pkih-C'/h • • •, U-C/r, • • •) which can be obtained from the definition. Let OO
Vim = 5 3 4Pk+m-i,
i = 0, . . . , m - 1
(14.4.2)
fc=o with some coefficients c£ being CQ = 1. Then Eqs. (14.3.3) and (14.3.4) hold and Eqs. (14.3.6-8) define solutions for any sets of coefficients c£'. In
238
Soliton Equations and Hamiltonian
Systems
particular, up to a non-important sign, 1
(m VOm (m Vim
2/0m
1
I'm —
Vim
(m-1) "m—l,m
(14.4.3)
ym—l,m y
Equation (14.4.2) implies
= £
7"m —
koy...,km
—\
-i)
c (0) c ko
Pko
Pko+m-l
Pki-1
Pki+m-2
- l
Pkm-i-(rn-
-1)
(14.4.4)
Pfcm-l
14.4.5. Proposition (Itzykson and Zuber [IZ]). Tau functions (14.4.4) have the stabilization property: terms of weight I do not depend on m when m> I. Proof. The diagonal terms of the determinant are pk0, Pklt- • • ,Pkm-iAll the terms of the determinant are of equal weights, namely, ko + ki + ••• + fcm-i = I- Let us consider a determinant of weight / and prove that all ki with i > I vanish unless the determinant vanishes. Suppose that there is some i > I such that ki ^ 0. The elements of determinant which are located in the ith column above pkt have the following subscripts: fco + i, fci + i — 1 , . . . , ki-i + 1. Together with ki, there are i + 1 positive integers with a sum fc0 + fa + • • • + fa-i + fa + J2]=i 3^ I + S ] = i 3 < i + 1 + J2)=i 3 = S f c i 3> i-e- l e s s t n a n t n e s u m °f t n e n r s t * + 1 integers. This implies that at least two of them coincide. Then the corresponding rows coincide, and the determinant vanishes. Thus, if a determinant does not vanish, then, starting from the Zth row, all the diagonal elements are equal po = 1, and all the elements to the left of the diagonal vanish. The determinant of weight / reduces to a minor of Zth order in the upper left corner, and the terms of weight / are Pko r(°)
1
. .. J'" )
Pko+l-1 Pki+l-2
Pki-1 Ph-i
that does not depend on m.
Pk,-x-(l-l)
•
Another
Chain of KP Hierarchies and Integrals Over Matrix
239
Varieties
The stable limit of r m will be called r, r = lim r m . In the next section of this chapter we closely follow the work by Itzykson and Zuber [IZ] and, partly, by Adler and Moerbeke [AvM92]. Maybe, the word "follow" is not quite exact here, since we rather go in the opposite direction. While they start with the Kontsevich integral and arrive at the chain (14.4.4) with some additional properties, we show how, imposing some requirements, to pass from (14.4.4) to the Kontsevich integral that emerges only at the very end (v kontse which is Russian for "at the end").
14.5
From the Stabilizing Chain to the Kontsevich Integral
14.5.1. Proposition (Itzykson and Zuber). periodic:
If the coefficients eg' are
4 j + n ) = <£> ,
(14.5.2)
then the stable limit T does not depend on the variable tn. Proof. Let us discuss terms of weight I taking m big enough, e.g., m > l+n. Differentiate them with respect to tn. One determinant transforms under the action of dn into a sum of determinants where the subscripts of the ith row are diminished b y n , i = 1,... ,m. Compare the differentiated ith row with the (i + n)th one: Pfei-2, •••
Pki+n-2,
Pkt-i,
Phi,
Pki+n-l,
Pki+n,
••••
The determinant is antisymmetric with respect to the indices fcj and fcj+n while the coefficients in front of the determinant .
JOJi+n) ft,!
A>t-j-71
_...J*)„(0 ft-!
A.t-|_n
are symmetric. Therefore, the terms cancel pairwisely.
•
Thus, the stable limit of solutions belongs to the nth GD hierarchy. In what follows, we are always assuming that Eq. (14.5.2) holds.
240
Soliton Equations and Hamiltonian
Systems
14.5.3. Looking at the determinant Pko
Pk0 + 1
Pfci-i
Pkx
Pfcm_i-(m-l)
Pk0+m-l Pfci+m-2
Pkm-i-(m-2)
(14.5.4)
Pfem-l
involved in the definition of the r-function, one can recognize (see, e.g., Weyl [Weyl39]) primitive characters of the group GL(m) or U(m) where
U
EJ fc
(14.5.5)
e
and efc are the eigenvalues of a matrix for which this character is evaluated; it is assumed that ko > k\ > • • • > km-i- The latter can always be achieved by a permutation and relabeling of indices. It is not easy to understand why r-functions happen to be related to characters. However, we can extract lessons from this relation. First of all, the r-function is given as an expansion in a series in characters. Hence it can be considered as a function on the unitary or the general linear group invariant with respect to the conjugation. In the end we will have an explicit formula giving this function which is called the Kontsevich integral. Secondly, the benefit of the usage of variables tk instead of U is obvious: the elaborated techniques of the theory of characters can be applied to the r-function as well. We shall also use an inverse matrix with the eigenvalues Aj = e~x. Thus, we have (14.5.6) This change of variables is often called the Miwa transformation. We have oo
oo
oo
m
..
Y^Pi{t)zl = e x p ^ i i Z * = exp^J2jekz' =1
i=l
fc=i
= ex P
ii,...,im
i.e.
pt(t) = llH
E birr
X>(l-e f c z) fc=i
fc=l
MM e
l
e
2
• • e
r
Another
Chain of KP Hierarchies and Integrals Over Matrix
241
Varieties
This is the Newton formula expressing complete symmetric functions of variables ek in terms of sums of their powers, tj. The determinant (14.5.4) is a coefficient in Xo°+m~1x'l1+m-2 • • • x^Zi in the generator (the summation is over all integer fco,..., fcm-i, not only non-negative):
E
ko+m-l x
fci+m-2 x
0
l
fcm-i x
"
m-l
fco,--,fem-l
Pk0
Pfco + 1
Pko+m-1
Pfcl-1
Pfel
Pk^+m-2
Pfcm_!-(m-l)
Pfcm_i-(m-2)
Pfc m -i
E X 0Pi
E /_^Xm-lPl
' xm-l
Z_^xm-lPl
x
'
Z_jXm
m-l
Y[-^—\xm-\xm-\...,x0\, where
xm~1
xm~2
.,x°\
=
™m-2 0
XQ
1
x1
x1
1
x
x
1
x
m-l
m-l
Furthermore, we use the Cauchy formula \em-\
em~2,..., e°| • \xm-1,xm-2, Ui,kO- - ekXi)
...,x°\
det
~"\-tkXi
The right-hand side has an expansion in a;,:
E
k0,.-.,k„
ko+m-l x
0
ki+m-2 x
l
fem-i
i
x
fc
' ' ' ro-l l
fc0+m—1 '
fci+m-2 t
,...,€
l
X\PI
-iPl
242
Soliton Equations
and Hamiltonian
Systems
from where T Tm
~
Vw 2
C C
r ,N (m—1)
(0)
ko
,
2_>
,.
n{m—l)
c
k0
c
km-i
fco,...,fcm-i
,fc x +m-2 ,fcm-il ic i•••) c 1
|e"»-1,em-2,...,e°|
fcm-i
ko,---,km~l
=
\fko+m-l lc
Ic^ o fki-l c
|
;
j - • • j
c
,fcm_i-(m-l)| |
m
1
|e°,c-i)...,e-( - )|
In t e r m s of Afc this is T
_
y-
c(o)...
C
km-i
|\0 \
fco,...,fem-l
im-ll
>>•••>
I
Finally, letting /-(A) = £ £ ° c^A"™, we get ]/o(A),/ 1 (A)A,/ 2 (A)A 2 ,...,/ m _ 1 (A)A'"- 1 |
„. Tm
=
|l,A,A>>...,A-i|
(14
•
-5-7)
14.5.8. Up to this point there were no restrictions imposed on the coefficients c£ except the periodicity (14.5.2). Now we try to satisfy the string equation. It is equivalent to the fact that r does not depend on the additional variable i l n + 1 j . The action of dln+11 on a r-function is given by
(Sec. 7.2.3), and, according to Eq. (6.5.8), this is
1I
d-n+i,i T = -
X ! ij'*i<J
+ 2
2 ~ ^ + n)U+ndi + (n- l)ntn
\ r.
Notice that the last, linear in time variables, term is not essential. Indeed, the function w(z) is defined up to a multiplication by a constant series in z. In terms of tau functions, this corresponds to a multiplications by exp ^2 ciU- The coefficients c» do not depend on time variables but they can depend on additional variable t*_n+11. Therefore, <9I n+11 T is determined up to a term linear in U times T. Using this remark, we write
\W%(T)
= ^ ( £ W +2 f>+ n )*<+« a *) r • yi+j=re
»=1
J
Another
Chain of KP Hierarchies and Integrals Over Matrix
Varieties
243
One must express this in terms of new variables, A's or e's. Equation (14.5.6) implies d
_
^
d\k
1 x
_L d ldX
^xt
1 +
v
K- »
Y4 " *
and
k
K
i
Now, up to non-important terms, (2) =
y> jj>0;i+j=n
V —V—-2V — T
r
S
s
k
—
k
We also can shift the variables: U = U + a* and pi(ii) = pi. More than that, the tau function can satisfy the string equation only after the shift which is determined by this condition uniquely (see below Sec. 14.5.12). We have oo
oo
oo
i
oo
izi
^TpiZ* = e x p ^ ( t j + ai)z = Y^P i=0
i=l
-exp^ajZ*.
j=0
(14.5.9)
i=l
Repeating the above calculations in new variables ii we proceed up to the determinant | Y^ xlPr xN~1 > Yl xlPl' = l ^ a ^ W • xN-\YxlPi
xN 2
• •' H xlPl |
~ >'
• xN~2> • • • ' H ^ l J ! e x p ^ a ^ a r f . i
Now, we have the expansion in powers of Xi of det(l — efcZi)-1 exp ^ . ajX?, instead of det(l — e^a;,) -1 . 14.5.10. Lemma. If Y^ bjXj is a formal series in x, then oo
/
oo
where the subscript "+" symbolizes positive powers of e. Proof. Indeed, both sides of this equality are equal to J ] ^ 0 X l L o bjt^x1. •
244
Soliton Equations and Hamiltonian
Systems
Applying this lemma, we have to get the expansion of the expression (det(l — CkXi)-1 expY^j ajekJ)+ w n e r e the subscript "+" refers to all efc. In the end, instead of (14.5.7), we have (l/o(A),/i(A)A,/ 2 (A)A 2 ,... Jm-iW^-'l 7"m —
n*exp£7. a ^ ) < m _ i
|1,A,A2,...,A—1|
(14.5.11) where the subscript "< m — 1" means taking powers of A, not larger than m — 1 (the numerator and the denominator were multiplied by fj A™ -1 ). Now, let the operator W_l act on r m . If the numerator of the expression (14.5.11) is Am, then denoting
^=W{^Am-l[(Xr-\s)-1, we get -2
V k
A
OAk
k
r>s
J r>s
v^n^-A-r x-(n-l)
T + 2AmJ2^ \\ =J 8
\~(n-l)
— Ar )A s A r
n^-^) -1 A,)" 1
r>s \i+j=n-2
r^s i,j>0;i+j=n
^r^K J
n /Xs
r
r s
r>s
i
d
( n - ^ En i - ^ A-^i n-l T X
r>s
X
8Xk
Am • 1[(K - Kr1.
u x d/dXkX {1 x -(n-l)/2. The operator in the brackets is — 2 £ f e A-(n-l)/2 "zd/d\kX ""\ the fcth fc ' k ' term of this sum acts on the fcth row of the determinant Am. Now, the
Another
Chain of KP Hierarchies and Integrals Over Matrix Varieties
245
string equation has the form
E
jk,--((nn--ll) / 2
A
8 0
.-(„_l)/2 A
Xk * 7nT dr k
*
(l[exP^2ajXi\f0(X),f1(X)X,f2(X)X2,...,fm-1(X)Xm-1\\ i
^
=0
i
'<m-l
or
E .-(n-l)/2 x
3
(„_i)/2
n « P Z ) a i A i • 1/oW' /i(A>A- /2(A)^2. • • • ,/m-i(A)A"- 1 | I
J
<m— 1 — n
Let
Dk =exp f - E ^ ) ^ M ) / , ^ ( " " 1 ) / 2 o t P E M i E^'Af" j
n- 1 1 2 A» + ^
A
1 9 n-iaA
Afe" -O-Afc
The string equation takes the form neE'a'A*£l>fe|/o(A),/i(A)A)/2(A)A2
,/m-i(A)Am-1| 1
=0
/ <m—1 —n
which is equivalent to (n
e E
'
a i A
'(
E^I/o(A),/i(A)A,/2(A)A2,
.,/m_1(A)Am-1|l
I
=0.
/ <m — 1 — n / <m —1 —n
=0
246
Soliton Equations and Hamiltonian
Systems
The operator Dk acts on the fcth row of the determinant. Finally, we can write ' m—1
J J e E , . ^ | J2 |/o(A),/1(A)A,/2(A)A2,...,DMA)Ai, 1=0
..../m-iCAJA™-1!!
J
=0.
(14.5.12)
/ <m —1 —n/ <m — 1—n
The last term, with / = m — 1, vanishes. Indeed, we consider stabilization when m —> oo. For the terms of a given weight p all /( with I > p must be considered as 1 since for these terms only those clk with I > p are relevant that have k = 0, see Sec. 14.4.5. Now, DA™"1 = ( m - l - ( n - l ) / 2 ) A m - 1 - " + J2 , a j j A ™ _ 1 + J . The first term vanishes because it is proportional to the entries of the column m — 1 — n. The second vanishes since it is of more than m — 1 — n degree. 14.5.13. Now we try to satisfy Eq. (14.5.12). It holds, for example, if the determinant |/o(A), /i(A)A, /2(A)A 2 ,..., / m - i ( A ) A m _ 1 | can be written as |/ 0 (A),D/o(A),£) 2 /o(A),...,£) m - 1 /o(A)|. Indeed, all the terms on the left-hand side of (14.5.12) except the last one vanish as determinants with two equal columns. The last term is also zero, see above. What does this requirement mean? First of all, *kfk = Dkf0, fc = 0 , l , . . . , n - l .
(14.5.14)
This determines fk for k = 1 , . . . , n — 1 in terms of /o. Then we recall the periodicity (14.5.2), or /» = /j+ n . This determines all / j . There must be Dnfo = A n /„ = A"/o, i.e. the series /o in A - 1 has to satisfy Dnfo = \nfo.
(14.5.15)
This suffices since it is easy to prove by induction that all Xhfk can be represented as Dkf0 + S o ™ "i -^'/o which permits to replace all Xkfk by
Dkh.
Equation (14.5.15) also determines the shift {aj}. The equation
^ =!
/
which must hold identically in A can be satisfied by some /o = X^o° cm A _ m if and only if £ . ajjV~n = A, i.e. £V a,j\i = A n + 1 / ( n + 1).
Another
Chain of KP Hierarchies and Integrals Over Matrix
247
Varieties
Thus, the functions (series) fk are, finally, found. They are determined by Eqs. (14.5.14) and (14.5.15) where „
x
n - 1 1 1 d n 2 X A^dA
and the r-function is (14.5.7). It is possible to perform some scaling transformation A H-» aX, D i-> a~xD. This is not so important, but just in order that our formulas coincide with those in [AvM92] we take a = n1^n+1'>. Then „ , n - 1 I d n + 71 1 D = X + 2nX -—nX - dX = A'"" 1 )/ 2 exp ( —Xn+1) 4 - exp (-^-Xn+1) *\ n+1 J dXn y\n + l
A"*"" 1 )' 2 . J (14.5.16)
The function go = exp(nA n + 1 /(n + l ) ) A _ ( n _ 1 ^ 2 / o satisfies the equation d , 9o = Xng ; (14.5.17) dxn 0 which, being written in the variable \i = A™, is a generalization of the Airy equation (the latter is a special case with n = 2). A solution can be found by the Laplace method. The solution is 9o=
exp I Xnm
— j dm,
where the path of integration should be chosen such that the integral converges. Indeed, I Qo = I m " e x P I Xnm ) dm. v dX ) y o J \ n + lj Integrating by parts (m™ exp(—m n+1 /(n + 1)) = -d/dmexp(—mn+1/(n 1))), one obtains the desired equation. Now, TTT n
/„ . ^ - ^ ( - - ^ A - ) /exp (AN»- = £ ) *.. It is easy to see that Dkf0 = A^-^exp (--JL-\"+A
fmkexp (xnm - ^ ^ ) dm.
+
248
Soliton Equations and Hamiltonian
Systems
For the r-function we use Eq. (14.5.7) and not (14.5.12), i.e. it will satisfy the string equation only after the shift of variables i n + 1 H-> f„ + 1 + l / ( n 4-1) and not in its original form. Tm = I I 4 " - > / ! exp ( - £
^ A J " ) / • • • / * » • • • • dm„
Let us rewrite this integral changing variables of integrationTO*,n- imk where i = y/—\:
rm = « « ( ^ > / » n A i - 1 ) / a e x p T i
.
TT
( - E ^ T ^
+ 1
m
V ^ /" ^ ™
r
_
ms
) (*"1fc)" + 1 \
/
•••ydm1...dmmn-x—^-exp^KAfcmfc-^T1-j. r>s
fc
(14.5.18) We do not discuss here the problems of the choice of contours of integration and the convergence. 14.5.19. In a sense, the problem is already solved, the solution is explicitly written. However, we remember the remark in Sec. 14.5.3 that it is natural to consider a r-function as a function on a matrix space invariant under conjugation, \k being eigenvalues of a matrix, since originally it was written as a series in characters of the group. If we want to restrict ourselves to real values of the time variables U, i.e. real Afc, then the function is restricted to matrices with real eigenvalues, e.g. hermitian. Our intention now is to write the formula (14.5.18) more directly in terms of matrices themselves, not of their eigenvalues. The theory of representations has a tool for that, the Harish-Chandra formula ([H-Ch]). Let &{X) be a function of hermitian matrices X, invariant under conjugations, i.e. depending only on eigenvalues of X. We consider fhm $(X)exp(—itr XY)dX, integrals over the space of hermitian matrices (h.m.), where Y is a hermitian matrix with eigenvalues Hk, and dX = \\i-dXij. If X = UMU~l where U is unitary and M — diag m i , . . . , r a j v then the function can be partly integrated, with
Another
Chain of KP Hierarchies and Integrals Over Matrix
249
Varieties
respect to the "angular" variables U and only integral over diagonal matrices M remains: (
${X)exp(itrXY)dX
«/h.m.
= ( 2 7 r i ) m ( m - 1 ) / 2 f • • • f dmi • • • dmm J
r>s
x———$(M)exp(iy^rrikfik] / i r - fj,s
\ t—'
Direct application of this formula with Hk exp ti(-(iX)n+1/(n + l)) yields
rm = ( 2 7 r ) - ( - D / 2 i - n 4 n " 1 ) / 2 e x p
X
Uf^f
J\
J
J
=
•
^JJ and &(X)
f-E^T^
+ 1
=
)
exp tr (W> - ffl^ )dX
r>3
*£K-x.
A.m.
V
n+i ;
. <*«)-<->/• ini,-1"- • n f ^ f «p (-* ^ A n + 1 ) /
exp tr f XAn
/
Xn+1
\
J dX ,
where A is a diagonal matrix with eigenvalues Afc, and Z = X — A, then
tr
(XY
- — - _!L_y(«+i)/n ra + 1 n+1 - t r [ ( Z + A ) n + 1 - (n + l)ZA n - A n + 1 ] .
There is an expression (Z + A ) n + 1 in the brackets minus its terms of the zero and of the first degree in Z, i.e. the nonlinear terms, which will be symbolized by n.l.(2T+A) n+1 . The integral transforms to / exp(tr(—n.l.(Z+ A ) n + 1 / ( n + l))dZ.
250
Soliton Equations and Hamiltonian
Systems
If only the lowest nonlinear terms, i.e. quadratic, are preserved, then the integral is /
exp(tr(-quad.(Z + A)n+1/(n
/ \n
2 -1/2 \n \ ~V
—n &^y r>s
N
+ \))dZ)
''
-n* —
-(n-l)/2
fe
This expression can be recognized as an inverse coefficient in r m . Now the final formula appears: Tm
"
C nSt
°
/ exp(tr(-n.l.(Z + A ) " + 7 ( n + l)))dZ / e x p ( t r ( - q u a d . ( Z + A)"+V(n + l)))dZ '
l
j
We repeat that subtle problems of convergence of integrals, in what sense they must be understood were just ignored. The expression (14.5.20) is called the Kontsevich integral (more precisely, its generalization from n = 2 to any n).
Chapter 15
Transformational Properties of a Differential Operator under Diffeomorphisms and Classical W-Algebras 15.1
Tensors with Respect to Diffeomorphisms and the A G D - Algebra
15.1.1. A classical W n -algebra is just another name of the second Poisson structure (AGD-structure) (see Sec. 3.1.5) generated by an operator L = Y^o Ukdn~k (following the tradition formed in the W-algebra theory, we relabeled coefficients of the operator: we write Uk where earlier we wrote un_fc), with the reduction u\ = 0. This theory differs from what we discussed earlier not by an object of the study but rather by its goals and specific problems. The main point is the possibility of an extension of the Virasoro algebra provided by the second Poisson structure. We have already seen that taking Hamiltonians w™ = / xm+1udx we obtain the Virasoro algebra. The first, and the most important, problem of the theory is the choice of proper generators. The choice of generators 'is dictated by the conformal theory. This theory is interested in transformational properties of field variables under diffeomorphisms. Generalizing Virasoro, it is natural to start with Fourier coefficients of all Uk (Virasoro relates to k = 2). However, transformational properties of these variables are very complicated, and they are inconvenient, except k = 2. It happens that there are linear combinations of Ui and their derivatives with coefficients that are differential polynomials of U2, having simple transformational properties, in fact, they are tensors with respect to the infinite-dimensional group of diffeomorphisms. They (or their Fourier coefficients) generate a Wn algebra, with respect to the same second Poisson bracket. The forthcoming exposition is based mainly on the works by Di Francesco, Itzykson and Zuber [DiFIZ], Radul [Rad91], and Bakas [Bak89, 90]. 251
252
Soliton Equations and Hamiltonian
Systems
The author allows himself a personal remark. When we wrote with I. M. Gelfand our first paper on integrable systems [GD75] we were surprised to find in an article by Lazutkin and Pankratova [LP] the same equation (3.7.2) on which we based all our constructions. Meanwhile, they dealt with a quite different problem: they studied transformational properties of a second-order linear differential operator under diffeomorphisms. The reason of this coincidence was not clear. Now I see that the link between two problems is established by the fact that the vector fields of infinitesimal diffeomorphisms are Hamiltonian in the second structure. However, this fact, maybe, also needs an explanation. 15.1.2. Let Lx = dn + u2dn~2 + • • • + un be an operator where d = dx = d/dx, and u/. = Ufc(x). Let x = x(t) be a smooth change of variables. How does this operator transform under a diffeomorphism x = x(t)? We have Lt = {^dtT
+ u2{x{t)){
= rndt - ^ - V - W -
1
+ •••
+ • • •,
where (/> = dx/dt and <$> = d<j)/dt. The subscript t in Lt, as well as later in J/t) Vx etc. is just a label, it does not symbolize a partial derivative. Two properties of the operator are lost: the absence of the term with n x d ~ and the coefficient 1 in dn. In order to preserve these properties, a more sophisticated transformation must be performed
Lx^Lt = ^"+1)/2[(r1at)n + ^ ( i w j r ^ ) " - ' + • • #("-1'/2. (15.1.3) This is a group action of the group of the diffeomorphisms, or a representation of this group in the space of all operators of the above form. If we pass from a variable x to t and then from t to s, the result will be the same as passing directly from x to s. We can also express this differently. We consider all possible independent variables with smooth transition from one to the other, and operators given for all variables, Lx,Lt,Ls,... connected with each other by the formula (15.1.3). Then we say that a covariant operator is given. Here is the complete analogy with the tensor analysis where a quantity (tensor) is given in all possible frames simultaneously with definite rules
Transformational
Properties of a Differential
Operator
253
of transition from one to the other. Therefore, a choice of an independent variable will be called a choice of a frame. The next problem: how do the solutions y to the equation Ly = 0 transform under the action of a diffeomorphism? If yx(x) is a solution to Lxyx = 0 then, evidently, yt(t) =
yx(x(t))d>-{n-1)/2
is a solution to Ltyt = 0. With some abuse of notations we will omit here the subscripts t and x just writing y(t) = y(x(t))(f>~^n''1^2, or y(t) = y(x)0-< n -W 2 . If for every variable a function is given and they are connected by the formula y(t) = y(x(t))<j>h
(15.1.4)
we say that a primary field of conformal weight h is given. They also call it a /i-differential since this is the way the formal differentials y(x)dxh transform under diffeomorphisms. The space of all /i-differentials is denoted by T^,. There are also other names for this object, e.g., conformal tensors. It is obvious that above defined covariant operators transform J--(n-i)/2 r into J (n+i)/2It is not difficult to find the law of transformation of the coefficient u2 of a covariant operator: u2(t) = u2(x)<j>2 + cnS{x,t),
(15.1.5)
where cn — n(n2 - 1)/12 and S(x,t) = x - X 2 / 2 , X — H
= l(x)4> + X,
X = M
(15-1-6)
15.1.7. Lemma. A quasi primary field / of weight 2 can be represented as / = — (l/2)7 2 + 7' where 7 is a quasi primary field of weight 1. Proof. First we solve the equation / = —(l/2)7 2 + 7 ' with respect to 7 in one frame, then define 7 by Eq. (15.1.6) in other frames; it will be a quasi
254
Soliton Equations and Hamiltonian
Systems
primary field of weight 1. It remains to prove that / i = —(l/2)7 2 + 7' is a quasi primary field of weight 2; then its coincidence with the given / in one frame guarantees that in all the frames. We have fi(t) = — (l/2)j2(t) + 7(i) = - ( 1 / 2 ) ( 7 W ^ + X ) 2 + 7 ' W ^ + 7 W ^ + X = / i ( * ) ^ + ( * - ( l / 2 ) x 2 ) =
h{x)
•
15.1.8. Lemma. The operator d — h-y maps the space Th on Th+iProof. Let / € Th and h = (d - h-y)f. We have fi(t) = (
a The operator V = d — /17 is called the covariant derivative. We deliberately do not label this operator by the index h, thus, it denotes different operators depending where they act. E.g., if / € Th then V2f = (d— (h + l)^) (d - h^f. 15.1.9. Lemma. The covariant derivative has the Leibniz property
v{hh) = {vh)-h + h-{vh). Proof. Let /1 e Th and h £ Tk- Then / i / 2 € ^,,+fc and £>(/i/ 2 ) = ( 0 - ( h + * b ) ( / i / 2 ) = ( 9 - / i 7 ) / i - / a + / i - ( 3 - * 7 ) / 2 = (P!i)-h + !i-CDh). D 15.1.10. Corollary. Let 7/ be a primary field of conformal weight 9, and let p be an arbitrary number. Then Vori = r)V+ {Vrj) where the left- and the right-hand sides are understood as operators Tp —• Tp+q+l.
15.1.11. Now, we discuss infinitesimal diffeomorphisms. Let a diffeomorphism be close to identity: x(i) = t + er(t) where e is a small parameter. Then
= y(t) + e (^-^Zlfy(t)
+ r(t)i)
+
0(e2),
i.e. the infinitesimal action of diffeomorphisms in ^ r _(„_ 1 )/ 2 is Py =
y(t).
(15.1.12)
Transformational
Properties of a Differential
255
Operator
Correspondingly, the operator L undergoes some infinitesimal deformation. It can be found just from Eq. (15.1.3) expanding it in e. However, we shall do it differently, in a more general context following [Rad91]. Let y have an infinitesimal change y + tPy where P is an arbitrary linear differential operator. The problem is to find an infinitesimal change of an operator L: L + eVr.(P) (where VT.(P), or simpler, V(P), is a differential operator of an order less than n) such that any solution y of the equation Ly = 0 after deformation goes to a solution of the deformed equation: (L + eV(P))(y + ePy) = 0(e 2 ). This implies (V{P) + LP)y = 0. Among solutions of this equation are all the solutions of Ly = 0. This means that the operator V(P)+LP is divisible by L on the right, i.e. can be represented as V(P) + LP = QL where Q is a differential operator. Now,
V(P) =
QL-LP.
This is a formula more general than the Lax representation we dealt in the first chapter (the latter corresponds to the case Q = P). Multiplying by L " 1 we have V(P) • L~l = Q - LPL~l. Taking the positive (differential) part of this equality, we get Q = (LPL~1)+ and V(P) = -LP + (LPL~l)+L
= -{LPL~-l)_L.
(15.1.13)
This formula gives a mapping of infinitesimal deformation of functions Py to deformations V(P) of the operator L, i.e. to the objects we called in Chap. 1 "vector fields" and denoted by dy(p)What deformations of the solutions do not change the operator L? For that, the operator P must transform solutions of the equation Ly = 0 to the solutions of the same equation: LPy = 0. This yields LP = QL where Q is a differential operator, i.e. (LPL"1)- = 0. From the formula for V(P) we see that such P form the kernel of the mapping P ^ dy(p), indeed. (Two extreme cases of the operator P € ker(P i-» dy(p))'- it preserves all the solutions, then P = const, or it destroys all the solutions, then P = QL.) Denoting [Pi, P2](i) = - [ P l . f t ] + dv{Pl)P2
- dy ( P 2 )Pi ,
one can prove [dy(Pi)>dv(p2)]
=<9
V([Pi,P2])
(check it!) which implies that the space of all differential operators P modulo ker(P >-> dv(P)) equipped with the bracket [Pi,P2](i) becomes a Lie algebra. We call it the Radul algebra 71.
256
Soliton Equations and Hamiltonian
Systems
15.1.15. Lemma. The Adler mapping X i-» d(Lx)+L-L(XL)+ *s a composition of a mapping X e R-/d~nRH-> Px = (XL)+ € 7£ and a mapping Px G 7£ M- ^v(p x ) € V. All these mappings respect corresponding commutators (for X's this is [XI,X2]L + ^ K ( P X )-^2 - dv(px )-^i> s e e Sec. 3.2.2). Proof. The inner subscript + can be omitted in the term (LPxL~1)+ = (L(XL)+L~1)+ since if it is replaced by —, the expression vanishes. The rest is clear. The correspondence of commutators is practically already proven. • 15.1.16. Proposition. The action of an infinitesimal diffeomorphism on operators, dy(p) where P is given by Eq. (15.1.12), is a Hamiltonian vector field with the Hamiltonian J r(x)u2dx. Proof. We have to find an operator X e i?_ such that P = (XP)+. Besides, the condition res[L, X] = 0 must be satisfied. This is X = d1~nr + d~n((n—l)/2)r' (we use x instead of t and r' instead of r). Both statements can easily be verified. If f fdx is the Hamiltonian of this field, then it must be Sf/Su2 = r(x). We can take / = ru2. D Let us find the corresponding vector field dv(p)- First of all, (LPL~l)+
=
1
{L[rd-
y
dn ( rd
- l
n
rd — n
1 -r
+nr
Hence, V(P) = dv(P)L
^2,ukdin—k
=-
rd —
n k
+ nr''f^ukd
.fc=o
-.
fc=0
Easy calculations yield dv(P)Uk = ru'k + kr'uk
k - 1 (n + 1 ,(*+!) Jfe + 1
k-l
£
n —m n —k — 1
m=2
n — 1 fn — m k m+1 rl - '>un n—k
(15.1.17)
The first two terms on the right-hand side represent the infinitesimal transformation of a fc-differential. The rest of terms represent the deviation of conformity.
Transformational
Properties of a Differential
Operator
257
Let us write a few formulas, for k = 2, 3 and 4. + -(n
dV{P)U2=ru'2+2r'u2
dV(P)U3 = ru'3 + 3r'u3 + [ ^
.
,
A
,
3
)r'", )riv + (n - 2)r"u2 ,
3fn+l\ (5)
9v( P )U 4 = ru4 + Ar u4 + - I
Irw
n + 5 fn-2\ ... 3, „. ,. + - 6 - - ( 2 y"u2 + -(n-3)r"u3. In the first formula the term with r'" descended from the Schwarzian term of the finite transformation. Our goal is to kill additional terms taking combinations of these equations. For example, for w3 = u3 — ((n — 2)/2)u'2 it is easy to check that dV(p)W3 = rw'3 + 3r'w3 which means that w3 is primary of weight 3. It is more difficult tofindthe next combination, of a conformal weight 4. This is ([DiFIZ]) wA = u4 - ((n - 3)/2)u'3 + ((n - 2)(n - 3)/10)u 2 ' - ((n - 2){n - 3)(5n + 7)/10n(n 2 - l))u\ . (Check!) Before we start doing this in a general form, let us study the conformal properties of the Miura variables (see Sec. 4.1). 15.1.18. Proposition. The operator L can be represented as
* \S-(~-A-l^---(o+'L^-l-\.
(15.1-W)
where 7 1 , . . . , j n are quasi primary fields of weight 1 with the only constraint Ylii21^ ~ 07* = 0- ^ n i s °dd, the middle factor of the product must be replaced by (d — l(n+i)/2) where 7( n + i)/2 is a primary (not "quasi"!) field of weight 1. Proof. The right-hand side can be written as T>\---T>n\ J--(n-i)/2 ~^ •^r(n+i)/2i hence this is the general operator L in its multiplicative form,
258
Soliton Equations and Hamiltonian
Systems
and ( Ii y^ — «)7i are nothing but its Miura variables (in Chap. 4 they were called — vn+i-i). • It is instructive to give an alternative proof of this proposition starting with a given factorization of L and proving that the Miura variables are changing under the diffeomorphisms in the needed way. Let L — (d —
i
Hence, fa = ((n + l ) / 2 — 2)7, where 7, is a quasi primary field of weigth 1. Di Francesco, Itzykson and Zuber [DiFIZ] have proven the following general and remarkable proposition: 15.1.20. Proposition. For every k > 2 there is a linear combination
where W2 = U2 and Ay.yr are differential polynomials in u-i, such that u>k's transform as fc-differentials. Functionals Wk[r] = Jr{x)wkdx for all r{x) are taken as generators of the Wn algebra with respect to the second Poisson bracket. (It suffices to take a complete system of functions r, e.g., xm+1, if x are on the circle \x\ = 1.) Proof of the proposition. The difference of two quasi primary fields of weight 1 is a primary field, therefore, Eq. (15.1.19) can be written as
d-l^±l-2)j
+ r,2)...(d+7^j
+ r,n) , (15.1.21)
Transformational
Properties of a Differential
259
Operator
where 7 is a quasi primary field we specify below and 77, are primary fields. A constraint
X>=0,
(15.1.22)
i
must hold, in order to have a reduced operator L. Let us choose 7 such that the terms with dn and dn~2 in L coincides with those in
MT->)#-(^-2>)-"(8+^) = Vn = dn + c n ( 7 ' - 7 2 /2)d"- 2 + • • • .
(15.1.23)
2
This leads to the equation 7' — 7 /2 = U2Jcn = u 2 which is solvable with respect to a quasi primary field 7 of weight 1, according to Lemma 15.1.7 above. Now, Eq. (15.1.21) can be written as L
= (v + m)(v + 7J2) • • • (p + vn) .
The variables r]i are not independent, they are submitted to two constraints: first of them is (15.1.22), and the second is
IE
n+1
.\
— —
1} im + Vi
= J2^Vrli
2
+ Y^(i-IK i
+ lvi^=0-
(15.1.24)
Using Lemma 15.1.8, we can open the parentheses and rearrange terms: L = Vn + a 3 P " - 3 + • • • + an , where a» are of the form
and, therefore, primary fields. They can be expressed as differential polynomials in U 3 , . . . , un and 7 while we need them to be differential polynomials in U3, . . . , u n and U2 = c n (7' — 7 2 /2). In order to achieve this goal, a triangular change of variables can be done: i-3
1=0 = Wi + a^hl{Vwi-i) WQ
= 1,
Wi — 0 ,
+ ••• + Q 3 , i - 3 ( © i _ 3 w 3 ) U>2 = U2 •
for i > 3 ;
260
Soliton Equations
and Hamiltonian
Systems
15.1.25. Lemma. The coefficients aki can be chosen such that Wi depend on 7 only in the form u^ = cn(Y — 7 2 /2). Proof. The new quantities Wi are determined by a triangular system n
n
Yl <*ki{Vlwk)Vn-k~l = Y, Uid71-' •
£
i=0 k+l=i
(15.1.26)
i=0
The relation w2 = ~l' + 7 2 /2 allows to express all the derivatives of 7 in terms of 7 itself and Ui and its derivatives. Then all the differential polynomials in U3,... ,un and 7 can be written as differential polynomials in «2, • • •, un and ordinary polynomials in 7. We have to find aki such that this latter dependence vanishes. To this end let us give a small increment 67 to 7, 7 H-> 7 4- £7 such that 112 remains unchanged. This means that 6j is constrained, — £7' + 7J7 = 0 which can also be written as (d + a)6~t = 67(8 + 0 + 7) for an arbitrary a. We must choose aki such that the system for wki does not change; then neither do its solutions, wk's, and, therefore they are differential polynomials in 1*2, • • •, u n . The right-hand side does not depend on 7 at all. Now we gather the terms of the left-hand side containing one of wk 's each: n—k
^k = Y
a
ki{Vlwk)
• Vn~k~l : ^ _ ( n _ 1 ) / 2 -» ^(n+D/2 •
(15.1.27)
1=0
(This formula also describes the term without wk's as a special case if we put k = 0 letting 0:0; = Sio-) Taking the operator S from the expression (15.1.26) and using the constraint, we arrive at s
71 —K
*Afc = „ kl I = -*y -57 y Y >OL 1=0
\m\l~l){Vl^wk)V-k-1
*•
{piWk)thjz3^±zJlVn-k-i-i
+
The requirement SAk = 0 implies a recursion relation for aki: akil(2k +1 1) = akj-i(n — k — I + l)(fc + / — 1) that can be solved and the result is akl
~
fc rr)C7 ) (2k+l-l )
Transformational
Properties of a Differential
Operator
261
This lemma completes the proof of the theorem.
•
15.1.28. Let us calculate a few Poisson brackets. (1) For two Hamiltonians, w2 [r] = J r(x)u2dx and w2\s}. We have already found Hamiltonian vector fields corresponding to these Hamiltonians, therefore {w2[r},w2[s}} = J fru'2 + 2r'u2 + \ \ ' ^ )
r
sdx
" )
— I w2(r's — rs')dx + cn I r1"sdx r /
n
= w2[r s~rs\ For example, r = xm+1
f
+ cn
in
n n2
i
(
r sdx,
c„ =
- 1)
—
.
and s = xl+1,
am = (2-Ki)'lw2{r),
at =
(2iri)~1w2(s),
then 27ri{a m , ai} = (TO - l)am+i
+ cnm(m2 - l)<5m+j,0 •
Thus, am are generators of Virasoro with a central charge n(n2 — 1). If r = 5(x — x\) and s = S(x — x2), then {w2(xx), w2(x2)} = S'(x2 -x^lw^xx)
+w2{x2))
+cn6'"(x2
-
xx).
In the same way we find { K ^ M , W 3 [S]} = / w 3 (2r's - rs')dx = w 3 [2r's — rs'). Taking r = 5(x — x\) and s = 5(x — x2) we have {w2(xi),w3(x2)}
= S'(x2 - xi)[w3(xi)
+ 2w3(x2)].
The computation of {w3(x\), W3(x2)} is very cumbersome. We shall take a result for a particular case n = 3 from [Bak89]: {w3(xi),w3{x2)}
= --<5 (5) (a: 2 - xx) - -(1*3(0:1) + u\(x2))5'(x2 0
6
5 - rr(u 2 (a;i) + u2{x2))5'"(x2 + ^ K ( a : i ) + u2\x2))6'(x2
- xi) -
n).
- x{)
262
Soliton Equations and Hamiltonian
Systems
It is noteworthy to mention that the right-hand side cannot be expressed in terms of w2 and w^ linearly, there are nonlinear terms. If n is big enough, n > k+l—2, then {wk, wi} is a linear combination of Wi and their derivatives for i < k + I (for detail see [DiFIZ]). For example, {w3(xi),w3(x2)}
15.2
= -2w'46(x2 - z i ) - - f
]S{5Hx2 - x i ) .
Another Construction of Primary Fields
15.2.1. This section deals with another choice of generators Wi, which will be called Wi. In some respect this new choice will be worse since Wi can be expressed in terms of ttj, in general, nonlinearly, while the old Wi's depended nonlinearly only on u2. However, the construction of new generators is more convenient for practical use. These new generators were introduced by Balog, Feher, O'Raifeartaigh, Forgacs and Wipf [Bal], Bauer, Di Francesco, Itzykson and Zuber [BDiFIZ], and Bonora and Xiong [BX]. We return to the Drinfeld-Sokolov reduction, see Sec. 9.4. We discussed there two gauges, [/-gauge and V-gauge. Now we add the third one, VF-gauge. Let J_ be the matrix we called before J, let J+ be the matrix whose only nonzero elements are under the principal diagonal and are equal to (J + )i t i -i = i(n — i), i = 1 , . . . , n — 1 (recall that the rows and columns are numerated from 0 to n — 1). Let Jo be a diagonal matrix with elements (Jo)a = ~((n ~ l)/2) + i- The commutation relations between them are [Jo, J±] = ± J ± and [J+,J-] = 2J 0 . These matrices generate a spinor representation of the algebra su(2). Let
On every TV-trajectory there is one operator of this form (which is clear from the number of free parameters, and can be rigorously proven), i.e. manifold Mw of all operators \w is transversal to all orbits. It is not difficult to show that Uk = (?kWk + (dif. polynomials in Wi with i < k) where Ok are constants. Therefore, Wk can be expressed as differential polynomials in Uj.
Transformational
Properties of a Differential
263
Operator
We can choose Wfc as parameters on the manifold of orbits of the gauge group TV, and a problem arises: what is the action of the Hamiltonian (with respect to the second structure) flows on these parameters. Especially interesting is the flow with the Hamiltonian J ru2
-1
3W2
9 4 + u2d2 + u3d + u 4 = det lw
12W3
0
0
9 - 1 0 AW2
d
-1
36 W4 \2W3 3W2 d 9 - 1 0 4W2
3W2 - 1 d + 12W3
d
12W3 3W2 d
+
4W2
-1
12W3
d
d+
-1
0
d
-1
d
-1
3W2
d
36W3 3W2 d d
-1
3W2
d
3W2 +
12W3
-1
36W4
d
= d4 + WW2d2 + (24W3 + lOW2)d + 36W4 + 12W3 + 3W2 + 9W$ from where we can find W2 = — u2 , W3 = — v(u3 - u'22), W4 = — 10 24 ' 36
1 ,
1 ,/ 100
We see that these generators are proportional to u2, w3 and W4 found earlier (one has to put n = 4 there). 15.2.3. Thus, we must express the action of Hamiltonian vector fields on orbits in terms of parameters Wk- We can use the routine procedure described in Sec. 9.4. Namely, let g e N be a gauge transformation such that \\j = glwg"1 and \u = —d + J_ + U; the matrix U has the only nonzero row, the bottom one (its last element also vanishes). We take all possible independent last columns: X = d~i~1Xj + d~nXn-\ where Xn-\
264
Soliton Equations and Hamiltonian
Systems
is to be found from res[L, X] = 0 and construct matrices Qx such that only the last row of [U/,Qx] does not vanish (the last element of the last row vanishes, too). Then we return to the W-gauge: Qx = g~1Qx9- We have: [lw, Qx} = 9~1\}-U, Qx]g is a strictly lower triangular matrix, as a product of three triangular matrices one of which is strict. However, this matrix cannot represent the flow on the manifold Mw since it does not have a form Yy{~ Pk+\J+- To improve this, we have a possibility to change the vector field [liy,<5jrl by a vector field tangent to the orbits. Such vector fields have a form [lw, v] where v are strictly lower triangular matrices. In other words, we can change the part of Q*x below the principal diagonal to make the commutator [Uy,<3x] to have a "good" form Y^i~ Pk+iJ+15.2.4. Lemma. Let A = (Ay) be a matrix with Aij = 0 for i — j < s, where s is a positive integer, then A can be represented as a sum of a "good" matrix, of the form YlPk+iJ+i a n d [K,J-\ where if is a matrix with Kij = 0 for i — j < s + 1. In other words, A is strictly lower triangular and K has one zero diagonal more than A. Proof. Subtracting a "good" matrix we can achieve that X^-i=Const -^O" = 0. Then it can be represented as a needed commutator which is easy to prove directly, moving from the diagonal i — j = s to the next one, i — j = s + 1, and so on. • 15.2.5. Proposition. A strictly lower triangular matrix can be added to the matrix Qx so that the new matrix Q*x has a property [lw)Qxl = Tn~l Pk+\J+Proof, (step by step) We have [\w;Qx\ = "good" + [Kx, J_]. Here Kx is strictly triangular. Then[lw,Q*x-Ki] = "good"+A'i + E Wk+i J+,Kx] = "good" + [if2, J-] where if2 has one more zero diagonal, i — j = 1, and so on. Finally, only a "good" matrix remains. D This remaining "good" matrix gives the Hamiltonian flow. However, it is not easy to obtain explicit formulas. Let us do this for the simplest case of the Hamiltonian F = / ruidx generating infinitesimal diffeomorphisms. We have X = d~n-lXn-2 + d^Xn-i where Xn-2 = SF/Su2 = r and X n _ i has to be found from 0 = res[L,X] = ies[dn,X]. This implies Xn-i = ((n - 1)/2)X;_ 2 = ((n - l)/2)r'. Then we find the matrix QXThe zero row is Q0 = {XL)+ = (Xdn)+ = - ( ( n - l)/2)r' + rd, i.e. there are only two non-vanishing elements in it, (Qx)oo = —((n — l)/2)r' and
Transformational
Properties of a Differential
265
Operator
(Qx)oi = T. For all the rows we have - {j?X)+L = rdi+1 + (-'^—^ + 1 ) r'di + lower terms.
Qi = d\XL)+
This means that the matrix Qx has the following structure: Qx = r J _ +r'Jo + strictly lower triangular. It is easy to see that the gauge transformation Qx = g~lQx9 does not change this structure, and Qx has the same shape. Now we must find the elements of this matrix below the principal diagonal such that the resulting matrix Q*x satisfies the requirement: [lw,Q*x] is a linear combination of j £ . This is not very difficult. Namely, taking Q*x* = rJ^ + r'J0 - (l/2)r"J+ - r ^ " " 1 Wk+1 J% we easily obtain -
-[lw,Q*xl
= ^'"J+
n— 1
+ 5 > W £ + 1 +(k +
\)r'Wk+1)Jk+
fc=i
which implies the following action of infinitesimal diffeomorphisms 6F on Wk: dFWk = (rd + kr')Wk + Uk,2r'" . We have proved 15.2.6. Proposition. The fields Wk with k > 2 are primary of the weights k, the field W2 is quasi primary, and u2 = ( " J 1 ) ^ The last equation follows from the comparison of central terms for u2 and W2. 15.2.7. Now we give an alternative description of the above procedure which does not use the {/-gauge. Matrices Q must be found such that [\\y, Q] have a form Y^i~ Pk+iJ+First, we write the equation [lw, Q)ij = 0 for elements of a diagonal j—i = s, in succession from s = n - l t o s = l . For every diagonal s this gives n — s linear equations with respect to n — s + 1 elements of the next diagonal. E.g., for s = n - 1 we have go,n-i +
• 1 • ( n - 1) = 0
266
Soliton Equations and Hamiltonian
Systems
etc. One variable in every set of equations remains free. The quantities Yk+i = tr JlQ,
k =
l,...,n-l
can be taken as free variables. On the principal diagonal we use another normalizing condition, tr Q = 0 which can be done since dtr Q = 0. Now all elements of the upper triangle of Q are uniquely determined as differential polynomials in Yjt where k = 2 , . . . , n. Passing to the lower triangle, s = i — j , s = 0, l , . . . , n — 1 we see that n — s elements of [lw, Q] on the sth diagonal involve only n — s — 1 new variables qij belonging to the next diagonal. The equation 71-1
\lw,Q} = YlPk+lJ+' l
where Pk+i are additional variables is uniquely solvable with respect to the elements of the lower triangle of the matrix Q and Pfc+i's as differential polynomials in Yfc. The matrix Q corresponding to given Yjt is denoted byQy. The mapping Y^ >-> QY permits to write the Poisson brackets. Given two functionals / and g expressed as differential polynomials of Wk let Y^f) = 6f/SWk and Y^g) = 5g/5Wk. Then n-l
/
tr{lw,QYU)]QY(g)dx
.
Pkf+iYk+idx •
=J2 J
k=i
Unfortunately, we cannot write the correspondence Yk i-> QY by a closed and simple formula like that for the f/-gauge. 15.2.8. We have seen that the finiteness of n seriously deteriorates expressions for Poisson brackets between uk and it;, or Wk and wi: instead of primary fields Wk+1-2 they contain nonlinear expressions in lower generators. There were many attempts made to embed iy„-algebras into a universal algebra, or Woo-algebra. Two ideas how to do this occur when you start thinking about it. First, send n to infinity in formulas for Poisson brackets with a proper normalization. This yields some limiting W-algebra (see below), however, this algebra, which is called Woo, is not the universal algebra. It is obtained not by an extension of Wn. Second, it is known that all nth KdV hierarchies can be embedded into one KP hierarchy. It is natural to assume that the symplectic structures of KdV are restrictions of one universal structure for KP. Unfortunately, the situation is much worse.
Transformational
Properties of a Differential
267
Operator
Each Poisson, or symplectic, structure for KdV can, indeed, be extended to a structure for KP. However, all these structures are, apparently, different, and there is no universal structure obtained in this way. See a discussion on the universal algebra in [FO'F(a), (b)]. The limiting transition with n —> oo was studied by Bakas [Bak90]. Here is his result. The generators are
E
( — T\ni-\
hrip + 1 u,1
ni + • • • + np
ni%i + ...npip=s
r
•••u n
p i
.
l
"
If ws (k) are their Fourier coefficients, then {ws(k),ws<(k')}
= [(s'-l)k-(s-l)k'}ws+s<-2{k+k')
+—k35k+k'fitsrfs',2
,
c is a constant. This is a well-known algebra of area-preserving diffeomorphisms of a torus, with a central extension. It is called Woo-
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Chapter 16
Further Restrictions of the K P ; Stationary Equations
16.1
The Ring of Functions on the Phase Space of the Equation
16.1.1. We know that the nth GD hierarchy is the restriction of the KP hierarchy to the manifold of the \tDE operators which do not depend on the "time" variable tn. We also know that if one more restriction is imposed: independence of the additional variable i!_„ +1)1 , then the string equation arises. What happens if two restriction: independence of two times tn and tm are imposed on the KP operator? We will consider only the "pure" case when m and n are relatively prime, otherwise the situation is confused by the fact that if p is common divisor of n and m, then the pth GD hierarchy is a part of the solution. (One can also consider a more general case when the KP operator is independent of two linear combinations of time variables.) We can formulate the problem in the following manner. If L is the nth order Lax operator, one has to solve the equation [L™ , L] = 0. Thus, this is a problem of searching two commuting differential operators, of the nth and mth order correspondingly. A similar problem can be formulated about other hierarchies, e.g., the matrix AKNS-D. We shall discuss this later. Another possibility: to impose only one restriction, independence of an additional variable. On this way the so-called constrained equations can be obtained. In this chapter we are talking about the first of these problems, concerning the equation [L™ ,L] = 0, or [P, L) = 0. We call it the stationary equation since this is one of the equations of the nth GD hierarchy for solutions independent of a time variable tm. The general fact is that the stationary equations arising from nonstationary integrable systems are integrable Hamiltonian systems themselves. Their Hamiltonian structure will be constructed, together with sets of first integrals in involution in an amount equal to the number of 269
270
Soliton Equations and Hamiltonian
Systems
degrees of freedom, i.e. to half of the dimension of the phase space. More than that, the integration procedure can be carried out effectively, and variables of the "action-angle" type can be constructed. In these variables the system becomes linear and thus can easily be integrated. The angle variables are obtained by means of the Abel mapping of the gth power of the Riemann surface (g is the genus of the surface) onto its Jacobi manifold. The return to the original variables can be done, after Riemann, with the aid of the ^-functions. The importance of the stationary equations is particularly great for the reason that the finite-dimensional manifold of solutions of the equation [P,L] = 0 is an invariant manifold for any equation dkL = [Pk,L]. Thus, finite-dimensional manifolds of solutions of nonstationary equations will be obtained having explicit form (as a rule, in terms of the 0-functions). These solutions are called algebraic-geometrical. We have mentioned them earlier but now we consider them from another point of view. The discovery of the algebraic-geometrical solutions is connected, in the first place, with the names of Its, Matveev, Novikov, Dubrovin, Lax, Mac Kean, van Moerbeke and Krichever. We interpret this problem as integration of a Hamiltonian system with sufficiently many first integrals in involution, according to the Liouville procedure. To this end, it is necessary (i) to construct a symplectic form and a Hamiltonian for the equation [P,L] = 0 (do not confuse these with the form and Hamiltonian for the nonstationary equation L = [P, L]\), (ii) to construct sufficiently many first integrals in involution, (iii) to find, according to Liouville, the angle variables corresponding to the first integrals taken as action variables, (iv) to write the formulas for the solutions in the original variables in terms of the ^-functions. This chapter contains some general conceptions of Hamiltonian structure of variational Euler-Lagrange equations, in addition to Sec. 2.3. First we consider the simplest case of the KdV-hierarchy for n = 2, i.e. the KdVhierarchy in the narrow sense. This study is based on the theory of this hierarchy developed in Sec. 3.7. In Chap. 17 we deal with the stationary equations for matrix hierarchies. 16.1.2. Stationary equations have the form H(Sf/SL) = 0 where H is the Adler mapping, whence 5f/SL € kevH. The kernel of the operator H is always very small, and we shall see that slightly changing / one can in every case write the equation simply as 5f/SL = 0. Thus, we deal with the variational Euler-Lagrange equations. These equations will be of the
Further Restrictions
of the KP; Stationary
271
Equations
Cauchy-Kowalevsky type, i.e. they have the form Qi=vlmt)+Fi(v)=0,
i = l,...,N,
(16.1.3)
where {vi} are some variables and the functions {i^} are polynomials in { « . - } , the order j of derivatives v\3' in all the terms being less than rrij. Let A be the differential algebra of polynomials in {vf }. We shall introduce a new system of generators for it. Any element of A can be uniquely written as a polynomials in two kinds of variables: (a)
{v\j)},i
= l,...,N,j<mi
(b)
Qff\
i = 1 , . . . ,iV, j = 0 , 1 , . . . . (16.1.4)
The first kind of variables will be called the phase variable. The variables (b) generate a differential ideal in the algebra A : JQ = {$2aijQi }> ciij 6 A. Let AQ = A/ JQ. The elements of the differential algebra AQ are classes of differential polynomials which are equal modulo the differential equations (16.1.3). We consider these elements as functions on the phase space. In each class there is one polynomial which is expressed only in terms of the phase variables (a). What is the purpose of this complicated approach? If the functions on the phase space are understood as elements of AQ, then the vector field corresponding to the system of Eqs. (16.1.3), i.e. the derivatives along the trajectories of this system, can be written extremely simple. Indeed, this system can be written as dvi = Vi,
dVi = v",...,
d(v\
') = v\ '
',
d(v\
') = v\ " ,
where v\m' must be eliminated with the help of Qi = 0. This sequence can be extended: dv\m = v\m' , . . . . Thus, the differentiation of any function along the trajectory is the application of the operator d modulo the equation {Qf = 0}, i.e. in the space AQ. The vector field corresponding to the system (16.1.3) is simply the operator d in AQ. What are other vector fields in the phase space? Any derivation in A is £ = ^2aijd/dvi
i
atj e A.
272
Soliton Equations
and Hamiltonian
Systems
The space of these fields is denoted by TA. This field can be transferred to AQ if and only if it preserves the ideal JQ, £JQ C JQ. The subspace of such fields we denote as TAQ. An example of such fields is a partial derivative with respect to a variable (a) when variables (b) are fixed. Another example isi = d = ^v\j+1)d/dv\j\ Vector fields commuting with d are, as we know, da = ^a^'d/dvf'. To give an example of a vector field which preserves the ideal and, at the same time, commutes with d is not at all easy. This amounts to no less than finding a symmetry of Eqs. (16.1.3). If such a field is found, the flow determined by it, dvi/dr = a^, transforms solutions of Eqs. (16.1.3) into other solutions. A symmetry can be found if a first integral is known. Right now we shall describe how to do this. An element / e A and its class in AQ we denote by the same letter. The meaning of equality signs must be pointed out precisely: the exact equality in A will be denoted simply as / — g, while the equality in AQ, i.e. modulo the equation will be denoted as / = g; this means that / and g differ by an element of the ideal JQ .
16.2
Characteristics of the First Integrals
16.2.1. This section is based on [GD75], [GD76(a)]. Let an element of the ideal JQ be represented as / = £ a ^
0 )
-
(16-2.2)
This representation is of course not unique. 16.2.3. Lemma. If
then all the ay belong to JQ. Proof. Let us express all the ay in terms of the generators (16.1.4). This representation is unique. If some ay £ JQ, then the term a^Q^ is linear with respect to the generators of the type (b) and there are no other terms to cancel with this one. • Thus, two distinct representations (16.2.2) of an element of the ideal have coefficients which differ by elements of the ideal. •
Further Restrictions
of the KP; Stationary
273
Equations
16.2.4. Definition. The characteristic of an element of the ideal JQ is the set of N elements of AQ:
f € JQ,Xf = {(x/)i} = I E(-l) 0 ) a!? | , i = l,.-.,N. Here cM are understood as elements of AQ. The characteristic is well defined by virtue of Lemma 16.2.3. 16.2.5. Proposition. If / = dg, g £ JQ, then Xf — 0. Proof. Let g = E « u Q ? } . then / = I X ^
{
oo
oo
E i - i ^ +B 3=0
+ I>,Q^+1)
j=0
and
1
1
) ^
=°-
D
J
Thus, the characteristic is a mapping JQ/OJQ —> AQ. 16.2.6. Definition. The first integral of Eq. (16.1.3) is an element / e AQ such that df = 0 (in AQ). 16.2.7. Definition. Characteristic of a first integral is an element of AQ which is constructed in such a way. Let / € A be a representative of the class of the first integral. Then df = g e JQ. The characteristic of the first integral is the characteristic of the element g of the ideal. The characteristic of the first integral / will also be denoted as x/> i.e. Xf = Xg- We must remember that Xf a n < i Xg have quite different meanings: the characteristics of a first integral, and that of an element of the ideal respectively. The characteristic of a first integral is well-defined, i.e. it is independent of the representative of the class of the first integrals. If f\ and fa are two distinct representatives, then f\ — ji = h G JQ. Hence df\ — 8/2 = dh and 91-92 = dh, Xgi - Xg2 = Xdh- According to 11.2.5, Xdh = 0. 16.3
Hamiltonian Structure
16.3.1. Recall that the equations we are dealing with have a variational form Qi = 6A/Svi = 0, where A is a Lagrangian. If the highest orders of derivatives entering the Lagrangian, v\m , are m* (and they cannot be
274
Soliton Equations and Hamiltonian
Systems
reduced by adding to the Lagrangian exact derivatives), the equations are N
Qi = ^2aijv^mi+mj)+Fi=0,
t = l,...,JV
(16.3.2)
the coefficients a^ and Ft containing ir- only with k <mi + rrij. We shall not consider the general case. In the examples below all the coefficients a^will be constant. Then, as it was already shown for N = 2 in Sec. 2.3 we can express the highest derivatives v\ m'' in terms of the lower ones: v<2mi) = MQ,v),
i = h...,N,
(16.3.3)
<j>i are differential polynomials in {Qj '} and in {ir- '}, k < 2mj. The system is of Cauchy-Kowalevski type, we have a system of generators (16.1.4) (where 2rrii must be substituted for m^), and all the definitions of the last section are valid. 16.3.4. The space V of vector fields consists of derivations £ = Y,iihijd/dv?). The dual space is the space of one-forms J2t j ajdVi where 5v\3' is the basis dual to d/dv\3'. Differential forms are multilinear functional on V. In terms of the basis they are
« = £$*,<*> A... A ft,g*>, °vleA> where (i) and (j) are multi-indices. For any vector field £ = ^li bijd/dv\ , the operation i{£) of the substitution of the field into a form is defined. This is agraded derivation: if ui^ anda;| &rep- and (/-forms, then j(£)(wf Aw|) = (i(Ow?)Awf + (-l) p a;PAi(£)wf and iffiSv^ = btj = ^j). In other words,
A
. . . A ^ ) + ... + ( _l)fc-i^ a g^) A ... A (^) ) .
The operation 5 : u i-> 5u is defined either by the Lee formula (2.1.3) or, equivalently, as a graded derivation such that 8{5vf') — 0 and it is Sf = T.{df/dv\j))5v\j) for any function / . Then, Sw = J2 d(o^)/dv^)Sv^
A 5v™ A • • • A 8v™ ,
52 being zero. Further, the Lie derivative L% is defined as in Chap. 2 by L^LJ = (5i(£) + i(^)5)uj. This is an ungraded derivation commuting with 6.
Further Restrictions
of the KP; Stationary
275
Equations
For any function / we have L$f = £/. Instead of L^LJ we shall write simply £w: A-
+ a«>Sv$]*8{tv%]) A... + .-.}. If a = ( a i , . . . , ajv) is a set of functions then da will designate the vector In field ^a^d/dv^. Particular d = T,ijvlJ+1)9/dvlJ) = dv,, where v' = (v[,..., v'N). We have du; = ^{{dafySv*}
A • • • A *,<*> + a$8v£
+1)
A *„£'> A • • • ,
•• + • • • } , "(i)""^) """fe) i.e. this is differentiation in succession of all the multipliers including those which enter under the sign 5. Proposition 2.1.5 and its Corollary 2.1.6 hold, with the same proof. Let us restrict forms to the phase space, i.e. consider them only on vector fields belonging to TAQ. Some of the forms become zero. One can easily see that these are the forms which, written in terms of the variables (16.1.4), have at least one multiplier Q]3' in any position in a^? or under
the sign 5. We introduce the relation of equivalence W\ = o»2 for forms which coincide by the restriction to the phase space. Some auxiliary role will be played by another notion of equivalence of forms. Two forms are equivalent in a stronger sense: LJ\ = u>2, if their values coincide as elements of AQ on all the fields from TA (not only from TAQ). It is easy to see that this means that Qf are identified with zero only when they are in coefficients ar?l, and not under the sign 5. o Q® Example. SQi = 0 but SQi ^ 0. 16.3.5. Recalling the construction of the symplectic form and the Hamiltonian of a variational Euler-Lagrange equation given in Sec. 2.3, let us rewrite <5A as <5A = Y^SA-/fok)6vk where Ct,^ is a one-form. Then Q = Stl^
+ dft ( 1 ) ,
(16.3.6)
is a symplectic form. Let
H = -A + i(d)Q,{1).
(16.3.7)
276
Soliton Equations and Hamiltonian
Systems
Then 6H = -i(d)il
- Y,(SA/6Vk)Svk
(16.3.8)
whence the equation {6A/dvk — 0} is equivalent to 5H = -i(d)il.
(16.3.9)
This can also be expressed thus: if H is defined by the formula (16.3.7) as an element of AQ then the identity (16.3.9) holds in AQ16.3.10. Proposition. The Hamiltonian ~K can be found from the equation
H' =
-J2v'i5A/Svi-
Proof. One must apply the operation i(d) to both sides of Eq. (16.3.8). • A vector field £/ can be assigned to every element / £ AQ, i.e. to every function in the phase space, with the property
It happens that £f can easily be found if / is the first integral. This procedure is based on the notion of the characteristic introduced in Sec. 16.2. First we prove two simple lemmas. 16.3.11. Lemma. If 6F Q= 0 then F = const + Fi, where Fi £ JQ and XF1 = 0. Proof. Let us write F in terms of the variables (16.1.4). Then F = F1 + F2, where Fi G JQ, and F2 is expressed in terms of phase variables (a) only. Let £ G TA. Then £F = i(£)6F £ JQ. In particular, when f £ TAQ, this yields £F2 £ JQ since £Fi G JQ. Let £ be, for example, a partial derivative with respect to a phase variable (with fixed (b)-variables). We shall see that F2 = c = const. Hence F = c + Y^aijQi• Now we take £ = d/dQ^' and find that all the
Further Restrictions
of the KP; Stationary
Equations
277
16.3.13. Proposition. Let / € Ao be a first integral with the characteristic Xf- Then the vector field dXf has the property dXfA = d( where d(
),
) is the derivative of an element of A.
Proof. We have dXl A = ^(Xf^dA/dv^
= Y^xfrSA/Svi
i,k
+ d(
)
i,k
+ d{
) = dF + d{
) = d(
)
according to the definition of a characteristic. This can also be written as (Adx
d
XF
•
= 0.
(16.3.14)
16.3.15. Proposition. The vector field dXF is a symmetry of the equation, i.e. dXf e TAQ. Proof. We act on Eq. (16.3.6) with the operator dXf: SdXfA = Y,dXs{5A/5vk)5vk
+ ^(<5A/«5Ufc)5(X/)fc +
According to 16.3.13, dXfA = d( £
dXf (SA/6vk)6vk
d(dXfil).
), hence
+ 5>A/Ju f c )*(x/)fc = d(
),
whence ^5 X / (JA/«5 V f c )^ f c Q = Q d( Lemma 16.3.12 yields dXf(SA/5vk)
£
JQ,
i.e.
).
9XJJQ
C
JQ
as required.
•
16.3.16. Proposition. For a first integral / the equallity
sf£i(dXf)n holds. That means that the vector field d(-Xf) corresponds to the first integral / with respect to the symplectic form fl.
278
Soliton Equations and Hamiltonian
Systems
Proof. Let / be the representative of the class of the first integral which is expressed in terms of only the phase variables. Then df = Y^(Xf)k?>A/5vk (the derivatives (SA/Svk)^ with i > 0 are not involved). Apply the operator 5 to (16.3.6): "£6(SA/$vk) A 6vk + dfl = 0. Further, di(dXf)n
= i([d,dXf])ii
+ i(dXf)dn
=
i(dXf)dn,
(here the equality [L^,i(rj)} = «([£,//]) is used, as well as the fact that LQ is d). Transform the right-hand side of this equation: i(dX/)dfl
= -i(dXf)^26(5A/5vk)
A 6vk = - y~] dXf
(SA/Svk)5vk
+ Y,(Xf)kS(SA/5vk) Q=Q J2(XfhS(5A/5vk). On the other hand,
d6f = 6j2(Xfh5A/5vk
Q
=° Y,(Xf)k6(6A/6vk).
Hence
d(5f-i(dXf)n)Q^o. Lemma 16.3.12 gives 5f — i(dXj)Cl
= 0. If another representative of the
class / is chosen then the sign = must be changed to =.
D
16.3.17. Corollary. The symmetry dXf preserves the form CI : dXf$l = 0. Proof. Apply 5 to 16.3.16.
•
Thus, a first integral generates a symmetry which preserves the Lagrangian (more exactly, the action), and the symplectic form. This is a conversion of the Noether theorem. Now we go from the general considerations to our particular integrable systems. We start with the KdV hierarchy (i.e. n — 2), see Sec. 3.7.
16.4
Stationary Equations of the K d V Hierarchy ([GD79])
16.4.1. A stationary equation is R'2k+\ = 0, i.e. i^fc+i = c = const, which can also be written as i?2fe+i —cR\ = 0. A more general differential equation
279
Further Restrictions of the KP; Stationary Equations (see 3.7.7) is n+l Q = Y d2k+iR2k+i o
= 0•
(16.4.2)
This is an ordinary 2 n t h order differential equation. 1 6 . 4 . 3 . Proposition, (i) If a function u(x) satisfies Eq. (16.4.2) then the n t h degree polynomial in C, = z2 n+l
fc—1
n
R(0 = Y rf2fc+i Y ^i+ic"-1-1 = Y *
i=0
1=0
where Rl =
n+l Yi fc=(+l
d2k+lR2(k-l)-l
and u ^ ( i ) are substituted for uW satisfies Eq. (3.7.2) for the resolvent. (ii) Conversely, if for some function u(x) there is a solution of Eq. (3.7.2) which is a polynomial in C whose coefficients are differential polynomials in u(x), then u{x) satisfies a stationary equation of t h e form (16.4.2). P r o o f , (i) Substituting R{Q for R into t h e left-hand side of Eq. (3.7.2), one has: n+l
Y
fc-1
< W i £ M + i + 4uR2i+i + 2u'R2i+1)Ck~i-1
fc=l
+ 4^ i+1 C fc_i ]
i=0 n+l
2fc+i -1
Y ^
fc-i
Y(R'2i+l +
4ui
4+l
fc=i
+ 2u'R2i+1 + 4^ i + 3 )C f c - i - 1 - 4R'2k+1
, -4 } jd2k+iR 2k+\ fc=i
(see (3.7.10)). This expression vanishes by virtue of (16.4.2). (ii) Let .R(C) be a solution of Eq. (3.7.2) having t h e form of a polynomial of n t h degree in £. Then R(Q also satisfies Eq. (3.7.5), where c(Q is a (2n4- l)-degree polynomial in £ with constant coefficients. Let \/c(C) = ^ ( 0 = S - o o ^2fc+iC fe+1 ^ 2 We have R(() = d(C)-R (1) (C) = E L - c o £ £ o d2k+iR2i+1Ck-i according to 3.7.6. On t h e other hand, this is a polynomial. T h e coefficient in £ _ 1 must vanish: Y^,o d2i-iR2i+i = 0. This is an equation of t h e type (16.4.2). D
280
Soliton Equations and Hamiltonian Systems
16.4.4. Proposition. For each I there is a first integral of Eq. (16.4.2), namely n+l hi+\
= / Jd2k+iPk,i k=0
(see 3.7.9). For I = 1 , . . . , n these first integrals are independent. Proof. We find n+l
df2i+i = Yl d2k+iR2k+iR2i+i
=0•
(16.4.5)
fc=0
Thus, all the f2i+i are first integrals. Now we shall prove the independence of these first integrals with / = 1,... ,n. The terms with the highest weight are contained in Pn+i,i, their weight being 2n + 2 + 21. There are quadratic terms among them which are obtained from the product of linear terms in R2n+3 and i? 2 ;+i' i.e. of u(2™) a n d t i ' 2 ' - 1 ) . We find u (2n) u (2J-l) =
(u(2n-l)u(2l-l)
_
U (2 n -2)„(2I) +
. . . ± I^n+J-D^V
Thus f2i+i has the term c • ( u ( n + ' _ 1 ) ) 2 . The multiplier u ( n + i - 1 ) can be involved in other terms only in the product with u^, where j < n + I — 1. On the other hand, in f2s+i, s < I, the multiplier u(™+'_1) can only be in the product with u^\ j < n + I — 1. Letting all the u ^ ' be zero except for that with j = n + I - 1, one gets f2s+i = 0 for all s < I and f2i+i ^ 0, which proves the independence of this quantity of the previous ones. D Notice that this argument does not work when I > n since then u^n+l~1' can be eliminated modulo Q. 16.4.6. Proposition. Equation (16.4.2) is of the Euler-Lagrange type, with the Lagrangian n+l
A = Y,d2k+1R2k+3/(-k
- 1/2).
fc=0
Proof. This immediately follows from 3.7.14.
D
The discussion of Sec. 16.3 is completely applicable to this equation. In particular, a Hamiltonian form of this equation can be written.
Further Restrictions
of the KP; Stationary
281
Equations
16.4.7. Proposition. The one-form Q,^ related to this Lagrangian A is n+l
n
J2d2k+1(R'SR/2R)k+1/2, fe=0
where the subscript A; + 1/2 means the coefficient in £~ fc_1 / 2 . Proof. According to 3.7.11 we have n+l
5A = - J2 d2k+i{R<;Su + d(RcSR'/2R - i2,c«iJ/2iJ)]fc+3/2/(fe + 1/2) fe=o whence n+l fi(1)
= - E d2k+l{RQ5R'/2R fe=o
- R'c5R/2R)k+3/2/(k
+ 1/2).
An arbitrary differential 6f, f e a, can be added to the form f^1' does not change the symplectic form Cl. Then one can write
which
n+l
"
(1)
= - E d2k+i(S(Rc/2R) fc=o
• R' - ^<5i?/2fi) fe+3/2 /(fc + 1/2)
n+l
= £
= - Y, d2k+i(R' • 5R/2R)k+1/2 .
•
fe=o This explicit expression for il^ [Alb76, Alb81].
and fi = Stt^
W as
obtained by S. Alber
16.4.8. Proposition. Let w = R!/2R. Then the forms Cl^ and O can also be written as £l(1) = (w5R)i,
fl =
(SwA6R)i,
where the subscript 1 denotes the coefficient in C _ 1 .
282
Soliton Equations
and Hamiltonian
Systems
Proof. We have R(Q = £ £ + J d2k+iCk~1/2 • R + 0(C _ 1 ), whence w = w + 0(C~" - 2 ) with w = R'/2R. Besides, w = 0 ( C - 1 ) n+l
n^ = k=0
(
/
oo
\
t=0
\
-J^d2k+1lw6'£R2i+1ci-1/2) n+l
oo
/fc+1/2 \
w-tf^dafc+i^/Jai+iC*-'- 1 ) fc=0
i=0
•
/ 1
The expression within the parentheses will change by 0{C,~2) if w is substituted for w and 5Zi=ro for X^So- This change does not affect the coefficient in £ - 1 . Now tl^ = — (w • SR)i. The second relation follows from this one. • The advantage of this new formula over 16.4.7 is that the form equals now the residue of a rational function, instead of a coefficient of a formal series. 16.4.9. Proposition. The vector fields which relate to the first integrals —/2/+1 with respect to the symplectic form O are 8R> . Proof. We have 8/21+1 = R'21+1 ' $A/Su (see (16.4.5)) whence the characteristic of the first integral f2i+i is R^i+i- ^ r e m a m s t o apply 16.3.16. • Recall the construction of the Hamiltonians (see 16.3.10). A Hamiltonian satisfies the equation %' = —u'SA/5u. 16.4.10. Proposition. The Hamiltonian of (16.4.2) is U = 4/ 3 . Proof. We have W = 4R'3 • 5A/5u = -u'SA/Su.
•
16.4.11. Proposition. If the first integral —f2i+\ is taken as a Hamiltonian in the phase space of Eq. (16.4.2) endowed with the symplectic form 0, then the corresponding flow is the restriction of the flow given by the nonstationary equation 3.7.7 d2i+\U = i? 2 /+i( u ) to the phase space of Eq. (16.4.2). Proof. Both flows are determined by the same vector field dw J
restricted
"-2I + 1
to the phase space. • This fact was established by Bogoyavlenski and Novikov [BN], and by Gelfand and Dickey [GD76(a)]. 16.4.12. Proposition. The first integrals f2i+i of the stationary equation are in involution.
283
Further Restrictions of the KP; Stationary Equations
Proof. Their vector fields commute.
•
16.4.13. Proposition. Equation (16.4.2) is integrable in quadratures. Proof. The equation of the nth order has n independent integrals in involution. The integrability follows from the classical Liouville theorem.
• In the next subsection we shall obtain effective formulas for such integration. 16.4.14. The first integrals will be written in a different form. The polynomial R{C) satisfies Eq. (3.7.2) by virtue of Eq. (16.4.2). Then it must also satisfy Eq. (3.7.5), which will be written now as 2R"R - R2 + 4(u + C)£ 2 = -P(C),
(16.4.15)
where the coefficients of the polynomial P(C) of degree 2n + 1 are not absolute constants but only constants by virtue of Eq. (16.4.2), dP(() = 0, i.e. these coefficients are first integrals of the equation. However, the higher n + 2 coefficients of P(() are absolute constants. In fact, they do not change if we replace R(() by Ri(Q = J^ktl du+i Y^=o ^ W i C f c _ i _ 1 = 2fc=i ^2fc+iCfc_1^2-R(C)i since R — R\ = 0 ( £ _ 1 ) and this replacement does not affect the higher n + 2 terms. Thus, the higher n + 2 terms of P(£) coincide with those of the polynomial C(X)™+1 ^2fc+iCfc-1)2- The only nontrivial first integrals are the n lower terms of P(C). Let 2n+l 1=0
The new first integrals are not independent of the old ones. 16.4.16. Proposition. n+l
Ji = 8 22
^2fe+i/2(fe-o-i > / = 0 , l , . . . , n - 1 .
k=l+2
Proof. We find (cf. the proof of 16.4.3) 3P(Q = 2{R'" + 2u'R + 4(u +
{)R')R
n+l
= - 8 ] T d2k+1R'2k+1 fc=i
• R = -8RQ'
= -8{RQ)'
+ 8R'Q .
284
Soliton Equations and Hamiltonian Systems
The term (RQ)' can be omitted (changing P(() by 8RQ = 0). Therefore the characteristic of the first integral Ji is equal to the coefficient of the polynomial 8R' in C,1, i.e. n+l
XJi =
8
22
d
U+lR2(k-l)-l
•
k=l+2
From x/ 21+1 = R'2i+i we obtain J, = 8 ^ = 1 + 2 d2k+if2(k-l)-i
as required D
16.4.17. Proposition. The Hamiltonian of the system is U =
-Jn-l 2^2n+3
Proof. This is a corollary of 16.4.10 and of 16.4.16.
a
The first integrals {.fo+i} can be expressed in terms of {Ji}: i-i
f2l+l = / ^C2(j_fc) + 1 J n - i _ f c ,
1=
1,..., n
(16.4.18)
fc=0
where the matrices (
/ d2n+3
c3
l2n+\
D =
c \
d5
"•
d2n+i
d2n+z J
\C2n+l
C5
C3/
are mutually reciprocal, D = C"1.
16.5
Integration after Liouville
16.5.1. Recall the Liouville theorem (in local form). Let U be a 2ndimensional symplectic manifold and ft its symplectic form. Let J o , . . . , Jn-i be n independent functions in involution, {Jk, Ji} = 0 for any k and /. Then a canonically conjugated system of the functions ipo, • • •, fn-i such that {ipk, fi} = 0, {Jk,
Further Restrictions
of the KP; Stationary
Equations
285
If there is a Hamiltonian "H then in the variables (J,
j
k
= -&H/d
(16.5.2)
If {Jfc} is a system of first integrals then dW/d
+ const.,
Jfc = const.
16.5.3. We describe the procedure of constructing the angle variables. Consider an n-dimensional level surface {Jfc = Cfc} in the 2n-dimensional phase space hi. The vector fields {£&} corresponding to {Jfc} are tangent to the surface, since £fcj; = {Jk,Ji} = 0. The number of independent vector fields {£fc} is equal to the dimension of the surface, therefore they form a basis to the tangent space in every point of the surface. These vector fields are orthogonal to each other with respect to the form fi :fi(£fc,£j) = {Jfc, J(} = 0. Thus, the symplectic form restricted to the tangent spaces at all the points of the surface vanishes. These spaces cannot be extended in such a way that the restriction of the form Q. remains zero, since the form is non-degenerate, and the orthogonal complement of a subspace of the dimension m > n has dimension 2n — m < n. An n-dimensional subspace of the 2n-dimensional symplectic space is said to be Lagrangian if the form Ct restricted to it completely degenerates. Thus, the level surfaces {Jfc = Cfc} are Lagrangian. Let us choose as coordinates in the phase space the variables {Jfc} and some functions {sfc} independent of {Jfc} and among themselves. Let us write the form £1^ in these coordinates: Cl^ = ^A,
ft(1) = $3(avva*ods< + ^2^dJi = dv + S o * - dv/dJi)dJi. Let
i = 0 , . . . , n - 1. Then n = dn{1) = ^
dJi A difi.
286
Soliton Equations
and Hamiltonian
Systems
This yields that the variables {
,Rn_i)
.
The highest derivative u^ entering Ri is u(2n-2l~2): a n d it is involved 2 -2 -1 linearly. In R! it is u^ ™ ' ), respectively. Thus, two coordinate systems (u) and (R, R') can be expressed, one of them in terms of the other, polynomially and triangularly. So, u is proportional to Rn-i- u' is a polynomial in Rn-i and R'n_i which is linear which respect to R'n-i, u" is a polynomial in -R„_i, R'n-i and Rn-2 etc. The next coordinate system will be (R, w) = (Ro,..., Rn-i, w\,..., wn) where w = R'/2R = Y^T ^iC - J - It is a l s o connected with the previous one by a triangular relation. This coordinate system is convenient since the symplectic form is written in it canonically, (see 16.4.8). Finally, following Dubrovin [Dub75] we introduce the coordinates (J, s). When ( J o , . . . , J n - i ) are fixed, the variables (s) are coordinates on the corresponding n-dimensional surface. If (J) are fixed then the polynomial .P(C) (16.4.15) is also fixed. The two-valued function y/P(Q is defined on a two-foliated Riemann surface Tj. Let Ci > • • • > C ^ e some points of the Riemann surface Tj over £ i , . . . , £„ € CP 1 where Ci> • • • i Cn are roots of the polynomial R(C)- In other words, Ci> • • • >C a r e t n e P o m t s Ci> • • • >Cn together with the indication as to which of two roots ±^/P(Q) should be taken. These points (Q) are taken as the additional coordinates s,. How the coordinates (R, R') can be expressed in terms of (J, £*) and vice versa? We have n
R(C) = (d2n+3/2)l[(C-<;i).
(16.5.5)
i=i
That means that Ro,.- • ,Rn-i of {(,•}. In particular, Rn-i/Rn t h a t Rn
— d,2n+3/2,
Rn-l
are the elementary symmetrical functions = -(Ci H h Cn)- Taking into account
= d2n+3R3
+ ^ 2 n + l - R l = -ud,2n+z/^
+ C?2n+l/2
Further Restrictions
of the KP; Stationary
287
Equations
we obtain u = 2d2n+i/d2n+3
+ 2(Ci + • • • + Cn) •
(16.5.6)
The expression of {-RJ} in terms of the new variables is given by the lemma below. 16.5.7. Lemma. # ( G ) = *V^(
1 = 1,--.,n.
Proof. One must substitute C = C( m t o (16.4.15).
D
Conversely, if the values of the coordinates (R, R') are given they determine the values of {J;} (which are polynomials in these coordinates). Solving the algebraic equation R(Q = 0 one obtains { 0 } , and R'(Q) determines the choice of the sign of the root y/P{Q), i.e. the point on the Riemann surface Q € Tj. 16.5.8. Proposition. (J, C) has the form OJi=0,
(Dubrovin) The system (16.4.2) in the coordinates 1=
0,1,...,n-l;
dQ(x) = ~(2i/d2n+3)y/P{^j
/jJ(C« - Q , 1 = 1,-.. n.
Proof. The quantities {J;} are first integrals, dJi = 0. Further, dRi = R\. It remains to use (16.5.5) and 16.5.7. D 16.5.9. Proposition. The form fi^1) in the coordinates (J, £*) is
nW = (i/2)j2^fp(&)K J
•
3= 1
Proof. According to 16.4.8, fiW = {R! /2R • &R)X. This is the residue in C = oo of the differential (R' J2R • 5R)d£. It is equal to the sum of the residues of the differential in the finite part of the Riemann sphere with the opposite sign. The poles are C = 0> a n d
"(1) = X 3= 1
i^3
3=1
a
288
Soliton Equations and Hamiltonian
Systems
16.5.10. Proposition. The variables {
(
l = 0,-..,n-l,
»<»
where Q i s a fixed point on the Riemann surface. Proof. According to Liouville's procedure one must find a function V( J, £*) such that 5V = fi' 1 ' on the surfaces {Jfc = c^}. The form fi^1) from 16.5.9 enables one to perform this integration very easily since the variables are separated. We have
v = (i/2)J2 / " , ^ / P ( 0 ^ . m=lJQ
Further, tpi = dV/dJi
(see 16.5.3; here we have /i; = 0). From P(£) =
YlT+1 Ji(l we n n d dv/dJt = (i/4) £
I
m
(c7v^(0K.
It is convenient to take Q = oo.
•
16.5.11. Thus, we have constructed a system of variables in which the solutions to Eq. (16.4.2) are linear functions of x. The integration can be performed either directly: using 16.5.8 we have n
d
dVl = (i/4) J2(CUy/P(G)) • ^k = (i/4) £ C, fc=l fc=l
x (-2i/d2„+3) / J | ( a - 0) = (l/2d2n+3) X I j#fc
fc=i
res
c=cfcUn
lr
J ^
_r\ *>
= (l/2d2n+3) • (27TZ)-1 / c 1 • I l « - 0)_1dC J
f 0,
Z < ra- 1
\ l/2d2„+3 ,
/ = n - 1,
i
whence ipi = ^ ( 0 ) +(Ji, n _i • (l/2d 2 „+3)z ,
Jj = J? ;
Further Restrictions
of the KP; Stationary
289
Equations
or by writing one of the canonical equations (16.5.2) as ip\ = dri/dJi taking the Hamiltonian from 16.4.17.
and
16.5.13. The finite-dimensional manifold of the solutions of Eq. (16.4.2) is invariant with respect to the flow generated by each of the equations 3.7.7, i.e. the solutions transform to solutions under the action of this flow. Let us look for the functions u(x,t2k+i) which satisfy Eq. (16.4.2) with respect to x and Eq. 3.7.7 with respect to x and <2fc+i- We find these solutions first in the coordinates (J,
d2k+i
= -c 2 (fc-H-„) +3
(i>n-k),
This system can be integrated and we finally obtain Ji = Jf,(fii =
6n-1,i(l/2d2n+3)x
- c2(fc+i-n)+3*2fc+i ,
i = 0,.. -, n - 1,
(16.5.14)
where Cj = 0 if j < 3, J? and tpP are constants. Note that a common solution of several equations 3.7.7 could be considered, depending on several "time" parameters.
16.6
Return t o the Original Variables
16.6.1. Now we have to return from the variables (J,
J=0,l,...,n-1
(16.6.2)
over paths on the Riemann surface. 16.6.3. Proposition. The differentials (16.6.2) are Abel differentials of the first kind (see 8A).
290
Soliton Equations
and Hamiltonian
Systems
Proof. The genus of a Riemann surface can be calculated with the help of the Riemann-Hurwitz formula (see [Spr]) g = ^2ur/2-N+l
(16.6.4)
where N is the number of sheets of the surface, vr is the degree of a branch point (the number of sheets joining at this point minus one). In our case the Riemann surface of the function y/P{C) has two sheets, N = 2, the branch points are simple: vT = 1, and their number is 2n + 2 since the polynomial P(() is of degree (2n + 1 ) . Thus, g = n. The differentials tjjj could have singularities only in branch points. However, at a branch point (,a ^ oo the substitution £ — (a = z2 transforms this differential into
j = 1, • - •, n
(16.6.5)
1=0
where / uij — <Sy, Qj are o>contours (see 8A). Then* ipj{x,t2k+i)
= l^rjm
= 2^
1=0
UJJ Jo
m
°
n-1 ~ Z2 r3l(V° 1=0
+ Sn-l,l(l/2d2n-3)x
- C 2 (fc + (_„) + 3i2fc+l) •
This is exactly the Abel mapping of the gth symmetrical power of the Riemann surface into its Jacobi manifold. We know how {ipj} depend on x and t2k+i- If the Abel mapping is converted we shall know the dependence of {Cm} o n x a n d *2fe+i- After that we find the dependence of u on x and tik+i from (16.5.6), which is our aim. We use the Riemann theorem (see 8.A.4). If n
-^
S
/
u
i
=
*i
Jo
m=l °
then {Cm} are the zeros of the Riemann theta-function £(C*) = * ( . A ( C * ) - t f - K ) ,
^=(V-l,...,^n).
Further Restrictions
of the KP; Stationary
Equations
Fig. 1.
To find u we must calculate the sum of zeros of 1?(C*)- The symmetrical function of the zeros can be found by integration over a contour. Let us cut the Riemann surface along all the contours a» and Pi. Instead of handles on the surface holes are formed. (See Fig. la, b.) The edge of a hole consists of four edges of the cuts: af, (3f, a~ and /?r, which are being passed in the positive direction (the orientation on the Riemann surface is induced by the orientation on the complex plain Q. We consider the positive directions of the contours o^ and fa coinciding with o+ and [if. Let 7 be the contour consisting of all edges of the holes. The function i?(C*) is single-valued on the cut surface. 16.6.6. Lemma. The value of the integral
is a constant independent of ip.
292
Soliton Equations
and Hamiltonian
Systems
Proof. The values of d on two edges of a cut are connected with each other by the following formulas. For Uj(C,*) = f Wj we have
UjiOlpr = Wait,* - f »i = uAOlpt ~ sv (see Fig. 1). Equations (8.A.3) yield tf(C%r=0(OI#, HC)\a-
= nC)\at
• exp{-7ri(B« + 2^(C*))} •
Finally, dln#(C)\a-
=d]nd(C)\a+
-2iriLJi.
The integral under discussion becomes (uji
(16.6.7)
which is independent of {ipi}.
•
Replace this integral by the sum of residues. The residues are at the points Ci> • • • iCn> which are zeros of the function #(C*) and these residues are equal to Ci> • • • > Cn, and there is one more residue at £ = oo. This can be found by the substitution £ = z~2. Then (f; = {fji})
3(C) =e(-i; = 6(i;-K
ftz-V/y/Piz-2)
+ K+ (t/4) f
+ (i/2d 2 n + 3 )f n _ 1 • z +
since P(z~2) = rfL+3-2_2(2n+1) + 0{z'22n). 0 ( 0 ) = 8(xl> - K) +Yid6/dPj
•
• (-2z~3)dz)
0(z3))
Now (i/2d2n+3)rj,n.1z
3
+ {l/2)Y,d20/dPidpj
• {i/2d2n+z)2ritn-1
= 6ty - K) + i~d(i> - K) • z - ^6(4
• rj:n-lZ2
- K)z2 +
+ 0(z3)
0(z3).
Further Restrictions
of the KP; Stationary
293
Equations
The constant K is not important, it can be included into arbitrary constants tfi '. We have ,Pe --2 d , r e s z = 0 z dzlnV((,)-
_ (deW/dxf
-Oj^dH^/dx2 ^
e 2
_ -
d2\n6(j>) Qx2
and a = Ci + • • • + Cn - (d2/dx2)
ln6{i;{x,t2k+1)).
Substituting this into (16.5.6) we obtain u{x,t2k+i)
= 2d2n+i/d2n+3
+ 2a + {d2/dx2)ln6(ip(x,t2k+i))-
(16.6.8)
This explicit formula for the solutions was suggested by Its and Matveev [IM].
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Chapter 17
Stationary Equations of the Matrix Hierarchy
17.1
First Integrals
17.1.1. The most complete and practically important result of this and the next chapters is Proposition 18.1.16 given by Dubrovin [Dub77]. Its proof is almost independent of the material of this chapter. Nevertheless we decided to precede this proposition with the reasoning ([Di81(a)]) which shows its linkage to the Hamiltonian formalism: the procedure of integration of stationary equations is actually the standard Liouville integration of Hamiltonian systems with sufficiently many first integrals in involution, as we did it before for the KdV and GD hierarchies. In Chap. 9 the matrix AKNS-D hierarchy was introduced (9.1.14). Let diag U = 0. The corresponding stationary equations have the form Q=[A,Rm+1]=0.
(17.1.2)
Here Rm+i = Rm+i' where RB is the resolvent whose expansion starts from a constant diagonal matrix B with distinct nonzero elements, other terms containing no constants (more general equations [A, Y^o+ ckRk k] = 0 can also be considered). Equation (17.1.2) is a system of n(n — 1) ordinary differential equations of mth order; the order of the system is mn(n — 1). 17.1.3. Proposition. Lagrangian
Equation (17.1.2) is a variational one, with the
A= -trARm+2/(m + l).
(17.1.4)
Proof. Equation (17.1.2) means vanishing of the off-diagonal elements of the matrix Rm+\. From 9.3.9 we have 6A/SU = Rm+i for the off-diagonal elements. Thus, Eq. (17.1.2) is equivalent to 5A/5U = 0. • 295
296
Soliton Equations
and Hamiltonian
Systems
This implies that Eq. (17.1.2) is of Hamiltonian type. 17.1.5. Proposition. The quantities
where Ra are basic resolvents (see 9.1.9), are first integrals of Eq. (17.1.2). The corresponding vector fields are fyi ,«•»]. Proof. Equation (9.3.13) implies dFg = -tr[A, R%]Rm+i = 0 (where = symbolizes an equality by virtue of Eq. (17.1.2)), and that the characteristic of the first integral Ffea is -[A, R%\. The rest follows from 16.3.16. • Thus, the flows which are restrictions of the flows given by the equations of the hierarchy (9.1.14) to the phase space of Eq. (17.1.2) correspond to the first integrals 17.1.5. 17.1.6. Proposition. The first integrals 17.1.5 are in involution. Proof. The vector fields {d[A,R<*]} commute.
•
It is clear that not all of the first integrals Fj? are independent since the phase space is finite-dimensional. We suggest now a second way to construct the first integrals. Let m
^Rkzm-k.
R = fc=0
(In Chap. 9 we used another notation, V.) 17.1.7. Lemma. R satisfies the equation R' + [U + zA,R} =
-[A,Rm+1}.
Proof. Let us multiply the recurrence relation 9.1.4 by zm~l and sum up from i = 0 to i = m. Taking into account [A, Ro] = 0 we get the required.
• 17.1.8. Corollary. Equation (17.1.2) is equivalent to the fact that the polynomial in z : R(z) satisfies the resolvent equation & + [U + zA,R] = 0. 17.1.10. Proposition. The coefficients of polynomials in z trRk,
k =
l,...,n
(17.1.9)
297
Stationary Equations of the Matrix Hierarchy
are first integrals of Eq. (17.1.2). There are mn(n — l ) / 2 nontrivial first integrals of this kind. Proof. If R is a resolvent (by virtue of Eq. (17.1.2)), then so is Rk. This implies d tr Rk = — tr[U + zA, Rk] = 0. Hence tr Rk are first integral. The quantity tr Rk is a polynomial in z of degree mfc. The higher m + 1 terms of this polynomial coincide with those of the series tr(z m i?) fc , while tr Rk are absolute constants (not only by virtue of the equation). Therefore the higher m + 1 terms of tr Rk are trivial first integrals. There remain mfc + 1 — (m + 1) = m(fc — 1) nontrivial terms. In all, there are 2fe=i m ( k — 1) = rnn(n — l ) / 2 nontrivial first integrals. • Below it will be proved that these first integrals can be expressed in terms of {Fj?} which implies that they are in involution. More or less cumbersome calculations prove that they are independent. Then the Liouville theorem guarantees that Eq. (17.1.2) is integrable. This integration will be performed effectively. Instead of tr Rk, the coefficients of the characteristic polynomials n
f(w,z) = det(R~wl)
n l
= Y^Mz)w
= Yl
(n-l)m
E
J
^zfe
i=o i=o fc=o can be considered. Here also, for each fixed I the higher m + 1 coefficients yield trivial first integrals; thus, the nontrivial ones are obtained for each of / = 0 , . . . , n — 2 when fc = 0 , . . . , (n — / — l)m — 1. In all, there are Y^i=o [(n ~ 0 m — m] = rnn(n — l ) / 2 of them. Notice, for the further use, the formula n
Y, Ji,(n-i)m™lz{n-l)m 1=0
n
= det(zmR0
- w • 1) = H(zmba
- w).
(17.1.11)
0=1
Let a point of the phase space U, U1,..., t/( m _ 1 ) be fixed. Then the equation f(w,z) = 0 determines an algebraic function w(z). Two distinct points of the phase space determine the same algebraic function w(z) if they belong to the same level surface {Jik — const.}. The Riemann surface of this function has n sheets. The points of the Riemann surface will be denoted by the letter P : P = (z,w(z)). We consider a generic case when the Riemann surface does not have singular points where / = fw = fz — 0. 17.1.12. Proposition. If all ba are distinct, the genus of the Riemann surface of the function given by the above equation f(w,z) = 0 is g —
298
Soliton Equations and Hamiltonian
Systems
mn(n —1)/2 — n + 1 . The number of the branch points, taking into account their multiplicities, is mn(n — 1). Proof. We use a very general theorem which is proved in Appendix 17A to this chapter. First of all, we must explain what is the Newton diagram, or the Newton polygon. Let a polynomial equation be given: F(w, z) = wn + iViOz)™"- 1 + • • • + P0(z) = ^aklzkwl
= 0,
where h
Pi(z) = ^2aMzk. fc=0
One marks all points (k, I) on the integer lattice such that aki ^ 0. This set of points will be called the frame of the diagram. Then the convex hull of the frame is the Newton diagram. The boundary of the diagram consists of two branches: the right-hand side branch is convex to the right, and the left-hand side is convex to the left. 17.1.13. Theorem. Suppose (i) The polynomial is non-singular, i.e. there are no points where F = FW = FZ=
0.
(ii) If a side on the right-hand branch of the Newton polygon contains more than two points, say ( & i , / i ) , . . . , (ks+\,ls+i) then the polynomial s s_1 aklht + afe2(2t + 1- ak,+lis+1 has no multiple roots. (Some of a^u may be zero.) Then the genus of the Riemann surface of the function w(z) determined by the equation is equal to the number of lattice points inside the polygon. Some examples, (a) The equation is F(w,z) = Ylakizkwl
= 2w3 - 3z3w2 + 2z = 0.
Mark the points (0,3), (3.2) and (1,0), and take their convex hull. Inside there are three points of the integer lattice, so the genus is 3. (b) A generic nth order curve J2k+i
Stationary
Equations
of the Matrix
299
Hierarchy
In the example (a) we use the condition (ii) since three points (3, 2), (2, 1) and (1, 0) belong to one side of the polygon. We have a^ — —3, a,2i = 0 and aio = 2. The polynomial —3t2 + 2 does not have multiple roots and therefore the theorem is applicable. The last example will be Proposition 17.1.12 we have to prove. The Newton polygon is in this case the triangle with the vertices (0,n), (ram, 0) and (0,0) (if JQQ = 0, then at least one of J\Q and Joi is not zero, otherwise z = w = 0 would be a singular point; this does not change the number of inner points in the Newton polygon). The condition (ii) of the theorem holds since, by virtue of (17.1.11), X2"=o Ji,{n-i)mtl = li/3=i(&a ~~ 0 which does not have multiple roots. The Ith row has (n —Z)m— 1 inner points, I = l , . . . , n — 1 . In all, there are Yl?=i [(n—l)m—l] = mn(n—1)/2—n+1 points. Thus, the genus is g = mn(n —l)/2 — n + 1. The number of branch points ]T)i/r can be found from the Riemann-Hurwitz formula (16.6.4) (N = n). It is mn(n — 1). • A one-dimensional eigenspace corresponds to the eigenvalue w(z) of the matrix R(z). Let g(P) be the spectral projector onto this space, so the relations g2(P)=g(P),
tvgk(P)
= l
hold. 17.1.14. Proposition. The projector g(P) can be written as
s(i,) = /; 1 EJiWE» k « l " 1 " l l 1=1
where fw =
fc=0
df/dw.
Proof. Let g(P) be a matrix defined by this equation. If £ is the eigenvector corresponding to the eigenvalue w(z) then
sTO = /j1£-M2)i>fe-«''-1-fc« n
300
Soliton Equations
and Hamiltonian
Systems
If £ is an eigenvector for another eigenvalue wi / w, then
i=l
= f^[f(w,z)
fc=0
- f(Wl,z)]/(w
1=1
-Wl)-£
X
= 0.
D
17.1.15. Proposition. g(P) is a resolvent by virtue of Eq. (17.1.2). Proof. R is a resolvent. The product of resolvents is a resolvent, so is the sum. The rest follows from 17.1.14. • 17.1.16. Proposition. asymptotic relations
If z —> oo then n branches of w(z) satisfy the
w(z) = zmba + Oiz"1),
a=l,...,n,
where ba are elements of the diagonal of R0 = B. Therefore, there are no branch points at infinity. Proof. Compare the equations: (a) det(.R — wl) = 0 and (b) det(zmR — wl) = 0. The highest m+1 terms in the asymptotical expansion in powers of z~l of the functions R and of zmR coincide, hence in the asymptotics of w(z) for z —> oo the highest m + 1 terms coincide too. As to the equation (b), the eigenvalues of the matrix (mR are exactly zmba. • We denote the infinite point on the sheet where w(z) ~ bazm by {a}. 17.1.17. Proposition. When P —> {a}, the asymptotics g(P) = Ra +
0(z-°°)
holds, by virtue of Eq. (17.1.2) (the latter means that the higher derivatives f/ (m) ) £/ (m+i) ) s h o u l d be eliminated with the help of (17.1.2)). Proof. The fact that the higher m + 1 terms of g(P) and Ra coincide, i.e. g(P) — Ra+0(z~m~1), can be established in the same way as in 17.1.16: they are spectral projectors of matrices R and zmR, asymptotically coinciding i n m + 1 terms. The fact that this coincidence holds also for the rest of the terms follows from the equalities gk = g and (Ra)k = Ra for any k and from a reasoning of the type of 9.1.7. For example, let m = 1. Temporarily denote Ra = RQ + RIZ^1-] , and g(P) = go + giz"1 H . We know that Ro = go and Ri = gi. From (g(P))2 = g(P) we have g0g2 + g2go + gj = 92, or i?o52 + 02-Ro + R\ = 92- Similarly, R0R2 + R2R0 + R\ = R2 where from Ro(92 - R2) + (92 - R2)Ro = (92 - Rt), or (bi + bj - 1)(52 - Ri)u = 0.
Stationary
Equations of the Matrix
301
Hierarchy
Now, 52 = R2, and in the same manner we can proceed to g% — R3 etc. If, by accident, bt = 1 — bj, we can consider (g(P))3 and (Ra)3. • As it is seen from 17.1.14, the elements of the matrices g(P) : g%j{P) are rational functions of z and w, i.e. rational functions on the Riemann surface. The poles are branch points, fw = 0, with the same orders. Their number is mn(n — 1). At infinity there are neither poles nor branching. 17.1.18. Proposition. If gtj(P) = 0 at a point P of the Riemann surface, then at this point all the elements either of the same row, gik(P) — 0,k = 1 , . . . , n or of the same column, gkj = 0,fc= 1 , . . . , n also vanish. Therefore the divisor of zeros of the element gik(P) in the finite part of the Riemann surface splits into two divisors: d\ + dS , the divisor of the zeros of the ith row and the divisor of the zeros of the jth. column. Proof. The projector g(P) onto a one-dimensional subspace is a matrix of rank one. Locally, in a neighborhood of a zero of g^ (P) a decomposition g%i =
V-'M )ij
=
OictKP'Ot Qj)
^aj
uaj\Oji
&<*) ^ia •
This implies the following. If i ^ j and a coincides with one of i or j , then at the point {a} there is a simple zero. If a 7^ i,j then at {a} there is a double zero. We have on all sheets 2 + 2(n — 2) = 2n — 2 zeros at infinity. If i = j , then there is no zero in {a} for a = i = j and a double zero for a ^ i. In all we have 2(n— 1) = 2n —2 zeros at infinity also in this case. • 17.1.20. Corollary. Functions gij have mn(n — 1) — 2n + 2 zeros in the finite part of the Riemann surface. Proof. The meromorphic function gij has the number of zeros equal to the number of poles. The latter is mn(n — 1). From this number 2n — 2 zeros at infinity must be subtracted. • 17.1.21. Proposition. The degrees of the divisors d\ ' and dj ' are deg(^ a ) ) = deg(4 2 ) ) = mn(n - l ) / 2 - (n - 1) (recall that the degree of a divisor is the number of its points taking into account their multiplicities).
302
Soliton Equations and Hamiltonian
Systems
Proof. First of all, deg(d\ ) does not depend on i (and deg(d^ ') on j). This follows from the fact that the meromorphic function gtj/gkj has equal numbers of zeros and poles. At infinity the numbers of zeros and poles coincide too. Therefore they are equal also in the finite part of the Riemann surface. Moreover, the zeros and the poles of gij/gkj originate from zeros of gij and g\.j, not from their poles which are common, zeros of fw, and cancel. Therefore d e g ^ ) + deg(d$2)) = d e g ^ ) + deg(df]) and deg« ( 1 ) ) = deg(4 1 ) ). Now, the relation deg(cQ ) = deg(e6 ') reflects the fact that rows and columns are equal in rights which is more or less evident. If the reader is convinced by this reasoning he can consider the proof as completed. The rigorous deduction can be carried out, for example, as follows. Let us pay attention to the dependence of R(z),w(z) and g(P) on the point of the phase space (U, U',..., U^m~^). This dependence is continuous. Therefore, if the point moves on the level surface {Jik = const.} the roots of g^ change continuously, they cannot appear or disappear (if two poles do not meet which can be avoided). A zero also cannot pass from the divisor d\ to divisor d^ '. Thus deg(cQ ') = const., i.e. is independent of the point of the phase space. On the other hand, it is easy to see that the resolvent equation (17.1.9) admits the following transformation (an involution) R(U(x)) i-> RT(U'V(—x)) where the superscript T denotes the transpose, U(—x) denotes changing the sign of odd derivatives: U,—U',U", —£/•'", This involution does not change R0, therefore R(U{x)) = RT(UT(-x)), and R(UT(-x)) = RT(U(x)), g(UT(-x)) = T r T T g (U(x)). The point U (-x) = (U , -U' ,...) of the phase space belongs to the same level surface {Jik = const.} as U(x). Thus, deg(d\ '(U(x))) = deg(^ 1 ) (C/ T (- a ; )))- F r o m t h e equation g(UT(-x)) = gT(U(x)) it follows 1) T 2) that deg(^ ([/ (-a;))) = deg(d\ (U(x))). We have obtained deg(4 1 } ) = d e g ( c ^ ) , and this is a half of the full number of zeros (see 17.1.20). • 17.1.22. Proposition. The relationship between the two kinds of first integrals, {Fg} and {J^}, is given by the formula
F? = -wa,k where wa^ is the coefficient in z~k in the asymptotics of w(P) when P -> {a}. (These coefficients can be easily expressed in terms of the coefficients {Jik}-)
Stationary
Equations of the Matrix
303
Hierarchy
Proof. We have F? = -tr(RmR%
+ ••• + -RoiC+fc) =
-tv(RRa)k
(the subscript k denotes the coefficient in z~k). In this relation Ra can be replaced by the expansion of g(P) when P —> {a}, see 17.1.17, then Fg = -tr(Rg(P))k, P ->• {a}. The matrix g(P) is the spectral projector of R, hence tx gR is equal to the eigenvalue w(P), and this gives the required relation. D 17.1.23. We want to explain why we have introduced two kinds of first integrals and established a connection between them. As a rule, {Jik} are more convenient to deal with. They determine a Riemann surface, algebraic function w(z) etc. On the other hand, the first integrals F£ are Hamiltonians of restrictions of the phase flows given by the equations of the hierarchy (9.1.14) to the finite-dimensional phase space of the stationary equation (17.1.2). Exploiting this fact, we can write explicit formulas of solutions.
17.2
Hamiltonian Structure of Stationary Equations
17.2.1. Proposition. The Hamiltonian of the stationary equation (17.1.2) is n
ri = ] P
aawa,2
(aa are the diagonal elements of the matrix A). Proof. One must verify the equality H' = —tr U'SA/SU where A is the Lagrangian (17.1.4) (see 16.3.10). Let RA = £ a a . R Q b e t h e resolvent which begins with R0 = A. We have (a) (itf)' + [U, itf] + [A, Rf} = 0,
(b) (R?)' + [U, Rf] + [A, R?} = 0,
being the first two of the recurrence relations. Equation (a) implies that the off-diagonal elements in Rf are equal to those in U. Then (b) yields U' = -[A,R£]. NOW - t r U'SA/SU - - t r U'Rm+1 = tr[A, R$}Rm+i = - 5 3 a a d F ? =d^2
= tr £
««w«,2 - dH .
Let us express the Hamiltonian in terms of {Jik}-
aa[A, R%}Rm+1
304
Soliton Equations and Hamiltonian
Systems
17.2.2. Proposition. The second and the third of the nonvanishing coefficients of the asymptotics of w(P) when P —> {a}: w(P) = bazm + Paz'1 + qaz~2 + 0(z~3) are n-2 Pa = ^2 Jl,(n-l-l)m-lba 1=0
J J (&a ~ bp)~l /3#a
,
n-2 Vac = 22 " ^ , ( n - i - l ) m - 2 & L J Q (ba ~ W 1=0 0jta
_ 1
•
Proof. Let us expand f(w, z) into a sum f(w,z)
= f0(w,z)
+ fm+l(w,
Z) + fm+2(w,
z) + • • • ,
taking together terms with the highest power zk for every I, the next to the highest, and so on:
f0(w, *) = £ Jl.in-DWz^"1
= J} ( ^ ^ - W) >
1=0 n-2
a=l
1=0 n-2
f2(W,z)
YJ-Jl,(n-l-±)m-2WlZ(n-l-1)m-2.
= 1=0 m
The series w = baz + PaZ"1 + qa(,~2 + ••• must be substituted into the equation f(w,z) = 0. We have fo(bazm,z) = 0, i.e. all the terms with the power zmn vanish. The next power is 2 " i n -("»+i) ) this yields Padfo(ba, l)/dw + fm+i(ba,
1) = 0
and the next, z"™-(™+2): qadf0(ba,
l)/dw + fm+2(ba,
1) = 0
whence n-2 Pa = -fm+l(ba,
l)/(dfQ(ba,
\)/dw)
similarly for the second formula.
= 2~2 Jl,(n-l-\)m--J>a 1=0
J | (pa - bp)'1 0^a
,
D
Stationary
Equations of the Matrix
Hierarchy
305
17.2.3. Corollary. The Hamiltonian is n /
n—2
a
H=Y^ <x^2
Jl,(n-l-l)m-2bla Y\_ (b<* ~ &/?)_1 •
a=\
/3#a
1=0
We pass now to the symplectic form. 17.2.4. Proposition. The one-form related to the Lagrangian A is
a=l
where i and j are any two values of the matrix indices (if another pair is taken Q^ will be changed by a nonessential differential). Proof. We use Proposition 9.3.6. From 6A = tr 5U • SA/SU + 89,^ (17.1.4) we find
and
«(1) =
— tr(<5<W* -
^iabaSipai\m+1
l,a
E , l,a
, \ ^ ^ , ,, ,, , 1- 2_^
•
We have Y^i tpai
•
17.2.5. Besides the above-mentioned involution transforming solutions of the stationary equation to other solutions, there is another obvious group of symmetries depending on n — 1 parameters. Namely, U^KUK-1,
AT = diag(fci,...,fc n ) = const.,
fc,^0.
(17.2.6)
306
Soliton Equations and Hamiltonian Systems
This transformation sends a solution to another. The related infinitesimal transformations are Ui->U + e[C,U],
C = d i a g ( C i , . . . , C n ) = const.
(17.2.7)
Thus, the vector fields {d[c,u\} correspond to the transformation group (17.2.6). Let us find the Hamiltonians of these vector fields. 17.2.8. Proposition. The Hamiltonian of a vector field d[c,u] *s n
n
hc = tr CRm+1 = Y, C*F? = ~ H c*wa,i n
n—2
==
^2Ca^2 a=l
J
l,(n-l-l)m-lbla Y[ (6« - &/3)_1
1=0
/3#a
(we draw reader's attention to the fact that tr CRm+i does not vanish by virtue of the stationary equation as it may seem at first sight; this expression involves only diagonal elements of Rm+i, while the stationary equation means vanishing of the off-diagonal elements). Proof. At first we take the first expression for he, he = t r C R m + i . We have dhc = tr CR'm+1 = - t r C[U, Rm+i] - tr C[A, i? m + 2 ] = - t r C[U, Rm+i] = - t r Rm+i[C, U] which implies that he is a first integral with the characteristic — [C, £/]; the corresponding vector field is <9[c,c/] a s required. Further, according to 17.1.5, n / J CaF^ a=l
n = —tr 2_^ Ca{RmR" a=l
+ ••• +
fio^m+i)
n
n
= —tr 2_^ Ca(Rm+lRo a=l = - t r RBRc\m+1
+
---
+ tr CRm+1
+ RoRm+l) +
= tr CRm+1
t r
Z^i CaRm+lRo a=l
= hc ,
i.e. we got the second expression for he- The third and the fourth follow from 17.1.22 and 17.2.2. •
Stationary
Equations of the Matrix
307
Hierarchy
17.2.9. We summarize the obtained results. The higher m + 1 quantities Ji,(n~i)m, Ji,(n-i)m-i, • • •, Ji,(n-i-\)m for each I are absolute constants; the next, J(,( n -(-i)m-ij is the Hamiltonian of a trivial symmetry; from the next ones: {Ji,( n -i-i) m -2} * n e Hamiltonian of the equation, %, is combined, 17.2.3. All the others yield nontrivial symmetries of the stationary equation. These are flows given by the equations of the matrix hierarchy (9.1.14) restricted to the finite-dimensional phase space of the stationary equation (17.1.2). The flows commute therefore the first integrals are involutive. 17.2.10. Proposition. Vector fields d\q,v] {Ji,(n-i-i)m-i
= Ci = const.} ,
are
tangent to the manifold Uc-
Z = 0,l,...,n-2.
(17.2.11)
Proof. This follows from the fact that the first integrals are in involution.
• From now on the stationary equation as well as the symplectic form and the Hamiltonian will be considered as restricted to the invariant submanifold Uc of the phase space which has the codimension n — 1. 17.2.12. Proposition. On the submanifold Uc the form fi degenerates on the vector fields corresponding to trivial symmetries. Proof. The Hamiltonians of these vector fields are constant on Uc since they are linear combinations of {Jit(n-i-i)m-i}D To restore the non-degeneracy of the form one must factorize the manifold Uc with respect to the trajectories of the trivial symmetries. In other words, points of a new manifold U'c will be the sets (U,V,... ,[/( m - 1 )) satisfying Eq. (17.2.11); two of such sets being identified if U^ = KU^K-1,
i = 0,...,m-l,
where K is a constant diagonal matrix. Evidently, dim Uc = mn(n — 1) — (n — 1),
dim Uc = mn(n - 1) - 2(n - 1).
The form CI can be transferred to the factor-manifold Uc since it degenerates just on the vector tangent to the trajectories along which the identification is performed. On U'c the form is non-degenerate. The dimension of the new manifold U'c is mn(n — 1) — 2(n — 1). The number of the remaining nontrivial first integrals is mn(n — l ) / 2 — (n — 1), i.e. half of the dimension, as it was earlier.
308
17.3
Soliton Equations and Hamiltonian
Systems
Action-Angle Variables
17.3.1. Proposition. On the manifold U'c the form il^ fi(1> = - £ t u ( i > ) (fy ' 9)ij 9ij
can be written as
= -5>(p) dgik9ij• 9kj
(17.3.2)
a,fc
(the subscript 1 means, as usual, the coefficient in z l of the asymptotic expansion P —> {a}; the sum is over all the sheets a = l,...,n). Proof. It must be shown that up to a complete differential this expression coincides with 17.2.4. We have
-E™( p )
{Sg • gh
= -]>>(P)|_
($g • gh 9i3
9ij
£™(p)l i •
m+l
(tg • 9h 9ij
(the subscript — m denotes the coefficient in zm). The first term coincides with the expression 17.2.4, owing to the asymptotics 17.2.2 and 17.1.17. As to the second term, we have w(P)\\ = pa which is constant on He, and S^ffjfc '9kj/9ij\o is a complete differential (see 17.1.17 and the expressions ka
for Ra in 17.1.19).
•
In this proof the restriction on Uc is essential and we have used the fact that pa is constant. We have another representation of O^1) given below. 17.3.3. Proposition. The form fit1' can be written as fi(D = _
Y, p*ed\
w(P*)8zP.+^2alk(J)SJlk, (i)
l,k
v « , the coeffiwhere the first sum is over all the points P* of the divisor d\ cients aik of the second sum depend only on the values of the first integrals {Jik},zp- is the projection of a point P* on the complex plain z. (An explanation: if a point of the phase space experiences a small displacement {SU^}, then the first integrals also change by SJik and the points of the divisor d\ ' are displaced by Szp-.)
Stationary
Equations of the Matrix
309
Hierarchy
Proof. The right-hand side of Eq. (17.3.2) can be written as the sum of residues of the differential n^
=
_ J2 res {Q} w{P){89
' 9)iidz.
(17.3.4)
Replace this by the sum of residues in the finite part of the Riemann surface with the opposite sign. There are poles at two kinds of points: at the points of the divisor d\ ' (but not of d\ , the zeros here being cancelled out) and at the branch points. Let P* € d\ '. In the neighborhood of this point, gtj can be represented as ifiipj where Y^'Pi^Pi = 1- We have
— — - = w(P )ozp- . ViVo
Now we consider the residue at a branch point. The coordinates of a branch point (WQ,ZQ) depend solely on the coefficients of the polynomial f(w,z), i.e. on {Jjfc}. Therefore the residue at the branch point can only be of the form ^ aik(J)5Jik, which completes the proof. • We provide more detail about the coefficients aik'. Let Po(^o,^o) be a branch point. The expansion of f(w,z) in its neighborhood is b(z — ZQ) + d(z - z0)(w - w0) + e(w - w0)2 H . For simplicity we consider now the generic case, b ^ 0, e ^ 0. Then w — w0 = ^/z — zo • ip{y/z — zo), where >p is holomorphic,
w(P)(5g(P)-g(P))ij/gij(P)dz = - r e s w(5fw/fw)(g
• g)ij/gijdz
+ res wf~l(Sg
•
g)ij/gijdz.
The first term is — res w(5fw/fw)dz = — (l/2)w (Po)Szo, and the second 1 2 term vanishes since iesZo(z — zo)~ ^ dz = 0, which can be verified by substitution z — Zo = (?• It remains to express 5ZQ in terms of {SJki}Let us vary the equation f(w,z) = 0 : ^Twlzk5Jik + fwSw + fzSz = 0. At the point Po we have SzPo = —fz'1(wo,zo)Y^ik'wozo^^i,k^ whence 1 +1 a,fc = ( l / 2 ) / - ( t « o , z o K ^ . 17.3.5. We consider now the Lagrange manifolds {J/fc = Cik = const.}. On each of these manifolds coordinates have to be chosen. The number of coordinates must be equal to the dimension of the manifold, i.e. to mn(n — l ) / 2 —(n—1). For all the points of such a manifold the function f(w, z) is the
Soliton Equations
310
and Hamiltonian
Systems
same, so is the Riemann surface. It is convenient to choose the coordinates in which the form £1^ is written with separated variables. According to 13.3.3 these are the points of a divisor a\ ' with an arbitrary but fixed i. On each submanifold { J = C}, the form U^ easily can be integrated:
nw = ^K w ,j), v(41),j) = - £
p"
/ w(P)dz.
17.3.6. Proposition. The angle variables conjugated to the action variables Jik,k < (n — I — \)m — 2 can be obtained by the Abel mapping of the divisor
e~
Proof. The angle variables conjugated to the action variables Jik are (see Sec. 16.5) eik = -dV/dJik+aikWe have eik = }
fP' dw(P)/dJ dz lk
/
+ aik
Jp0 p*
zkdz + ojfc .
(17.3.7)
It remains to note that there are written here integrals of Abel differentials of the first kind, i.e. of the holomorphic differentials. Indeed, at the branch points fw = 0 the differential is holomorphic because it has the form (z — ZoY^dz, where v is the multiplicity of the branch point. At infinity we have w ~ zm,fw ~ wn-x ~ z"^"- 1 ), therefore f~1wlzk grows as zk+m(l-n+i)_ Taking into account that k ^ (n — / — l)m — 2 we conclude that at infinity the differential is z~2dz. • Note that the above written differentials of the first kind are not normalized. 17.3.8. Proposition. In the variables (J, 6) Eq. (17.1.2) can be integrated: Jik = Jik >
Oik= {
0° fc -5>« 6 « n ^ - 6 " ) " 1 - * '
k=
6°k,
otherwise.
Proof. This follows from 6lk = (dH/dJik)x
(n-l-l)m-2
+ 6°lk, see (16.5.2), and 17.2.3.
•
311
Stationary Equations of the Matrix Hierarchy
The constants aik can be included into initial constants 6°k, therefore they play no part here. 17.3.9. Recall that in the phase space the vector fields Ylca®[A,R?] commute with the vector field of the stationary equation. If t is a parameter along the integral curve of such a field then we can find the dependence of the point of the phase space on this parameter, i.e. integrate the nonstationary equation in the finite-dimensional manifold of solutions of the stationary equation. 17.3.10. Proposition. In the variables (J, 6) the dependence on x and t is given by the formulas J
lk = Jlk i fyfc = tfk ~ &k,(n-l-l)m-2
-^2cadwa
' 2 _ , a « & a ' I I ^a ~ M
_ 1
"x
-t,
where 6^ are constants independent of x and t. Proof. c
It suffices to recall that the Hamiltonian of the vector field
J2 ad[A,R?]
IS J2 crF"
= - YJ CaWa,r-
•
17.3.11. To obtain the inverse formulas it is more convenient to use the normalized Abel differentials of the first kind. If a system of contours aj and Pi is chosen (see 8.A.1), linear combinations n-2 ( n - ( - l ) m - 2
w» = ^2 1=0
Yl
citikf~1wlzkdz,
i=
l,...,g
fc=0
can be found such that / variables
Uj = dij.
Instead of 6ik we must consider
e< = 5 Z Ci.Jfcftfc
(17.3.12)
l,k
which also depend on x and t linearly. 17.3.13. The points of the divisor d\ were taken as coordinates. The points of another divisor
312
Soliton Equations and Hamiltonian
Systems
zero (on the Jacobian, i.e. modulo the lattice periods). If D is the divisor of branch points, and S = ^2{i} t n e n A(4X) + 4 2 ) ~D + 2S-{i}-
{j}) = 0.
If we denote r?f'2) = A{df'2)) then IJ<2) = -rjV constant vector on the lattice of periods.
17A.
(17.3.14)
+ C y , where C^ is a
Appendix. Genus of the Riemann Surfaces and the N e w t o n Diagram
17.A.1. This is the outline of the proof of Theorem 17.1.13. This theorem can be found in [Khi]. However, the exact condition (ii) is not formulated there, so the theorem holds just in a generic case. The tools in [Khi] are very subtle and sophisticated. We give here a quite elementary proof. If F(0,0) = aoo = 0, then at least one of aoi and
D(z)=H(wi(z)-wj(z))2, i<j
where Wi(z) are values of w on different sheets over z. According to the theorem on symmetric functions, the discriminant is a polynomial in z. If there is a branch point of order p over a point z0, then w(z) on p +1 sheets have the form Wi(z) = w0 + b(z -2;0)1/(p+1) H where different branches of the (p+ l)th root correspond to different i. There can also be other branch points over the point z0. For all of them wo will be different. The contribution of the branch point to the discriminant is (z - z 0 ) ( 2 / ( p + 1 ) ) ( p + 1 ) p / 2 = (z — z0)p. The proven fact has the following corollary: 17.A.2. Lemma. The sum of orders of all branch points over some Zo equals the multiplicity of ZQ as a root of the discriminant. Thus, the sum of
Stationary
Equations of the Matrix
Hierarchy
313
orders of all branch points in the finite part of the Riemann surface equals the degree of the polynomial D(z). If we find the asymptotic of all branches Wi(z) at infinity, we shall know the degree of D(z) as well as the number of infinite branch points. All this information is contained in the right-hand part of the Newton polygon, i.e. the part which is convex to the right. All asymptotic formulas have a form w = \zp + (terms of lower orders). One must find p such that at least two terms of the polynomial F(z, w) have the same degree to be able to cancel, and all the other have smaller degrees. If a,kizkwl is a term of the polynomial, then it has the order zlp+k when z —> co and w(z) follows the above asymptotic behavior. The equation Ip + k = const, determines a straight line on the (k, I) plane. At least two points of the frame of the diagram must belong to this line, and all the others must have smaller Ip + k, i.e. to lie to the left of this line. Thus, all p's can be obtained as the slopes of the segments of the right part of the Newton diagram, p = (fc' - k")/(l" - I') where (fc'.Z') and (k",l") are two points of the frame belonging to one segment. Even more is true: this procedure provides exactly n asymptotic expressions, and all of them have distinct leading terms, see below. 17.A.3. Lemma. Let Ai(ki,li),... ,As+i(ks+i,ls+i) be all lattice points belonging to a straight segment of the diagram. Then a branch point of order AZ —1 where AZ = l\ —l^ = h~h = • • • is assigned to each part A1A2, A2A3,..., AsAs+i of the segment, all told s branch points and sAl = h — ls+i sheets, with the asymptotic behavior w ~ XiZlp+k where all coefficients Aj are distinct. (In particular, when Al = 1, the branch points are of order 0, i.e. separate sheets.) The first and the last of the points Ai,..., As+i necessarily belong to the frame, the others must not. Nevertheless, we can consider all of them as belonging to it, with the coefficients ajy, maybe, zero. Proof. The numbers |fcj+i —fcj|and Zj — Zj+i for two neighboring points are relatively prime and the fraction p = (fc,+i —fcj)/(Zj— h+i) is reduced, hence the order of a branch point with the asymptotic behavior w ~ zp is AZ-1. The main terms of the asymptotic expression of Y^\ akitliZkiwli must cancel, i.e. s+l
^2aki>liXhzliP+ki=0. 1
314
Soliton Equations and Hamiltonian
Systems
Dividing this by Xls+1 and letting t = XAl we have s+l
5> f e i ,^ s + 1 -* = 0.
(17-A-4)
1
By assumption (ii), all the roots of Eq. (17.A.4) are distinct, we obtain s distinct values of t. A branch point corresponds to each of them. Then there will be sAl = l\ — Zs+i distinct values of A. • 17.A.5. Corollary. Newton diagram method gives n branches of asymptotic expressions, all with distinct leading terms. Proof. Indeed, a straight segment of the diagram gives the number of branches of the asymptotic expressions equal to the projection of this segment on the vertical axis. If all the segments are taken into account, one obtains n branches. • 17.A.6. Lemma. The degree of the discriminant is deg D(z) = 2 X a ~ ^(0 where k = k{l) is the equation of the right-hand part of the Newton diagram. Proof. The degree of the discriminant can be found by its asymptotic behavior when z —>• oo since the behavior of all Wi(z) is already known. Let k = k(l) be the equation of the right-hand part of the diagram. The number k(l) is not necessarily integer. A sheet of the Riemann surface in the neighborhood of z = oo is assigned to each I = 1 , . . . , n, with the asymptotic behavior wi(z) = inzVl + • • • and pi = k(l — 1) — k(l). If the exponents pi for two Vs are equal, the coefficients pi are distinct. The term with a smaller / in the difference (w^(z) — wi2(z))2 can be omitted since it is of a lower or equal order and in the latter case the leading terms do not cancel. Thus, one has
D(z) ~ f[Wl(z)2V-V ~ JJz2(fc(i-i)-*(D)('-i) i=i
l=i
and n
n —1
deg D(z) = J2 2(fc(/ - 1) - k{l))(l - 1) = 2 53 kV) 1=2
(since k(n) = 0).
1=1
•
The obtained result can be linked to the number of interior points of the diagram. First, note that if k(l) is integer, it is equal to the number of
Stationary
Equations of the Matrix
315
Hierarchy
interior points of the diagram with the same I, plus 1. Second, if k(l) is not integer then its integer part is equal to the number of interior points with the same I. This happens when we have a branch point, say of the order q. A sum of all fractional parts {k(l)} = k(l) — [k(l)} (where [k(l)} denotes the integer part) corresponding to the same branch point is equal to (q — l ) / 2 . This follows from the formula: if p and q are relatively prime then 9-1
r
^
'Si'!}- 1 (Indeed,
*!-<«-'>; ~.p I—
-
[ft-Of]
is an integer. It is less than 2 and larger than zero, i.e. 1.) The total number of all involved non-integer k(l)'s is ^2(q — 1), which is a sum of orders of all branch points at infinity. The number of integer fc(Z)'s is, correspondingly, n — 1 — YKQ ~ -0- Denoting by I the number of the interior points of the diagram, we have deg D(z) = 21 + 2(n — 1 — E ( 9 - 1)) + E(<7 - 1) = 2 / + 2(n - 1) - £((? - 1) and deg D{z) + Y,(V - 1) = 2(1 + n - 1). The left-hand side of this equality is nothing but / , the number of all branch points taking into account their multiplicities. The Riemann-Hurwitz formula for the genus g of a Riemann surface
g=l-n
+ l,
where / is the sum of orders of all branch points and n is the number of sheets, i.e. the degree of the polynomial F(w, z) with respect to w completes the proof of the theorem, 9=-
2(J + n - l ) , r g --n+l=/.
It is also not difficult to prove the following proposition:
•
Soliton Equations and Hamiltonian
316
Systems
17.A.7. Proposition. All holomorphic Abel differentials are given by zk-iwi-i
Uki =
p
dz ,
where (k, I) are coordinates of all interior points of the Newton diagram. All told, there are g such differentials.
Chapter 18
Stationary Equations of the Matrix Hierarchy (Cont'd)
18.1
Baker Function. Return to Original Variables
18.1.1. In the previous chapter the variables were suggested in which solutions of the stationary equation were linear in x and tka. We will get it again and even in a more straightforward way, and also we will show how to return to the variables ( £ / , [ / ' , . . . , [7^ m_1 ^). We closely follow here the work by Dubrovin [Dub77]. So let us have a solution U(x) of the stationary equation (which can also depend on parameters t; we shall not write these parameters explicitly). R becomes a function of x, so does the projector g(x, P), P is a point on the Riemann surface. The divisors d\ and
ipR = wip.
(18.1.3)
Substituting this into the equation g' + [U + zA, g] = 0 we obtain
whence Vi + 5 Z M»«V" +
Za
Wi
=
Vt '
~^'j + / L ^aUai
+
Za
3^i
=
^3
A depends on x and z but not on % and j . By the substitution exp(/ \dx)ifi, tjjj H4 exp(— J Xdx)ipj the terms with A can be removed without violation of the property gij = iptt/jj. Thus, the vector-column 317
318
Soliton Equations and Hamiltonian
tp = (
Systems
and the vector-row ip = (ipi, • • • ,ipn) satisfy the
V?' + (U + zA)
-ip' + ip(U + zA) = 0.
(18.1.4)
The functions ip and ip satisfying Eqs. (18.1.3) and (18.1.4) are called the Baker and the adjoint Baker functions. We have g = ipip, ip(p = trg = l. The functions
xf^dlniPi
(18.1.5)
(cf. 6.4.5 and 6.4.6). Indeed, Xi
= dlliipi = ip'J(pi = - (y^Uialfa
+ ZCLitpAhpi
= - {^2 Uiatpaipj + zanpiipjjjipiipj Xf
= {g{U + zA))ij/gij,
= - ( ( [ / + zA)g)ij/gij
Vi.
,
Vj ; (18.1.6)
Evidently, the relation X{p+xf]=d\ngij
(18.1.7)
holds. Now let H^\x,y,P)
=ipi{x,P)/ipi(y,P)
= exp /
$\x,y,P)
=i/>j(y,P)/il>j(x,P)
=exp(-
Xi1)(s)ds,
J
xf)(s)ds^j
.
The behavior of these functions on the Riemann surface will be studied now. 18.1.8. Lemma. The divisors of the functions /j.\
and ^
in the finite
part of the Riemann surface are, respectively, d\ ' (x) — d\ (y) and dj (y) —
d
Stationary
319
Equations of the Matrix Hierarchy (Cont 'd)
18.1.9. Proposition. If P -> {k}, then X? = -zak X?]=zak
+ O(l), + 0{1).
Proof. This follows from (18.1.6) (where j = k is taken), the asymptotics 17.1.17 and the formulas 9.1.5: Xi
~ ~Uik9kk/gik - za,i ~ zuik(ai - ak)/uik - zat ~ -zak,i
x[
~ -(Ug)kk/9kk
~ zak
i= k
zak
etc.
D
18.1.10. Corollary. If P -> {fc}, then £'2){x,
y, P) = exp[-a fc (a; - y)z] • (1 +
0{z~1))
holds. 18.1.11. We consider the following Abel differentials (see 8.A): (1) the normalized Abel differentials of the first kind Wi\ (2) the Abel differentials of the second kind fi{fc}, the only singularity of which is the point {fc} where it can be represented in the local parameter Q = z~x as d^'1 + (holomorphic dif.); (3) the Abel differentials of the third kind SlpQ having two singularities, simple pole in P with the residue + 1 , and that in Q with the residue —1. The differentials 0^fej and HPQ are unique if they are normalized by the condition J" fi{fc} = J QPQ = 0, V«. For simplicity we shall write fifc = ft{k}- Recall Proposition 8.A.6: / ftpQ 'dk
J
'0k
= 2"7U /
Wfc ,
JQ
(18.1.12) il{i} = -2iriipk{{i})
where wfc = ipkdC, in {i} .
Let, as above, 7]\ '(x) = A{d\ ') be the Abel mapping of the divisor 18.1.13. Akhiezer's
lemma.
fc
JPr
where the subscript r symbolizes the rth component of the vector.
320
Soliton Equations and Hamiltonian
Systems
Proof. Take the Abel differential dz Infif^x,y), where dzf = (df/dz)dz. It has poles at the points of the divisor d\ '(x) with residues +1 and at the points of the divisor d\ '(y) with residues — 1. Besides, it has poles of order 2 at all the points {k} where it can be represented as — ak(x — y)dQ~1 + (holomorphic differential). This is why the differential can be written as dz\nfi(:1)(x,z)
= - ^ak{x
- y)Slk + ^
£lPk,Qk +
^hkujk,
where Pk € d\ (x), Qk e d\ '(y). We integrate this formula, at first, over the contours {ar}. On the one hand / d z ln/zj ' = hT, on the other hand fi\ is a single valued function which implies that / dzln(i\ ' = 2irimr, mT £ Z. Thus hr = 2mmr. Then we integrate this formula over the contours {/3r}: /
dzlnfi[1) = -Y]ak(x-y)
/
J/3r
ilk
J0r rQk
+ 2wi
z2
^r + Y] hkBrk , Jpk
where {Brk} is the matrix of/3-periods of the differentials {tjj} (see 8.A.1). On the left-hand side we have 2irinr, nr e Z. Thus, A{df\x)
- d^(y))
= -J2ak(x
- y) [
nk/2ni.
The equality holds on J, i.e. modulo the lattice of periods.
•
The Akhiezer lemma yields another proof of the linear dependence of the Abel mapping of the divisor d\ ' on x. The next proposition contains the main result. 18.1.14. Proposition. The relation nP(x,y,P)
=exp
y
(x - v) 5Z afc ( / ^k ~~ £ fci I
9(A(P) - y?\x)
- K)fl([t] - V?\y)
- K)
0(A(P) - V?\v)
- K)0([x] - V?\x)
- K)
Stationary
321
Equations of the Matrix Hierarchy (Cont 'd)
holds where £fci = ffi^ ftfc(fc ^ i), &» = lim P ^. { i } (/ P o ft, - z), K is the Riemann constant vector (see 8.A.4), and [i] = A({i}).a Proof. At first we prove that the expression is a single-valued function on the Riemann surface. Provisionally we denote this expression as F(x, y, P). If the point P makes a tour along a contour ar, then to A(P) the vector 6r = ( 0 , . . . , 1 , . . . , 0) is added. The function 9 is not changed, neither the integral / flfc in the exponent. If the point P turns along a contour (3r, the vector B r = (Brl,..., BTg) is added to A(P). Recall (8.A.3): 6{px + Bri, ...,pg + Brg) = exp[—2iri(Brr/2 + pr)]Q(pi, • • • ,Pg)- The quotient of ^-functions is multiplied by exp{27ri[^ 1 \x)-rij1 \y)]} = exp[- £ f c ak(x-y) j 0 r Qfc]. The exponential function is multiplied by its reciprocal, and they cancel out. The divisors of/^ \x,y,P) and F(x,y,P) coincide. Indeed, according to the Riemann theorem, 8.A.4, the divisor of the zeros of the function 6(A(P) - ^(x) - K) is dj1}(x) and that of 9{A{P) - ^(y) - K) is Let P —> {I}. We have seen 18.1.10 that fi\ '(x,y,P) — exp[aj(:r — y)z](l + 0(z~1)). The asymptotics of F(x,y,P) is the following. If I = i then rP
fP
{lk-tki= JPa
JPo
fP
r{i)
tok-
nk = 0{z~1),
Vk= JPo
k^i;
J{i}
[ Qi - & = z + 0(z~l). JPo
Therefore if P -> {i}, then p,[1)(x,y,P)/F(x, then for k ^ i, k ^ I fP
/ JPo
fP
ftk-£ki=
f{i}
^fc - / JPa
JPo
y,P) -> 1. If P -> {1} j= {«}, fP
/•{*}
^fc = / fifc -> / J{i}
Hfe = const.;
J{i}
for k = I, JPg Cli-€u = z + 0(z~l), and for k = i, JPg tti - £« -> JpJ fi, £,u = const. Thus, F(x,y, P) = exp[(a; — y)aiz] • [c 4- 0(z~1)] where c ^ 0 and depends only on x and t but not on z. We have obtained that the a T h e definition of the right-hand side of the formula is not quite obvious, ^-functions and the integral are not uniquely determined on the Riemann surface but they are singlevalued on the Riemann surface cut along the a,, /?;-contours. Proving this proposition, we, first of all, make sure that the expression on the right-hand side is, in fact, singlevalued on the entire surface.
322
Soliton Equations and Hamiltonian Systems
quotient fi\ '(x, y, P)/F(x, y, P) is a bounded holomorphic function on the closed Riemann surface, i.e. is a constant. The asymptotics for P —> {i} implies that this constant is 1. D 18.1.15. Proposition. nf\x,y,P)
= exp (x - V) X ) «fc ( /
ttk-
tkj
e(A(P)-Vf)(y)-K)e([j}-V\2)(x) ]
K)
)
9(A(P)-Vf (x)-K)e(\j}~r)f (y)
•K)
D
The proof is the same. 18.1.16. Proposition. For the elements mj of the matrix U the formula
Uij(x) = Uij{y) exp (x - y)^2 ak(£kj - &»)] ^e{\j\-n?\x)-K)0{\i\-^\y)-K)
e([j]-vl1)(y)-K)e([i\-r]P(x)
K)
holds. Proof. We have M11}/42)
=9a(x)/9ij(y)-
If P —>• {j} this becomes Uij(x)/mj(y). It remains to substitute the expressions for (i\ ' and fj,^ ' from 18.1.14 and 18.1.15. The value y can be considered as an initial value for x, e.g. y — 0. The formula can be written in a simpler form Uij(x) = kij exp U ^ a f c t e y - £ki)
e([i
vl1](x)~K)
(18.1.17)
We recall that all the divisors d]1' can be expressed in terms of one of them, see 17.3.13. The coefficients fcy do not depend on x. Unfortunately, this formula cannot be considered as the formula for a general solution with arbitrary constants fcy; they are not independent since their number is too large. Let us count how many independent constants must be involved in a general solution. The order of the system is mn(n— 1). Some of the constants determining a solution are already fixed: mn(n—1)/2
Stationary
Equations of the Matrix Hierarchy (Cont 'd)
323
values of the first integrals {Jik}, and mn(n - l ) / 2 - (n - 1) arbitrary constants in the linear functions 6ik- There are mn(n — 1) — mn(n — l ) / 2 — [mn(n — l ) / 2 — (n — 1)] = n — 1 constants which remain arbitrary. Thus, all the kij cannot be independent. In accordance with the group (17.2.6), with respect to which we have factorized the system, arbitrary multipliers can only be of the type kjkj1. Dubrovin has found the constraints for k^; they are however very complicated. It is better to change the point of view and to present the process of integration in the following succession. (1) Equation (17.1.2) and initial conditions U = U0,U' = U1,..., C/(m-1) = Um-\ are given. (2) One finds the coefficients of the characteristic polynomial f(w, z) by substituting into det(.R — wl) the initial conditions. (3) On the Riemann surface which is determined by the equation f(w,z) = 0 the normalized differentials of the first and of the second kind as well as £jy must be found, (4) From the formula 17.1.14 for the projector g, one must calculate the points of the divisor d\ ' for the initial values of {U^}. This is probably the most difficult part of work. The Abel transformation of this divisor is taken for r/> '(0). Now the formula 18.1.16 will determine the solution. Here rji = Y^ Ci,ik^ik a n d Oik should be taken from 17.3.10.
18.2
Rotation of the n-Dimensional Rigid Body
18.2.1. An interesting application of the above theory concerns an ndimensional rigid body with a fixed point. If n = 3, then the motion of such a body in the absence of external forces is governed by the well-known Euler equation. In the works by Arnold [Arn66, 69] it was shown that this equation is one of numerous "Euler equations" which can be written for any Lie algebra. The infinite-dimensional generalization includes also the Euler equation for the ideal fluid. Mishchenko [Mis] considered the "Euler equation" in the algebra SO(n) which can be called the equation of rotation of the n-dimensional rigid body. He showed that besides the obvious first integrals, energy and momentum, there is a collection of other first integrals. In a note of the author [Di72] it was shown that these first integrals are in involution with respect to the Poisson bracket, which always exists for the Euler-Arnold equations and is none other than the Lie-Poisson-BerezinKirillov-Kostant bracket (see 2.4). However, the number of Mishchenko's first integrals is sufficient for integrability only when n < 4. The decisive step was made by Manakov [Mana] who invented the trick of multiplying the set of first integrals by introducing a parameter into the equation. He
324
Soliton Equations and Hamiltonian
Systems
also noticed that after that procedure the equation of rotation of the rigid body become a particular case of the stationary equation (17.1.2). 18.2.2. Recall the Euler equation for the rotation of a three-dimensional rigid body about a fixed point. We have a vector oj{t) which is called the angular velocity (in a frame attached to the body). There is a symmetrical positive operator J : R 3 —> R 3 called the inertia tensor. The vector m = JUJ is the kinetic momentum. The Euler equation is dm/dt = <JJ x m . It is convenient to choose a coordinate system such that the matrix J is diagonal: J = diag(Ji, J 2 , J3) (Jj are the principal inertia momenta). Then rrii = JitJi. To generalize this equation to the n-dimensional case we can write it as a matrix one using the isomorphism between the Lie algebra of vectors of R 3 (with respect to the vector multiplication) and the algebra of skew symmetrical matrices, SO(3). This isomorphism is given by the equality w^ = J2£ijk^k where e^ is skew symmetric and £123 = 1. The vector m is related to the matrix rrnj = ^2k£ijkJk^kLet J\ = I2 + h, J2 = h + h and J 3 = Ix +12, then rriij = X)fc(i» + Ij)etjk^k = {h + Ij)uij, i.e. m = lu + wl,
I = diag(I1,I2,h),
(18.2.3)
and the Euler equation takes the form m=[u!,m].
(18.2.4)
Now one can generalize this equation. Let ui and m belong to so(rt), the Lie algebra of skew symmetric matrices. Let / = diag(7i,..., /„) = const. Equations (18.2.3) and (18.2.4) will remain unchanged. 18.2.5. Equation (18.2.4) is Hamiltonian in the following structure. With the aid of the scalar product {X, Y) = tr XY, algebra so(n) can be identified with its dual (coalgebra). Let us consider orbits of the coadjoint representation of the group SO(n): g e SO(n) (-* Tg : so(n) ->• so(n); TgX = g~xXg, (X G so(n)). Infinitesimal operators form the coadjoint representation of the algebra so(ra) on itself: Y e so(n) H-» Ty: so(n) —> so(n); TyX — [X, Y]. The quantities trm fc are first integrals of Eq. (18.1.4). The values of these first integrals determine an orbit {g~1mg} of the coadjoint representation. Therefore the orbit is invariant with respect to the equation. The codimension of an orbit is equal to the number of the first integrals tr mk.
Stationary
325
Equations of the Matrix Hierarchy (Cont 'd)
One must take only even integers for k. If n is even, then the codimension is n / 2 . If n is odd, then the codimension is (n— l ) / 2 , i.e. it is always [n/2]. The dimension of the orbit is n(n — l ) / 2 — n / 2 = n(n — 2)/2 if n is even and n(n - l ) / 2 - (n - l ) / 2 = (n - l ) 2 / 2 if n is odd. A symplectic structure on the orbits is introduced in Sec. 2.4. Briefly recall the construction now. The general form of a vector tangent to the orbit at a point m is £x(m) = a,d*(X)m = [m,X], where X € so(n). The tangent spaces to the orbit are canonically embedded into so(n), thus £x (wi) e so(n). The commutator of ^xim) and £ y ( m ) as vector fields has nothing in common with their commutator as elements of so(n). To distinguish the first of them we denote it as [[£y (m),^Y{rn)]}18.2.6. Lemma. [[&(m),fr(rri)}]
= f[x,v],
where X and Y are fixed. Proof. See (2.4.4).
D
Thus, the vector fields £ x ( m ) yield the right representation of so(n). The form $7 is defined as n(tx,ZY)
= (Zx,Y)=tT[m,X)Y.
(18.2.7)
In Sec. 2.4 it is proved that this form is symplectic. If f(m) is a function on an orbit, then df(m) is an element of the dual to the tangent space to the orbit. This dual can be identified with so(n) modulo X € so(n) such that [m, X] = 0. It is not difficult to find df(m) if f(m) is a linear or a quadratic function. If f(m) = tr^4m, A e so(n), then df(m) is an element of so(n) such that for any £x G TmM the relation (£x, df(m)} = £xf(™>) = tr A£x holds, which implies df(m) = A. If f(m) = tr J ( m ) m / 2 where J is a symmetrical operator, then (6c, /(m)> = Zxf(m)
= ttJ(m)£x
;
d/(m) =
J(m).
To every function f(m) taken as a Hamiltonian, a vector field £df = [m, df] is assigned. 18.2.8. Proposition. Equation (18.2.4) is of Hamilton type with the Hamiltonian Ji = -(m,u)) =
-trmui.
326
Soliton Equations and Hamiltonian
Systems
Proof. Taking into account that UJ is linearly expressed in terms of m we have dM = u>. Therefore the vector field related to the Hamiltonian is
[m,w].
•
The Hamiltonian is a quadratic first integral of the system. Mishchenko [Mis] has suggested a series of quadratic first integrals. All of them are contained in the set of the more general first integrals constructed by Manakov. 18.2.9. Proposition. Equation (18.2.4) is equivalent to — (m + zl2) = [UJ + ZI, m + zl2}, at where z is a parameter, and the equation must hold identically in z. Proof. One must only make sure that [I,m\ + [w,/ 2 ] = 0. But this immediately follows from (18.2.3). • 18.2.10. Corollary. The coefficients of the expansion in powers of z of the quantities tr(m + zl2)k
,
k =
l,2,...,n
are first integrals. Let us count the number of these first integrals. If k is even, then only even powers of z give non-vanishing first integrals. Besides, tr mk are trivial on the orbit. Only zl with / = 2 , 4 , . . . , k — 2 remain, i.e. there are (k — 2)/2 first integrals. If k is odd, then the same calculation yields (k — l ) / 2 first integrals. Altogether we have: for n even 1 + 1 + 2 + 2 + h (n - 2)/2 + (n - 2)/2 = n(n - 2)/4 first integrals, and for n odd 1 + H h (n - 3)/2 + (n — 3)/2 + (n — l ) / 2 = (n — l ) 2 / 4 first integrals. In all cases, the number of first integrals is equal to half of the dimension of an orbit, i.e. half of the differential order of the system. 18.2.11. Proposition. The constructed first integrals are in involution. Proof. Let fk = tr(m + zl2)k. The vector field f#fc related to this first integral is £,dfk — [m, dfk] = k[m,(m + zl2)k~1}. One must prove that the Poisson bracket {fk, fi} = kltr[m, (m + zl2)k~1](m + zil2)1'1 vanishes (more precisely this is a generator of Poisson brackets which can be obtained by expansion into a double series in z and z\). We calculate tr[m, (m + zl2)k]{m - z(m + zj2),
+ zxI2)1
= {zx - z)'1 tr[ Z l (m + zl2)
(m + zl2)k]{m
+
zj2)1
Stationary
327
Equations of the Matrix Hierarchy (Cont 'd)
- z{m + zj2), = (zj - z)'1{zl
(m + zl2)k}(m
+
zj2)1
tr[(m + zl2), (m + zl2)k]{m
+ ztr[(m + zj2),
+
zxI2)1
(m + zJ 2 ) ( ](m + z/ 2 ) fe ]} = 0.
•
18.2.12. Corollary. Equation (18.2.4) can be integrated in quadratures. The procedure of integration can be reduced to that in the previous section with the aid of the following proposition. 18.2.13. Proposition. Equation 18.2.9 coincides with the stationary equation (17.1.2) where m = 1, A = I and B = I2, restricted to the subalgebra so(n). Proof. First of all one can see that Eq. (17.1.2) admits a restriction to the algebra so(n). The recurrence equations Sec. 9.1.4 imply that if U £ so(n), then all R2k are symmetric and all i?2fc+i are skew symmetric. Therefore [A, R2} is skew symmetric, and if the initial values for U are skew symmetric they remain so for all the values of x. The equation U' = [A, R2} is, as we know, equivalent to —R'i = [U + zA, Ri], where Ri = Bz + Ri. If we put I = A, B = I2 and U = w, then (-Ri)jfc = \(I2 - Ik)/(J3 ~ Jk)}ujk = (Ij + h)ujk (see 9.1.5), i.e. #1 = m, and the equation takes the form —(m + I2z)' = [UJ + zl,m + I2z] which coincides with 18.2.9 if t = —x. 18.2.14. We draw the reader's attention to the fact that the Hamiltonian structure of the equation under consideration constructed in this section does not coincide with those from Chap. 17; this can be seen e.g. from the fact that the Hamiltonian is now | (m, u>), which is equal to tr AR2. In the theory of Eq. (17.1.2) this expression was not the Hamiltonian of the given equation but of a trivial symmetry. This also means that the symplectic forms are different, too. Now this form is defined only on orbits. It would be interesting to discuss the relation between two structures.
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Chapter 19
Field Lagrangian and Hamiltonian Formalism
19.1
Introduction
19.1.1. In the preceding chapters the space coordinate and time played quite different roles: the space coordinate was merely an index labeling degrees of freedom, and the time coordinate was the usual physical time in which the system evolves. Such a theory is satisfactory unless we turn our attention to relativistic invariant equations, e.g. chiral fields, sine-Gordon, and others. Also, considering the KP-hierarchy (5.1.5) for arbitrary m and n, the variables tm and tn are quite equal in rights and there is no reason to prefer one to the other by choosing it as time. In such cases a new field theory is useful which involves many equal in rights variables. (Sometimes, for convenience of speech, we call all of them "times", though physically this does not make much sense.) This will be done in this chapter. The above symplectic form and the Hamiltonian (if one of the variables is fixed as time) will now be merely one of the components of a vector or tensor quantity (e.g. the old Hamiltonian is one of the components of the energymomentum tensor; however a new Hamiltonian which is a scalar will also be called the field, or scalar, Hamiltonian). One subtle problem is the relation between first integrals and vector fields which will enable us to construct a symmetry if a first integral is known, and another is the problem of Poisson brackets. Here the situation is more complicated than in the single-time theory. The field formalism is of interest even for our old examples (e.g. KdV or GD) where the variables are involved in a distinctly asymmetric way. The first component of the compound symplectic form appearing in this theory coincides with the form introduced earlier and the second one is in a direct relation to the stationary equation. Thus, the theories of stationary and of non-stationary equations turn out to be more closely connected with 329
330
Soliton Equations and Hamiltonian
Systems
each other. In particular a new interpretation can be given to the result by Bogoyavlenskii and Novikov [BN] and Gelfand and Dickey [GD76(a)] about the relation between first integrals of stationary and of non-stationary equations and the associated vector field. 19.1.2. The multi-time canonical equations are written in a book by de Donder [deD]. Let, for simplicity, a Lagrangian A depend on independent variables xi,...,xn, and some other variables qi,...,qm and their first derivatives qtj = dqi/dxj. Then the variational equation {6A/5qi = 0} can be reduced to the canonical Hamiltonian form n
dqk/dxj = dH/dpkj ,
^dp^/Oxj
= ~dV./dqk ,
(19.1.3)
3=1
where pkj = dA/dqkj are "momenta" (thus, n momenta are assigned to one coordinate), and W, is a Hamiltonian. If the Lagrangian depends on higher derivatives the canonical form of the equations is more complicated, see below (19.5.18). We shall write canonical equations in an invariant form which does not require usage of the canonical variables "coordinatemomentum". A formal framework for such study is represented by the so-called "variational bi-complex". 19.1.4. The multi-time calculus of variations is of course not new. The earlier works were by Caratheodory [Car], Weyl [Weyl35], de Donder [deD] and the more recent by Dedecker and Tulczyjev [Ded, DT, Tul], Vinogradov (see survey [VKL]), Vinogradov and Kupershmidt [VK], Tsujishita [Tsu], Goldshmidt and Sternberg [GS]. For the connection with the completely integrable systems see also Manin [Mani78(b)]. The latter author noted that in this study essentially different languages can be used, namely those of analysis, algebra and geometry. We are close to the algebraic concepts of Dedecker and Tulczyjev's works. The so-called variational bi-complex is used, i.e. a complex with two differentials d and 5. A similar complex was introduced by Gabrielov, Gelfand and Losik [GGL]. Some of our results appear newer: the construction of the symplectic form and of the energymomentum tensor as well as the relation between first integrals and vector fields (theory of characteristic of first integrals similar to that in Sec. 16.2). Martinez Alonso [Mar] has used a similar notion (integrating multipliers). However, he did not have symplectic forms and theorems like the one we call "main proposition about first integrals". Our point of view is close to that in the works of Shadwick [Sha80(a)-82(b)] who used the field formalism in the form of Goldsmith and Sternberg.
Field Lagrangian and Hamiltonian
19.2
331
Formalism
Variational Bi-Complex
19.2.1. Let /C be a differential algebra ("algebra of coefficients") with commuting derivations dj, j = l , . . . , n . An example: smooth functions of variables Xj, dj = d/dxj. A second example: constants K. = R or C, djK. = 0. This example will play the main role. 19.2.2. Let us extend this algebra adding independent generators u\, , k = 1 , . . . , m, (i) = ( i i , . . . , i n ) being a multi-index. It is assumed djii£ — uf+ei where (i) + ej = (h, ...,*,- + 1 , . . . , i„). Let 0 « = 9J1 . . . dfc. This o
differential algebra is denoted as A. Let A be the subalgebra comprising the differential polynomials in {itfc} having no terms free of the generators {ujt}, i.e. "polynomials without free terms". 19.2.3. The space
comprises formal sums
»P'9 = E /S),(i) Ju £° A ' • • A 6ut]
A
<*"* A • • • A dxiq ,
where (i) = ((h),... ,{ip)), (ix) = (in,... ,hn),... ,(k) = (ki,...,kp), and (j) = ( j i , . . . ,jq). The sums are called p, g-forms. All the differentials {$Uf!' ,dxj} are anticommuting. The exterior product is as usual. Now we have a Grassmann algebra with the following generators: (1) elements of fC, (2) u£ , (3) 6uf,* , (4) dxj. The differentials 5u^' will be called variations. Let A** = {A^}. Now we define operations d : A^q) -> A{p'q+1) and {v q) +1 j :A ' -> ^ ( P ' 9 ) . (i) They are (graded) derivations d(wf '91 Awf , 9 2 ) = dwf1 '91 Awf' 9 2 + ( - l ^ + ^ w f ' 9 1 Arfwf'92 , 5(wf '91 A w£2'92) = <Wf'9l Awf2'92 + ( - l ) P l + 9 1 w f , 9 1 A Suff '92 . (ii) On the generators:
4f = E d i f d x J 6f = Y,df/duf d(5uf)
= ~5duf
=
^2(df/dxo • 5uf,
+ dflduk]
• Uk)+£J)dxJ
5{dxj) = 5(5u{*}) = d(dXj) = 0,
= - J ] 5uf+e> A dr,-.
This determines the action of d and S on any form. 19.2.4. Lemma. d2=S2
> / G -4.
= 0,
dS =
-6d.
332
Soliton Equations and Hamiltonian
Systems
Proof. By virtue of (i) this must be verified on the generators which can be done easily. Here is the mapping diagram
«t
5
t
0 A .4(2.0) ^ _4(2,i)
«t
s
•
•
•
•
d _4(2,n) - >
t
0 4 .4(1.0) 4 . .4(1,1)
*t d -
»
«t *t o A >.o) 4 >•!) t t 0 o.
0
.4(1.")
Ai0'n)
t
0
.
o
where A( ,q' consists of forms whose coefficients belong to A- (When we speak about the bi-complex {A^p'q^} we always mean that the bottom line is { > • ' > } . ) 19.2.5. The bi-complex {A^p'q'} generates an associated complex with elements A^ = © p + q = r A^p'q^ and a derivation d + S since (d + 6)2 = d2 + dd + Sd + S2 = 0. 19.2.6. The dual to the space of one-form A^ fields" which are the formal sums
is the space TA of "vector
The coupling of vector fields and forms, or the substitution of vector fields into forms is the following. Only i(d/du^ )Su^ — 1 and i(dj)dxj = 1 do not vanish. The substitution of a vector field into a form of any rank is determined as usual: in succession into all the differentials Suf.* , dxj, changing the sign if the differential stands on an even place, e.g. i(dj)5u^' A dxj A dxi = — Su^ = — SU[
A dxi,i(d/du^
)6ui'
A 6u% A dxj
A dxj .
The operation of the substitution of the vector field is a (graded) derivation t ( 0 K 1 ' 9 1 A ^ 1 * ) = ( t ^ M 1 ' * 1 ) A uP2'92 + (-l)Pl+giuPl'qi
Ai(0u%2'92 •
Field Lagrangian and Hamiltonian
333
Formalism
Apart from the operation i(£) there is another operation between vector fields and forms: the Lie derivative Lz = (6 + d)i(t;) + i(z)($ + d). As it is easy to check, an anticommutator of two graded derivations, each of them changing the parity (—l)p+g of the form, is a not graded derivation, L^(cJi A w2) = (L^uJi) A w2 + Wi A L^u>219.2.7. The special vector fields
play an important part. Let us calculate the action of the vector fields dj and dj as the Lie derivatives on the functions, i.e. on the elements of A:
Ldj f = i(dj)(6 + d)f = i(dj) (J2 df/duf V
= mi^df/duf
• Suf +J2dkf-
• 5uf + £ d k f
• dxk) = djf,
dxk) = Y.df/du^
• v!?+<> .
Thus, Lfyf is a total derivative of the function / with respect to Xj, Lg is the total derivative minus the partial one. (The latter takes into account only the explicit dependence of / on Xj.) In other words, when calculating Lg., the elements of fC are considered as constants. We further use the simpler notations LQ^ = djf, Lg.f = djf and, in general, L^ui = £w. Partial derivatives will be denoted as djf = djf — djf. 19.2.8. Proposition. The action of Lg on a form u> is taking a derivative with respect to Xj of all the coefficients and all uk which appear as factors 5uki] (e.g. djifdu^ A dxr) = (djf)5ulm) A dxr + f6ulm)+ej A dxr). The action of Lg. is the same but the elements of K. are not differentiated. Proof. It is sufficient to verify this on the generators, which we leave to the reader. • 19.2.9. Proposition. Vector fields form a Lie algebra with respect to the commutator defined by the rules: if £,7/ € TA; f,g G A, then [£,rj\ is an element of TA bilinearly and skew symmetrically depending on
334
Soliton Equations and Hamiltonian
Systems
£ and r) being [fZ,!m] = (
f9[t,Ti]+m9)ri-9{rif)Z,
m)
ld/du j;\d/dul ] = {dj,dk} = o
[d/duf^]
0,
if ij = 0
' " d/duk
3
,
otherwise
e (the last rule can be formulated simpler: \d/du£ , dj] is always dfdu£ ', this expression being zero if the multi-index (i) — ej contains negative components).
Proof. One has to prove that A = [[£, 77], C] + (cyclic) = 0. A is a vector field with coefficients which are differential polynomials of the coefficients of the vector fields £, 77 and £, determined by the above relations. These polynomials do not depend on which algebra fC we consider. Now it is convenient to take the algebra of all smooth functions. In this case, if it is proven that Af = 0 for all / € A, it will follow that / = 0. Indeed, Af = E fc ,(o AkAi)d/duff + Y,j Ajdjf = 0. First we take / and arbitrary element of fC. Then the first term vanishes, and we find Alj = 0, then, from arbitrariness of / G A, we conclude that all Ak^ = 0. It is easy to verify that the commutator thus defined has the property
[£,ri]f = Z(Tif)-v{Zf) for any f E A. identity.
Then ([[£,?/],£] + c.p.)/ = 0. This implies the Jacobi •
19.2.10. Lemma. Si(dj) = -i(dj)5,
di(dj) = —i{dj)d.
Proof. It suffices to check this on the generators which we leave to the reader. D 19.2.11. Corollary. djuj = (di(dj) + i(dj)d)w,
BjU = (5(dj) + i(dj)5)u>.
19.2.12. Lemma. dw = 2_\ dxi A djtJ .
Field Lagrangian and Hamiltonian
335
Formalism
Proof. Check on the generators.
•
19.2.13. Lemma. The operation
8i{dlduf)+i{dlduf)5 acts as d/duk
applied to the coefficients of a form.
Proof. Check this on the generators.
•
19.2.14. Definition. The vector field related to the set h — (hi,.. •, hm) € Am is dh = ] T hf d/duf
,
h« = 3 « hk = d[> • • • dt hk .
19.2.15. Lemma. [dh,dj}=0,
j =
l,...,n.
Proof.
[Y: hfd/duk\ dj]=- Y: hi^d/du^+Y: hfd/duf-^=o. D 19.2.16. Lemma. di(dh) + i(dh)d = 0,
dhd = ddh ,
dh5 = 5dh .
Proof. According to Lemma 19.2.12, d = ^dxj A dj whence (dhd — ddh)uj = dhY^dxj A djW — YL^xj A dj(dhu). From d^dxj = 0, taking Lemma 19.2.15 into account we obtain the second equality. The first one can be checked on the generators: di(dh)8uf
+ i^dSuf
= dhf
- i(dh) 5 3 5uki)+F'r A dxr
= E hki)+erdxr - Y, hf+erdxr = 0. Finally, dhuj = Ldhuj = (5 + d)i(dh)u) + i(dh)(5 + d)uj , SdhUJ = 5di(dh)uj + 5i(dh)8uj + 5i(dh)du) = 5(di(dh) + i(dh)d)w + 5i(dh)5w = Si(dh)Suj, dhSuj = (5 + d)i(dh)5u) + i(dh)(5 + d)5uj = 5i(dh)5u).
•
336
Soliton Equations and Hamiltonian
Systems
19.2.17. Proposition. dhi(dg) - i(dg)dh = i([dh, dg}). Proof. It is sufficient to verify this on the generators Su£'. The left-hand side is
dk9^ - d.h® = £ hMagjp/du™ It is easy to see that the right-hand side is the same. 19.3
-{hog). •
Exactness of the Bi-Complex
19.3.1. Proposition. All the sequences, both vertical and horizontal of the bi-complex are exact. Proof for the vertical sequences. This is the easiest part of the proof of the proposition. In fact, this is precisely Poincare's lemma and can be proved by a construction of a chain homotopy, i.e. of an operator / such that IS + SI = 1. Then if a form w is 5-closed, Sui = 0 we have u) = SIUJ and it is exact, as required. Note that we can restrict ourselves to a case when coefficients of forms are homogeneous of the degree r with respect to the generators u£'. As above, p is the degree with respect to Su^' and q that with respect to dxj. It is easy to check that the operator
' = ^£4%v^)
( 19 - 3 - 2 )
has the needed property. Indeed, SIu = — ^ - J2
Su }
r
A
i{d/duf)w
The form Su> is also homogeneous of the total degree p + r, hence the last term is —ISui. The first term is (p/(p + r))u. The second one in accordance with 19.2.13 is the operator (p + r)~l ^2uk d/du^ applied to the coefficients of the form u>. All of them are homogeneous of degree r;
Field Lagrangian and Hamiltonian
337
Formalism
the Euler theorem about homogeneous functions yields: the second term is r(p + r) _ 1 w and in all SIOJ = u> — I6w as required. Let now 5uip'9 = 0. Then w™ = */
iw= f Vtijjftya/au^M* 1 *)* -1 *Jo Each term will automatically be divided by its degree. Our goal now is to prove the proposition for the horizontal sequences. 19.3.4. Proposition. All the horizontal sequences in the bi-complex are exact for p > 0. Proof. An operator D ("Tulczyjev's operator") will be constucted having the property Dd + dD = p • 1 for the pth row of the diagram. This will yield the required assertion for all rows except the last row, p = 0. For the latter the proof will be completed using the "diagram chasing" technique. The construction is however not very simple. Let (m) be a multi-index. We construct an operation #(m) having properties: (i) it is a derivation a a
i
(m)\^\
(Pi,9i
A
P2,92\ _
Ao;2
/a
,Pi,9i\
) — \V(m)U)1
A
, ,P2,92
)AW2
,
Pi,91
+UJ1
A
a
P2,<22
AB(m)U2
,
(ii) on the generators it is denned as: for / £ A, 0(m)f = 0) 0(m)dxj = 0,
•^HSR'-'MH) - CO -COThis expression is zero if m r > ir for some r = 1 , . . . , n; then 6^m)5u^' = 0. A derivation is uniquely determined by its values on the generators. It is obvious that for (m) = 0 w e have 9owp'q = pu)p'9. The explicit expression for the operator #(m) is
^ ) = E( ( ( 7 l ) ) )^ i) - (m) ^w4 i) )-
338
Soliton Equations and Hamiltonian Systems
19.3.5. Lemma. [#(m),<9Q] = 0(m)_eQ • Proof. If 0(m) is a derivation then so is [0( m ),d a ]. The equality of two derivations can be verified on the generators. The only not obvious case is that one:
'(«) + ea\
( (i) (m),' '
(TO)
(i)-(m)+e a _
W
K"—-«<-,^K
D
Now let 7Q, a = 1 , . . . , n be a set of multi-indices (TO) such that m Q > 0 and vn.fi = 0 if (3 > a. Let cra = -
(-l)Md{m)-e°e{rn),
Y,
\m\=m1
+
---+mn.
(m)e/„
19.3.6. Lemma. dpa°
a<(3,
<Jadfi = { 7
a> (3.
,0, Proof.
*Qfy = - E
(-l)W3(,n)"ea*<»*)fy
(m)e/„
= ^aa-
J ] (-l)l",|0(m)-e"0(Tn)-e„ •
1. If /? > a, (TO) G / „ , then ^( m )_ e ^ = 0, aadp = dpaa, 2. If j3 < a, then aadp = dpa" - Y,{-^m]dpd{m)-ea~ef,e{m)-e?
= dpaa - d0aa = 0,
Field
Lagrangian
and Hamiltonian
Formalism
339
3. If 0 = a, then aa3a
= daaa - £ ( - l ) | m | d ( m ) - e Q < ? ( m ) - e Q
= daaa - Yl a7(j7 + 0° = °° " 5Z V 7 • 7
D
7
An arbitrary p, q-form w = u)p'q can be written as w=
^2
i(dai)---i(dan^q)dxi
A---AdxnAaai...an_q,
oci<--
€ Ap'°. Let
where aai...an_q Dw=
^
i(dao)i(dai)---i{dan_q)dxiA---AdxnAa°"'aai...an_q
.
Qo
This is Tulczyjev's operator. If q = 0, then D = 0. 19.3.7. Lemma. The relation dD + Dd = 0O holds. Proof. 71
dDw = ^
dxp A dpDw = ^
dxp A i(dao) • • • i(dan_q)dxi
A • • • A dxn
0=i n—q
Ad0aa°aai...an_q=YJ{-Vr r=0
E
i(dao)---i(dar)---i(dan_q)
ao<-i-,
• dxi A • • • A dx„ A dolra°"'aai...an^q dw = E (3
^
dxp Ai(dai)---i(dan_q)dxi
; A---Adxndi3aai...an_q
ai<-
n —q
= ^(-l)r-
1
T=\
A oara,ai...anq
Yl ai<--
;
i{dai)---i(dZ)---i{dan_q)dx1A---Adxn
340
Soliton Equations
and Hamiltonian
Systems
n—q
DcLu = J2(-~>-)r-1 r=l
ato<-
E
W°0)i(dai)---i{dar)---i(da„_q)
-
•dxiA---AdxnAa°">daraai...an_q+
^T
* ( # a i ) " •*(#<*„-,)
ai<-
Aaaidaiaai...an_g
•dxi A •• • Adxn
(the reason for the appearance of the last term is the following). If r = 1, then the index of summation ao runs from 0 to 02 — 1. When a 0 > <*i, then aa°dai = 0 (see 19.3.6). When ao = a i , this additional term appears; the rest, for ao < oti, is involved in the first term. Now dDcu + Ddu) =
YJ
i(dai)-•
•i(dan_q)dxi
A •• • Adxn
ao
Adaoo-a°aai...an_q +
E
«( 9 «i) , -'( 9 <>»-,)
ai < - < a „ _ q
•dxi A ••• Adxn Aaaidaiaai...an_q
.
Taking into account 19.3.6 (Taidai = 60 — Ylao
we
obtain the •
Now we finish the proof of Proposition 19.3.4. If du)p'q = 0, then dDw™ = 0ocJp'q = pcJp'q, i.e. uip>q = dip'1Dujp'q). The closed form is exact. • 19.3.8. Remark. It is evident that Du) = 0 if the form u> contains only 8uk with zero multi-index. We have not proved yet that the bottom row is exact. Here we use the method of diagram chasing. 19.3.9. Proposition. If for a form wp'g G -4(p,
(a form
assumed to be in A.( ' ^-
are
341
Field Lagrangian and Hamiltonian Formalism
Proof. We construct a sequence of forms wp+r'q property Sujp+r'q-r
r
, r = 0, l,...,q
with the
= -dwP+r+1'«-r-1.
We already have two first terms of this sequence, for r — 0 and 1. Let w p+ r ,g-r
b e built
.
^P+r,q-r
=
_^p+r-l,,-r+l)
t h e n
Sdu)P+r,q-r
=
Q^
dSujP+T'i-r = 0. Proposition 19.3.4 yields (since p + r +1 > 0) 6vp+r
+ 5(fip+n-l,q-r
(19.3.10)
for some forms {
i JgjP+fo,«-r 0 -l
Now, dwP+ro+i,q-r0-i = ^^p+ro^q-ro from where we get <%? p + r °-''- r ' 0 - 1 = -Su)p+r°'q-ro, i.e. 5(d
exists s u c h t h a t
dipp+r0,q-rQ-l
_(Jp+r0,q-r0
=
1 q ro
—5
+ ^ - i , « i.e. ^ = {d + s) EP+g=r-i vp'q-
n
342
19.4
Soliton Equations
and Hamiltonian
Systems
Variational Derivative
1 9 . 4 . 1 . We turn now to the last column of t h e bi-complex. T h e problem is t o find t h e image of t h e mapping d : A^'71'^ -> A(p'n). Let E(p'n) = ^(P,n)/rf_4(p,n-i). T h e augmented bi-complex has one more column. 5 St <5 t t 0 _> .4(1.0) ^ . . . 4 A(hn) ^ ^(O.n) ^
Q
«t 5t « t (*/««) 0 ->i(0,0) 4 . . . 4^(0,n) ^ £(0,») _> 0
t
T
t
0
0
0
The anticommutativity of d and 5 implies that 6 can be considered in t h e quotient space Ep'n. The meaning of t h e symbol (S/Su) will be clarified later. 19.4.3. Proposition.
The augmented bi-complex (19.4.2) remains exact.
Proof. We denote the elements of E[p'n' by ujP'n, and ujp,n will be a representative of the class (coset) u>P'n. Let 5wP'n = 0. For a representative u>p'n, this means that StJp'n = -du)p'n~x for some u)p'n~x e A^'71'1^. By p n l vitue of Proposition 19.3.11 forms ip > ~ and tpp~l
If u°'n = fdx± A---Adxne
«4 ( 0 ' n ) then 6uj°'n can be
n
6cj°-n = ^
Ak5uk
A d i i A • • • A dxn +
du)l'n~l,
fc=i
where w 1 , n _ 1 e A^1'n~1^/dA^1,n~2^. The coefficients Afc are uniquely determined. They will be denoted as 5f/5uk and called variational derivatives with respect to Wfc. T h e form a; 1 '" - 1 is uniquely determined up t o a closed form (or a coboundary, which is the same since the bi-complex is exact). Let 5f/5u = {6f/5uk}. P r o o f . Transform the expression Sf A dxi A • • • A dxn = J2(df/duk))5uk)
A dxi A • • • A dx„
Field Lagrangian and Hamiltonian
343
Formalism
by means of repeated integration by parts
= J2(-^Hd{i)df/du^)
^df/du^Su^
• 5uk + J > B
a
,
where Ba are forms. It remains to put Sf/Suk = ^(-V^df/duP
(19.4.5)
and W 1 '" - 1 = ^{-l^Badxx A • • • A dxZ A • • • A dxn. The uniqueness can be proved thus. Suppose 0 = V ] AkSuk A dxi A • • • A dxn + duj1 If D is the Tulczyjev operator then Ddu1^'1 + dDw 1 -"" 1 = w1-™"1 1 n 1 i.e. CJ1'71-1 =d(Duj ' - )-DY^Ak6ukAdx1A---Adxn. According to 19.3.8 1 -1 the last term vanishes, and w '™ = d( ) whence J2 Ak5uk A dxi A • • • A dxn = 0 and {Ak = 0}. • 19.4.6. Proposition. The variational derivative vanishes, 5f/Su = {Sf/6uk} = 0 if and only if the form w 0 '" = fdxi A • • • A dxn is exact, UJ°'n = d(f°'n~1. Proof. The equation 6f/6u = 0 is equivalent to 8uj°'n = doj1,n~l for some Conversely, if <5w0'™ = dw 1 -"" 1 w i,n-i_ T h i s i s t h e c a s e i f wo,n = ^ o , n - i then u>°'n = dip0'"-1, according to 19.3.9. • Thus, the operation of the variational derivative can be transferred to 0,n) -> I t c a n b e c o n s i d e r e d as the mapping 5 : E[
E(o,n) = A(o,n)/dA(o,n-i)_
E[ ,nK If fdxi A • • • A dxn is a representative of an element in E{ ' , then ~Y^,{5fk J5uk)8uk A cfaji A • • • A ckr™ is a representative of the corresponding element in E\ . The fact that the variational derivatives Sf/5uk vanish if / is a divergence / = dgk/duk where gk are functions of {UJ} and their derivatives is well known in the common (non-formal) calculus of variations. It is a corollary of the fact that the integral / fdx\... dxn can be expressed in terms of boundary values of the functions uk and their derivatives and therefore does not change when the functions are varied inside the domain. The converse theorem is not so trivial since one has to prove that if the variational derivative vanishes, then / is a divergence of a vector consistent of differential polynomials.
344
Soliton Equations and Hamiltonian
Systems
19.4.7. Proposition. A set {Ak}, k = 1 , . . . , m, is a variational derivative of some / € A if and only if ^) SAk A 5uk A dx\ A • • • A cfan is a ^-differential, i.e. there is a form cj2'n~1 € A2'71"1 such that y ]
d9h =
{dghk}=\Y,9iii)dhk/duli)\
is linear with respect to both h and g. This expression can be considered as the result of application of a matrix differential operator to the vector 9 • Qh9 = U2(i),i dhk/du^d^gi}, k = 1 , . . . ,m. The operator Qh is called the Frechet derivative. Thus, dgh — Qh9- In the vector space Am there is a scalar product (h, g) = "^2 hk9kdxx A---Adxn£
E[°'n).
(19.4.8)
(The fact that the scalar product is considered as an element of E\ , i.e., up to an exact differential dw 0 , n _ 1 , in analytical terms means that the scalar product is the integral J ^ hkgkdxi... dxn.) 19.4.9. Proposition. A set g = {gk}, k = 1 , . . . , m is a variational derivative gk = 5f/5uk, k = 1 , . . . , m if and only if Qg is a self-adjoint operator with respect to the scalar product (19.4.8). (This theorem for n = 1 was proved by Gelfand, Manin and Shubin [GMSh] one way, and the converse by Dorfman [Dorf].) Proof. According to 19.4.7, the set g is a variational derivative if and only if a form w2'™'1 exists such that ^ &uk A 6gk A dx\ A • • • A dxn = du)2'n~1. This is, in its turn, equivalent to the fact that after the substitution of two arbitrary vector fields dht and dh2 an exact differential will be obtained,
Field Lagrangian and Hamiltonian
Formalism
345
nO.n
i.e. zero in h,x : d(w 2 ' n - 1 (g h l ,g h a )) = dw 0 ' n - 1 = Yl^dhiUk^dh29k^ ~ {dh2uk){dhx9k)}dxi A • • • A dxn = {dhlu,dh2g) (dh2u,dhlg) = (hi,dh2g) - (h2,dhlg) = (hi,Qgh2) - {h2,Qgh{). This equality expresses the fact that Qg is a self-adjoint operator.
•
The simplest way to reconstruct a function / knowing its variational derivative is given by the following proposition. 19.4.10. Proposition. If the condition 19.4.9 is fulfilled and {gk} homogeneous polynomials in {uk1' } of degree r, then
f= satisfies the equations 5f/Suk
are
(r+l)-1Y/uk9k = gk, k = 1 , . . . , m.
Proof. We find (all the computations in E\ 'n'): for all h € Am dhfdxi A • • • A dxn = (r + 1 ) _ 1 \~] hkgkdxi + (r + l)'1 ^uk(dhgk)dxi = (r + l)~1^2hkgkdxi
A • • • A dxn
A • • • A dxn
A • • • A dxn + (r +
l)"1(Qgh,u)
= (r + i ) " 1 O C A*»* + Yl hkdgk/du^ • u, w ) dXl A • • • A dxn . Applying Euler's theorem about homogeneous functions we obtain dhfdxi
A ••• Adxn = ^ ( r + 1) _ 1 (1 + r)hkgkdxi
A ••• A dxn =
(h,g).
On the other hand, dhfdxi A • • • A dxn = 2~] df/duk%'hk%'dx\
A • • • A dxn
= 2^ Sf/5uk • hkdxi A • • • A dxn — (Sf/5u, h) where h is arbitrary which implies g = Sf/Su.
•
346
Soliton Equations and Hamiltonian Systems
19.4.11. Remark. It is possible to do this without separating homogeneous parts of {gk}- The same result will be obtained substituting puf' for uf' into {gk} an integrating: / = 2^ /
19.5
ukgk{pu)dp.
Lagrangian-Hamiltonian Formalism
19.5.1. An arbitrary element of E{°'n) = A(°'n)/dA{0'n~V can be taken as a Lagrangian (or action, to be more precise). Let A = Adx\ A- • • f\dxn be a representative of the class (in physics this is the density of the Lagrangian). Then (see 17.4.4) a form w1-™-1 exists such that 6 A = Yl&uk A 8A/5uk + du; 1 , n _ 1 ; 5A/5uk = 8A/5ukdx\ A • • • A dxn. The form a;1'™-1 is especially important and therefore deserves a special notation — Q^. Thus, 5A = ^2 $uk A 5A/5uk - dfi ( 1 ) .
(19.5.2)
The form fi'1' is determined up to a form of the type do;1'™-2. If an arbitrary do;0'™-1 is added to A, the variational derivative does not change, but il^ will: the form Su0'71'1 is added to it. Let Q. = SSI1-1) e A^'"'^ (this form is determined up to a form d5u)l'n~2). The form will be called symplectic corresponding to the given Lagrangian. Evidently dfl = ^2 5uk A 6(5A/5uk).
(19.5.3)
Put + i(dj)n(1),
Tj =-i(dj)A
j = l,...,n.
(19.5.4)
The set of these forms is called the energy-momentum tensor. It does not change when A is replaced by A + du°'n~1 and, at the same time, fl^ by Q(l)
+
fo,0,n-l ( g e e
aboye
)
19.5.5. Proposition. The relation
dTj = - 53(5 iUfc ) • SA/6uk - d'jA k
holds (recall that <9'- is the partial derivative arising on account of explicit dependence of the Lagrangian on time).
Field Lagrangian and Hamiltonian
Formalism
347
Proof. dTj = -dt(a,-)A + dt(flj,-)n(1) = -djA
+ di{dj)n®
= -a$A - a,-A + dtfa^nw = -ajA - t(a,-)*A + di(a, )n(1) = -fljjA - t(flj) 5 3 5ufc A ^A/5ufe = -0JA - ^ ( ^ - " f c ) ^ / * " * •
D
The following proposition expresses, as we show below, the main property of the energy-momentum tensor. 19.5.6. Proposition. The relation 5Tj = -i(dj)n
+ t(fy) 5 3 Suk A <5A/<5ufc + dJ(d,)fiW - ajn (1 >
holds. Proof. By virtue of 19.2.10 JTj = i{dj)6A + Jt(5j)n (1 > = »(8j) ( 5 3 Jufc A <*A/<5ufc - d " ( 1 ) ) + &(fy )0 ( 1 )
= *(8j) 51 5ttfc A JA/(5wfc - (.Kfy)* + dt(fi|,-))0(1) + dt(a,-)n(1) + («(9j) + t(s,-)*)n(1) - i(aj)<jn(1) = i(dj) 5 3 <5ufe A SA/Suk - a,-n(1) + a,-n(1) + d*(aj)n(1) - t(a,-)fi.
•
19.5.7. Corollary. If the Lagrangian does not depend on time explicitly, then STj =
-i(dj)Sl
modulo a d-differential and by virtue of the system of equations {5A/Suk = 0}. This proposition generalizes the main relation of the usual (singletime) mechanics: 8H = —i(dt)£l is equivalent to the variational system SA/Suk = 0 where U = - A + i(dt)n^\ 19.5.8. Definition. The field Hamiltonian is the form
H = 5 3 dxo
AT
i + (n- 1)A
19.5.9. Lemma. It can also be written as n
n = - A + 5 3 dxoA *(fy)n(1) •
348
Soliton Equations and Hamiltonian
Systems
Proof. This immediately follows from (19.5.4) if the equality dx3- A i(dj)A = A is taken into account which holds for any A € A^0,n\ 19.5.10. Proposition. n
m
6H = Y^dxj
A
n
SA/5uk + ] T d x j A d'}Q.{1).
i(dj)Sl -^2SukA
3=1
k=l
j=l
Proof. SH = -SA - Y^ dxj A Si(dj)£l(1) = - J2
Su
= -YJSU*A - Ydxi
k A SA/Suk + dttW - J2 dxj A 6i(dj)SlW 6A 5u
/ k + Yldxi
A
ty + WA6 + **(^;))n(1)
A <5z(9j)J7(1).
One gets the needed expression after cancellation.
•
19.5.11. Corollary. If the Lagrangian does not depend on times x3explicitly, then the equation {5A/5uk = 0} is equivalent to
We call this equation the Hamilton equation. 19.5.12. Remark. Below we shall have an example where a Lagrangian depends on times but the additional term Yl dxj Ad'jfl^ in 19.5.10 vanishes and the equation has a Hamilton form. 19.5.13. The "coordinate-momentum" variables. Here the canonical variables will be introduced. Let C(») = | i | ! / i i ! . . . i n ! (if one of ik is negative, then C(j) = 0 ) . It is easy to verify that n C
(i) = ] C C W - e i 3=1
(Generalized Pascal's triangle). Let
*M° = £(-i)|m Wcw+(^(m)dMi)+(m) • (m)
Field Lagrangian and Hamiltonian
349
Formalism
If (i) = 0, this coincides with 5/5uk. Expressions
are called momenta. The variables {uk%'} are "coordinates". 19.5.14. Lemma. The equation n
Pk,(i) = c^dA/duk%)
- 5Z^' Pfc .W+ e i j=i
holds. Proof.
(m)#0j=l
c - ^ A / 9 ^ - ^
£
(-ljM-V^Mo+^-KnO-e,
j = l (m)#0 x
=
C
d(™)-^.dA/dt4 i ) + e j + ( m ) ~ e j
(i)l5A/5ufc
x
"
E L (-1)|rn|^C(m)/c(i)+e;,. J=l(m)#0
+
(m)
d^dA/duki)+ej+{m) n
19.5.15. Proposition. The form fi^1) is
"(1) = EEE c w fa i ,)A «( s jKw+«j • fc=ij=i
(»)
Proof. One must verify that Eq. (19.5.2) holds for this form:
dnt1) = - ^ c (i) aj(54 4) Ap fc , (i)+e .) = - ^ 5 4 ! ) A ( a A / a 4 8 ) -c w p f c , ( i ) ) fe,j',(t) fc,(i)
~ X ! "^fc
A c
(i)Pfc,(«)+ej = SA
+ ^25ukA
6A/5uk .
D
350
Soliton Equations and Hamiltonian
Systems
19.5.16. Corollary. In coordinate-momentum variables: m
fi
n
= m i £ CWJull) A *(^»)*Pfc,W+ei ' k=ij=i
(i)
T
J = 5Z c(i)tifc)+ej*(^)P*,(i)+«i - * ( ^ ) A »
w
=
S
c
W u fc ) + e i Pfc,(i)+«i- A -
19.5.17. Proposition. The Hamilton equation 19.5.11 is now Sn = ^2C(i)Uk)+ejSPk,(i)+ej
~ X^O*"!0
Ad
iPk,(i)+ej
•
Proof. The expression obtained for ft must be substituted into 19.5.11. • If some C(j)ujj/ and the corresponding Pk,(i)+e- c a n D e taken as independent variables in the phase space, then the last equation is equivalent to the system d c
j (i)uk}
= fih/6pkt{i)+e. ,
^2djpk,(i)+ej i
= -Sh/Sc{i)u{kl)
,
(19.5.18)
where H = hdxi A • • • A dxn and Pk,(i) — Pk,(i)dxi A • • • A dxn having the canonical form (19.1.3). Notice that there are n momenta corresponding to one coordinate, and the canonical equations (19.5.18) are not of Cauchy-Kovalevski type, i.e. are not solved with respect to the time derivatives (see [deD]). 19.6
Variational Bi-Complex of a Differential Equation. First Integrals
19.6.1. Up to now we considered a free variational bi-complex. In this section constraints in the form of differential equations Qk=0,
k = l,...,m,QkeA
(19.6.2)
will be imposed. The elements Qk generate in A a differential ideal JQ . Let AQ — A/ JQ. We assume that the system of generators {uk} of the algebra A can be replaced by another system of independent generators such that the set {d^Qk} is a part of the system, call them "generators of the type
Field Lagrangian and Hamiltonian
351
Formalism
(a)", the rest of generators, whatever they are, will be "generators of the type (b)". For example, this is the case if the system of equations is of the "pseudo-Cauchy-Kovalevski type" (pCK) (with respect to the variable xi) (see [Tsu]) which means that Qk = d(kuk + Ql and Q*k do not depend on {u\j*'} with n > jiIf a system is of the pCK type, then the set of elements of two kinds (a)
flWQfe,
V(i)ifc (19.6.3)
(b) l,{v%'},
where ii < j k
can be taken as generators of AIt is clear that instead of x\ one can take any other variable. Examples. (1) The equation Q = uxx — uyy = 0 can be considered as pCK both with respect to x and with respect to y. In the first case the set of generators of the type (b) is 1, {uy...y}, {uxy...y}, in the second case they are 1, {ux...x}, {ux...xy}. This example shows that the choice of generators (b) is not unique. (2) The same equation in the cone variables, u^n = 0. It is no more of pCK type, however the needed property holds. The generators of type (b) are 1, {ui..<},{ur,...r,}. (3) The equation uxuy + 1 = 0 does not have the property we are talking about. An element / G A belongs to JQ if and only if each term in its representation in terms of the generators (19.6.3) contains at least one factor of the type (a). Now we develop a theory of characteristics quite similar to that in Sec. 16.2. Let f e JQ then
f =Y1
a
Ui)d(i)Qk .
«fc,(i) € -^ •
k,(i)
Put
*/,* = {j2(-l^dii)a^))Q
. * = 1, • • •, m.
The subscript Q denotes the natural projection A -> AQ: The set Xf = (X/,i,...X/,m) ^ An is called the characteristic of the element / .
352
Soliton Equations and Hamiltonian
19.6.4. Lemma. If J2ak,(i)d^Qk the same as in 16.2.3.
= 0 then all ak^
Systems
£ JQ. The proof is •
19.6.5. Lemma. If / = djg, g € JQ, then Xf = 0. Proof. See 16.2.5.
•
19.6.6. Lemma. The characteristic is well defined. Proof. See 16.2.4.
D
19.6.7. A vector field £ is said to be tangent to AQ if £JQ C JQ. A tangent vector field is defined on AQ. Two tangent vector fields are equivalent (modulo the equation) if (£ — r))A C JQ. Then the coefficients £(k),j and £? of these vector fields are equal as elements of AQ. The linear space of the equivalence classes is denoted as TAQ. It will be given two notions of equivalence of forms, a strong one and a weak one. 19.6.8. Definition. A form u> is equivalent to zero modulo the equation in the strong sense, w = 0, if all its coefficients belong to JQ. A form is equivalent to zero in the weak sense UJ = 0 if the result of substitution of tangent vector fields into this form is zero in AQ. It is easy to see that this condition can be expressed thus: if the form is written in terms of generators (19.6.3) each term contains a generator of type (a) either in the coefficient or under the sign S while a form which is equivalent to zero in the strong sense has the elements of type (a) in each coefficient. Q
__
QQ
Example. 5Qk = 0 but 5Qk ^ 0. If ua; Q== 0, 0. then t(f)w Q= 0 for each £ e TA. If u = 0, then *(f)w = 0 for £ G TAQ. The classes of equivalence of forms make complexes AQ' = {AQ 'q>} and AQQ = {AQQ,}. The complex AQQ has the only differential d. Let Aip'q) = A^/dA(p'q-l\ A^ = {A^}. There is a differential S in this complex, and all the sequences are exact with respect to this differential (the proof is the same as in 19.4.3 where q = n). The mapping d can be considered as a mapping Ap'q -> Ap,q+l. Vector fields dh can be transferred to A^ since they commute with d, see 19.2.16. Finally, we introduce the complex A^'q) = A$q)/dA{™-1], A^ = {A^}.
Field Lagrangian and Hamiltonian
353
Formalism
19.6.9. Proposition. All the sequences AP
A(P,
A(P>I+2)
are exact if p > 0. Proof. The proof of 19.3.4 remains valid since the operator D has the property: if all the coefficients of a form w belong to JQ, then so do the coefficients of Dw, i.e. the operator D can be transferred to AQQ. • Note that if p — 0, then the sequence is not exact (otherwise there would be no first integrals). The proof of exactness by diagram chasing cannot be carried out in this case since there is no differential 6 in the complex AQQ. 19.6.10. Definition. {
First integral is an element F € AQ'™-
such that
n)
dF = 0 in A Q' . Let us give an explanations to this definition. Suppose we have a partial differential equations, for example, Q = ut — ux = 0. A first integral is an expression Je = J f(t,x)dx such that dtJ = 0 by virtue of the equation, or J ftdx = 0. With appropriate boundary conditions this means that ft = gx, or d(fdx + gdt) = 0 mod Q. The differential form F = fdx + gdt is what we call the first integral. Further, there are trivial first integrals, e.g. / Qdx. It satisfies the above definition of the first integral, however, it is not interesting. It vanishes itself by virtue of the differential equation, and no symmetry of the equation corresponds to it, as we shall see below. We must identify these integrals with zero. This explains the subscript Q in the relation F G AQ '"~ '. If F is a representative of the class then dF = adx\ A • • • A dxn where a e JQ. 19.6.11. Definition. The characteristic of the element a of the ideal is called the characteristic of the first integral F. According to Lemma 19.6.5 the characteristic of a first integral is independent of the choice of a representative. 19.6.12. Lemma. If a representative of the class F is taken in the reduced form i.e. contains only generators of the type (19.6.3b) then a has the form ^2akQk and XF = {a-k}The proof is obvious. •
354
Soliton Equations and Hamiltonian
Systems
19.6.13. Proposition. Let Qk = SA/Suk, i.e. let the equation have a variational Euler-Lagrange form. Let F be a first integral with characteristic XF = {XF,k}- Then the vector field dXF has the property dXFA = 0 in A(°>n\ Conversely, if a vector field da preserves the action functional, i.e. daA — d( ), this vector field corresponds to a first integral with the characteristic a (Noether's theorem). Proof.
dXFA=J2
X%dA/du^ = J2 XF,kSA/Suk in A^
k,(i)
.
k
On the other hand, dF = YJ aui)d^5A/5uk
= J^ XF,kSA/Suk
in
A{0'n).
k
Therefore dXFA = 0 in A^°'n^ as required. Conversely, let daA = dFx where Fi 6 A^^'^. dFx = Y^ a^dA/du^ k,{i)
= J2 akSA/6uk
We have + dF2 .
k
Then d(Fi - F2) = 0 and F = Fi - F2 is the needed first integral.
•
19.6.14. Remark. Noether's theorem can be formulated in a more general form considering more general vector fields £ = da + J2a*di. If £A = dF\, i.e. di(£a*di)A + ^ak(5A/5uk) + dF2 = dFi, then d(»(£a:0<)A + *,2-.Fi)£o, and i(%2a*di)A + F2 — F\ is the first integral. For example, if n = 1 (classical mechanics), A = Ldt, and L does not depend explicitly on t: d'tL = 0, we have (dt — dt)A = 0, i(dt - dt)(d + S)A + (d + S)i(dt-dt)A = 0, di(dt)A-i(dt)SA = 0, d i ^ A + i ^ d f l 1 = 0, l and, finally, dt{L — i(d)£l ) = 0. This is the conservation of energy law. In the canonical variables "coordinate-momentum", this equation reads dt(L — ^LiPiQi) — 0- The equality =, "by virtue of the equation", means that the derivative dt is taken along the trajectory, i.e. this is the total derivative d/dt.
Field Lagrangian and Hamiltonian
19.6.15. Lemma. If ^Suk 1-
355
Formalism
A ipk + dw 1 ' n 1
"^QQ then <^fc = dw 1 '"" 2 .
= 0,
0 and a form w '™
-2
n
exists such that u){ ~
G =
Proof. The proof of uniqueness in 19.4.4 with the aid of Tulczyjev's operator D remains valid since this operator acts also in AQQ. • 19.6.16. Proposition. The vector field dXF preserves the ideal JQ, i.e. it belongs to TAQ. (In this sense the vector field is a symmetry of the equation: it transforms solutions to the solutions.) Proof. Let us act with this field on the equality SA = ^ 5v,k A 5A/Suk —
dnw-. SdXFA = ^<5ufc A dXF(6A/Suk)
+ ^ ( 9 X F 5 u f c ) A SA/Suk + d( ) .
Proposition 19.6.13 yields 5dXFA = 0 in A. Hence YlSuk/\dXF(SA/5uk) £X
+
J2Suk/\dXF(SA/6uk)Q^d(). Then Lemma 19.6.15 gives dXF(5A/duk) dXFJQ C JQ.
= 0, i.e. dXF(SA/5uk)
C JQ and O
19.6.17. The main proposition about first integrals. The relation SF = -»(&•)« + dul'n-2 (where w 1 , n _ 2 € A]f~2
,
£F = -3XF
is a form) holds.
Proof. We have F € An'n~ and dF = 0. Let F be a representative of the class having the reduced form. Then dF = ^ XF,fc<5A/<Sufc (see 19.6.12). We have di(dXF)£l
= —i(dXF)d£l = —i(dXF)2_\^uk
A d(SA/6uk)
= - ] P XF,kS(5A/6uk) + Y2
SukdXFdA/6uk
356
Soliton Equations and Hamiltonian
Systems
(19.6.5 is used). On the other hand, Q
d5F = -SdF = -Sj2xF,kSA/Suk
=? -
Y,XF,kS(5A/5uk),
whence d(6F - i(dXF)Cl) = 0. It remains to apply Proposition 19.6.9.
•
It may seem that it is proven more than stated, the equality = instead of =. However this happened only because a special representative (reduced) was chosen as F. 19.6.18. Example. The energy-momentum tensor is a set of first integrals Tj if the Lagrangian does not depend on the times explicitly. Proposition 19.5.5 shows that the characteristic of the first integral Tj is XTj = {-djUk}, k = l , . . . , m . Hence dXT. = -Y,ul' <3/d«£ = -&,. The same was obtained earlier in 19.5.7. In physics these symmetries are shifted in time and space, and the first integrals represent the densities of the energy and the momentum.
19.7
Poisson Bracket
19.7.1. Let V c -4Q'™ _ ' be the subspace of first integrals. As it was proven, a vector field £p — —dXF € TAQ corresponds to each F £ V such that 5F = -»(fr-)fl in ^ g ' " _ 1 ) ' . For two elements F, G G V we define the Poisson bracket {F,G} = £FG.
(19.7.2)
19.7.3. Proposition. The space V is a Lie algebra with respect to the Poisson bracket and £{#,G>
=
[£F,£G]-
Proof. From dG = 0 we obtain (,FdQ = 0 since & G TAQ, and d£,FG = 0 since £F commutes with d. This proves that {F,G} G V. Applying the operator I(£G) to both sides of SF = — i(£F)Cl (in AQ) which can be done by the same reason as above we get: teF = - n ( f c . , f c )
(in^)
357
Field Lagrangian and Hamiltonian Formalism
which implies the skew symmetry of { , } . Now, using Proposition 19.2.17 and the fact that the form ft is closed, 5ft = 0, we have (calculations in
0 = i ( f r ) t ( & ) * n = ifofan
-
ifoMtaW
= fci(fr)« - i([ZG,ZF})ft ~ i(ZF)6i(ZG)ft = -&5F
- i({ZG,tF])ft + i(ZF)66G
= -SZaF - *(KG, &•])« =
=
Z{Pto}H=[ZF,ZG]H
= tF{G, H} - £ G {F, H} = {F, {G, H}} - {G, {F, H}} .
U
Two elements of V are said to be in involution if their Poisson bracket vanishes. It is not clear how to extend the notion of the Poisson bracket to more general objects then the first integrals.
19.8
Relationship with the Single-Time Formalism
19.8.1. Let n
ftW = ^nr1]
A
i(dr)dx1
A---Adxn,
r=l n
ft = \~* ftr A i{dr)dxi A • • • A dxn , r=l n
Tj = \ J Tjri(dr)dxi
A • • • A dxn
r=\
be the forms written in coordinates. Let one of variables x\,..., xn be taken as a time, say t = xn. The term with i{dn)dx\ A • • • A dxn = (—\) n ~ x dx\ A • • • A dxn-\ in the statememt of Proposition 19.5.6 for j = n is 5Tnn A dx\ A • • • A dxn-i
= -i(dn)ftn
A dxi A • • • A dx„_i + d^'^w
,
358
Soliton Equations and Hamiltonian
Systems
where Sn ^ means ]T™ dxidi, a differential with respect ton— 1 independent variables, and w is a (1, n — 2)-form with respect to these variables. As usual, = means that the non-phase variables must be eliminated with the aid of the equation {SA/Suk = 0}. Then the differentiation dn is the vector field £ associated with the equation. This equality can also be written as 5 / ••• /
Tnndxi A • • • A dxn-i
= -*(£) /
" / ttndxi A • • • A
dxn^i.
Thus, J • • • J Tnndx\ A • • • A dxn-\ is the Hamiltonian, and / • • • J Cln A dx\ A • • • A dxn-\ the symplectic form of the single-time formalism. 19.8.2. Conversely, if one knows a Hamiltonian and a symplectic form in the single-time formalism then the Lagrangian and all components of the multi-time formalism can be restored. It follows from (19.5.4): A = -dxjATj+dxjAiidj)^^
= -{Tjj-iid^n^dxiA-
• -Adxn
(19.8.3)
If Xj = t is the chosen time variable, Tjj and J r ' are supposed to be known up to dj( ). Then A is determined up to d( ) , just as it must be. 19.8.4. The first example. The KdV equation u = Quu' + u'". Recall that the forms Cl^ and Q are defined on the vector fields da', and Vl^l\da') = \ Juadx, £l(da>,db') = \ Ja'bdx. In order to write these forms in terms of the basic differentials we use a trick, which will be systematically discussed in the next chapter, namely, we make a substitution: u = if'
(19.8.5)
which embeds the differential algebra A = Au into Av. This is the socalled "potential representation", see Witham [With]. The equation takes the form
oo
a(i+1)d duii)
da, = Y, i=0
/
= 5> i=0
oo
(m)
d/V
i+1)
= Y
These fields, originally defined on Au, can be extended to the whole of Ap by oo t=0
359
Field Lagrangian and Hamiltonian Formalism
19.8.6. Lemma. The forms fiW and fi can be written as ftC1) = - / ip'Stp Adx,
Q, = - I5
Proof. £l^\da')
= i(da)-
/ ip'S
to(da.,db.) = i{db)i(da)-
uadx,
f 6
as required.
D 3
2
The Hamiltonian of the KdV-equation is H = J ( u - (u') /2)dx
=
Now we pass to the multi-time formalism. Q^ and Q. will now denote forms in this formalism. Let x\ = x and £2 = t. We know the following components of the forms il^ and {Tj} : Cl\ ' = fi2 f'Sf/2 and Tit = 3 2 T 22 = -<^' +
- i(dt)n[l))dx
A dt = - (-ip'3 + -tp"2 + -
Now all components of the forms il^1' and Ct, and of the energy-momentum tensor {Tj} can be found. We have SA = -(Gip'tp" +
ipSip) A dt I ,
whence fi(1) = n[1} Adx- ftW Adt = -ip'Sip A dx + (3
(19.8.7)
Further, —5\/5
Tt = -i(dt)A +1(4)0(1) = - ( v 3 + \p'a + \<S
360
Soliton Equations and Hamiltonian
v«
_ \>v\
= -Tttdx Tx = -i(dx)A
+ (3(p>^
dx
+
Systems
y"'^ _ ^/y _ i^A ^
+ Ttxdt, + i(5x)SlW = (-v'3
+ \p'n
+ | ^ V ) dt
+ \yndx + ( V 3 + ^'V - y"2 - \
3
+ ' - ^ v " 2 J dt = -Txtdx
+ Txxdt.
The field Hamiltonian is H = dx /\TX + dt /\Tt + A = ( 2
i(dt))Sl
(see 19.5.11). 19.8.8. Exercise. Using the obtained expressions for ft and % in terms of ip check that this equation is, indeed, equivalent to the KdV equation (recall that i(dx)6
= 3(V 2 + lOip'" .
Field Lagrangian and Hamiltonian
361
Formalism
It is also instructive to verify the relation SF = i(d30v'2+10ip>n)n + d( ) directly. The momenta (see 19.5.3) can be found: pi,o = 6 A/Sip' = ( 3
-ipjdxAdt,
P2,o = 6A/5
and Proposition 19.5.13 can be verified. However, no one set of c^ip^ and the corresponding P(j) +e can be chosen as independent variables, thus the representation (19.5.18) has no sense here.
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Chapter 20
Further Examples and Applications
20.1
KP-Hierarchy
20.1.1. At first we discuss the KP-equation in the proper sense: -3uyy
+ (4u - u'" - Gun')' = 0.
Letting u = ip' we get -Z
(20.1.2)
The new equation contains not all solutions of the original one since we integrated it with respect to x and put the constant of integration which can depend on y and t to zero. In order to obtain the Lagrangian of this equation, we multiply the equation by tp and divide every term by its degree in
dt
(20.1.3)
(we have used our right to add to A a complete differential). The variation of this Lagrangian is <5A = - ( 3 ^ y y - \
dilw
where Q,^ = -3
,
364
Soliton Equations and Hamiltonian
Systems
This yields 6A/dip = -3
Sip" A Sip' + Sip'" AStp + Qip'Sip' A Sip)
AdyAdt.
T h e field Hamiltonian is U = - A + {dx A i(d) + dyA i(dy) + dt A
i(dt)}fl{l)
= ( -ip2y - 2ipip' - -ip"2 + 2ip'3 + tp'ip'" )dx
AdyAdt,
and t h e Hamiltonian form of t h e equation is SU = {dx A i(d) +dyA
i(dy) + dt A
i(dt)}£l
(check t h a t this equation is, indeed, equivalent to (20.1.2)). T h e e n e r g y - m o m e n t u m tensor is t h e set of three forms + i(d)ft ( 1 ) = -3
Tx = -i(d)A + (-\ip"2 Ty = -i(dy)A
2
+ 2
= -2ip'ipydx
A dy
- \%p2y + 2ipip' + ^ip"2 - ip'A dx A dt - (2
Tt = -i(8t)A + i(dt)nW = - (%pl - -ip"2 + f>'A dx A dy - Zipyiptdx Adt-
(2ip2 + ip" ip' - tp'"tp - 3ip'2ip)dy A dt.
T h e Hamiltonian of the single-time formalism is
h= f
Tudx A dy = J j (- ^if2v + ^ip"2 - ip'3 j dx A dy
Further Examples
and
365
Applications
(which coincides with the one usually used, see e.g. [Case]). The symplectic form of the single-time theory is w= I
/ Qtdx A dy = 2 / / 5
Jt=const. J
J J
20.1.4. Now we go to the general case. As it became clear earlier one must use a new set of variables {tpi} instead of variables {ui} generalizing the substitution u = ip'. This is the dressing substitution (see (6.1.2)): L = V^/T 1 , where = 1 + J2T ^ d - * - 1 . As we have seen, an operation here is of great importance: division of all the terms of a differential polynomial in {ipi} by their degrees in {(pi1'}. This operation can be performed thus: JQ p~1f(jpip)dp, cf. 19.4.11. Therefore we introduce the notation
A = res | - J p- 1 [(^ ro 0; 1 ) + ,
(^d^U^dp
+ dn(/>-1 • d(j)/dxm - dm(j)-1 • d(p/dxn \dx A dxm A dxn . Proof. First the formula 5res Jo
fPp-l[{
= -res[(
fad"*;1)^
• 0 ; 1 + 8(
)
(20.1.6)
will be verified. It is sufficient to verify the equation obtained by differentiation with respect to p: 5 r e s [ ( ^ 9 m 0 - 1 ) + , (0 p 9'>; 1 ) + ]; 1 m 1 l 2 res[(0 i>ppd9m<]>C )+),+,{cf>pd ( ^ O n+
- r e s ( p ^ K ^ c F V ; 1 ) ^ ( ^ p 5 " ^ 1 ) + ] ) 8
)
366
Soliton Equations and Hamiltonian
Systems
(it was taken into account that pd(f>p/dp = 0 P — 1). We transform
-[4>;1,(4>Pdm4>;1)+]+,
= P
- ( ^ a > p - 1 ) + = -[0p-1,(^>p-1)+]+, S(<^dm4>;1)+
= \HP • 4*;1,
(4>Pdmcl>;1)+}+,
6{
6
res{([[T, U}+, V] + [U, [T, V]+})S + [U, V]TS - [U, V]ST -[[S,U}+,V}T-[U,[S,V}+}T}
= d(
).
Two of the terms are (we use the general property res[v4, B] — d{
))
res{[C7, [T, V]+]S - [[S, U}+, V)T} = xes{[S, U] • [T, V}+ + [T,V\-[S,U]+}
+ d{
) = res{[T,V]-[S,U]}
+ d(
)•
res{[[T, U]+, V)S - [U, [S, V]+}T} = res{[T, U] • [V, S]} + d(
).
Similarly
The expression takes the form res{-[T, U] • [S, V] + [T, V] • [S, U) + [U, V] • [T, S}} + d( = te8T{[U,[V,S\]
+ [V,[S,ir\] + [S,lU,V]]} + 9(
)
) = d(
which proves Eq. (20.1.6). Put p = 1, then 6 res f Jo
p-1[(0P5mp-1)+]^1rfp
where w\ is a form. Then we calculate the rest of the variation: 5ies{dn(j>-'1d4>/dxm -
dm^~1d<j)/dxn)
= (d/dxm)Tes(dn(j>'15(l))
- (d/dxn) res{dm (j)'184>)
),
Further Examples and Applications
367
+ Tes(dn4>-1(d(f>/dxm)(j>-18(f> - dm(j>'l{d<j)ldxn)(j>-15(j> -
{dldxn)xes{dm<j)-l5(j))
- dLn+/dxm)5<}) + dw2 ,
where w2 is another form. Thus, 5\ = xes{<j>-\-dLn+/dxm
+ dL™/dxn + [£+,£"]) - 5
A dx A dxm A dxn + d{—LJ A dxm A dxn
+ res(d7V"1<^> A dx A dxn + dm4>-18(j) AdxA dxm)} , where LJ — u>i -\- LJ2. derivative:
(20.1.7)
This implies the expression for the variational
5A/6ct> = {
+ dL™/dxn + [L™,Ln+})}+ .
Equating this to zero one gets an equation which is equivalent to Eq. (5.1.5). This completes the proof. • 20.1.8. Proposition. The one-form corresponding to the Lagrangian is f2(1) = w A dxm A dxn - ies(dn(l>~1S
= 5u) A dxm A dxn + xes(dn<j>~l54> A
dxm).
The field Hamiltonian is H = - \ + {dx A i(d) + dxm A i(dm) + dxn A i(d n ))0 ( 1 ) = + res J
(i{d)u
p-1[(())pdm(l>-1)+,(^pdn4>-1)+}4>-1dpJdxAdxmAdxn.
The Hamiltonian form of the equation is, as usual, 5% = (dx A i(d) + dxm A i(dm) + dxn A i(dn))Cl. Proof: All this immediately follows from (20.1.7). Now we go to the matrix hierarchies.
•
368
20.2
Soliton Equations and Hamiltonian
Systems
The Zakharov-Shabat Equation with Rational Dependence on the Spectral Parameter
20.2.1. We will discuss Eq. (10.3.13). We have the equation (for convenience, the notations are slightly changed): Uv-Vt
= [U,V],
(20.2.2)
where Afi
U^Uo + J^Uk,
N2
V = V0 +
fc=i
J2Vi i=i
and Tifc
tffc = £ r=0
mi
r 1
Ukr(z - ak)- - ,
Vt = J2
Vl z
^
~ 6')_r_1 •
r=0
Some 6/ can coincide with some of ak's, the other frj's and ak's have no counterparts. One of poles can be at infinity. Then z — ak (or z — bi) must be replaced by z~x. The equation must hold identically with respect to z. It we verify that the principal parts of the left-hand side and the right-hand side coincide at all the poles, then these sides differ by a constant in z. It remains to check that this constant is zero. For example, if there are no poles at infinity, then this condition gives the equation Uov - Voi = [U0, V0].
(20.2.3)
It is easy to see that the number of matrix equations is less than the number of unknown matrices Ukr and Vkr by 1. One of the matrices remains free. Writing the equation in the form [d^ + U, dv + V] = 0 , one can see that there is a possibility of a gauge transformation d% + U = g(d^ + U)g~x and d{ + V = g(d£ + V)g~l. Using this transformation, one can, for example, destroy the matrices Uo and Vo, if there are no poles at infinity. Although the number of unknown matrices becomes equal to the number of the matrix equations, some freedom of choice still remains. The situation is analogous to that in the case of the equation [A, X] = 0 where the number of unknowns equals to the number of equations, and, nevertheless, in the frame where A is diagonal (in the generic case) any diagonal matrix X is a solution. Our equation also has a freedom of choice of several diagonal matrices.
Further Examples and Applications
369
We do not specify here how many diagonal matrices remain free, this will become clear later. We shall see that Eq. (20.2.2) is equivalent not to one Hamiltonian system but to a series of them parametrized by the choice of some diagonal matrices. Each solution of (20.2.2) is a solution of one of these Hamiltonian systems. The differential algebra generated by elements of all matrices Ukr is denoted by Au, that generated by elements of Vkr by Av, and by A the algebra of all differential polynomials in elements of both, Ukr and VkrLet us fix a point ak where U has a pole. The rational function U(z) can be expanded into a Taylor series in a neighborhood of the pole U(z) = r l S-'JXJ Ukr{z — a,k)~ ~ . For r > 0 the coefficients Ukr are the same as above, elements of all matrices Ukr belong to A\j. We consider a generic case when the matrices Uk,nk can be reduced to a diagonal form: Uk,nk = 9koAk,nk9ko: with distinct diagonal elements of diagonal matrices Ak,nk • 20.2.4. Proposition. There exists a transformation di + U = gk{di + Ak)g-k\ where gk and Ak are formal series in z — ak '• oo
(20.2.5)
nk
9k = ^2gkr{z-ak)r
,
Ak =^2,Akr{z
0
-
ak)~r~l,
-oo
and the elements of the matrices gkr and Akr belong to Ay, such that the matrices Akr are diagonal. Proof. Let d< + U = gko(dz + U)g^ . Now U = Y^-ooUkr{z - afc) _ r _ 1 where Uk
5fc0,£ + Uk,nk9k,nk
+1 H
+ Uk,nk-l9k0
r Uk,~igk0
— 9kl-^k,nk
~ 9k,nk + lAk,nk
- gkoAk,nk
~ 0,
~
- ••• - (jkoAk,-l
= 0 ,
370
Soliton Equations and Hamiltonian
Systems
Let gk0 = I while the diagonals of all gkr, r > 0 be zero. Taking the diagonal part of the second equation one can find Af.,nk-i while the off-diagonal part determines gki, etc. After that put gk = gk9koD The matrices gk and gk = gk
satisfy the equations
9k,(, + Ugk - gkAk = 0, (20.2.6) -9k,i +9kU - Akgk = 0 . One must remember that all quantities here, gk, U, and Ak are considered as series in powers of z — ak, i.e. locally, in a neighborhood of ak. Absolutely the same can be done to V in neighborhoods of V s . There will be equations hi,v + Vhi - htBi = 0, (20.2.7) -hi,„ + hV-Bihi
=0,
where hi = Y^'Q' hir(z — bi)r and hi = h^1. With some abuse of notations, the subscript k will always denote a coefficient in the expansion in powers of z — ak while the subscript / that in the expansion in powers of z — bi. When ak = 6j, the subscripts k and I mean the same. 20.2.8. Proposition. If U, Ak and gk are as in Proposition 20.2.4, and U and V satisfy Eq. (20.2.2) then, undressing the operator dv + V in the neighbourhood of ak with the help of gk: dv + V = gk(dn +
Bk)g^
we get Bk = Yl™'ocBkr(z - ak)~r~1 if V has a pole bt = ak, or B = Y^ZX Bkr(z — afc) _ r _ 1 otherwise, where all the matrices Bkr are diagonal. Moreover, Ak,v - -Bfc^ = 0. The elements of the matrices Bkr belong to AProof. We have AktTI-BkA
= [Ak,Bk}
Further Examples and
371
Applications
whence Akr,n ~ Bkr,£ =
2_, s+t=r-l
l^ks,
-Bfct] •
Here Bkt with the smallest number t is Bk,r-i-nk in the term [Ak,nk, Bk,r-i-nk]- If it is already proven that all Bkt with t > r — l—rik are diagonal, then the off-diagonal part of this commutator vanishes which proves that Bk,r-i-nk is also diagonal. Since all diagonal matrices commute, one obtains that AkyV — Bfc,£ = 0. • 20.2.9. Corollary. Matrices Uk can be represented as u
k =
{9kAlg^l)ak
where the superscript "—" means the principal part at the pole afc. In this formula g^ is a nfc-jet: only terms with (z — ak)r with r
(20.2.10)
20.2.11. Remark. The equality (20.2.10) is the only condition imposed on matrices Ak and B\. Therefore we have min(nfc,m/) 4- 1 diagonal matrices (when afc =6;) depending on £ and r\ that can be freely chosen. This is the "diagonal" freedom we spoke of above. If V has at a point hi a pole which is neither of afc, then we switch the roles of U and V. For simplicity, consider here the case when there are no poles at infinity. Let us take the following expression as a Lagrangian:
{
JVi
N2
Y^resak g^idrj+rH^igkAk) +
fc=i
- j S * * * / i f 1 ^ + t~H{)[hiBi)
'52'jLteSattekAkgk1)ak(hiBihl-%
i=i
fc=i;=i
~ S ( 5Z \ak=bi
Tes
^[(9kAkg^^-^
(hiB^^gkogk1
H d£ A d»j, / )
(20.2.12)
372
Soliton Equations and Hamiltonian
Systems
where nk
tni
Ak = S£J Akr(z - a f e ) ^ - 1 ,
Bi = ^2 Blr(z - 6 , ) ~ P - 1 ,
r=0
r=0
nk
mi
9k = '^2l9kr{z-ak)r,
hi=^2hir(z-bi)r,
r=0
r=0
all coefficients Akr and Bir are diagonal matrices considered as given functions of £ and 77 which satisfy (20.2.10). It is also assumed that gk = hi if ak = bi. The last summation runs over all pairs (k,l) such that ak = bi. It remains to explain the operator 5. All expressions are polynomials in elements of matrices gkr or hkr, r > 0 and their derivatives. The operator 5 divides each monomial by its degree (cf. Sec. 19.4.10). Notice that (1) the terms in the double sum vanish if ak = b; since resafc(z — ak)r = 0 when r < —2. (2) The formula seems to be asymmetric: the double sum contains residues at ak but not at 6;. Actually, instead of resafc, one could write —res;,, since if meromorphic function has only two poles ak and bi and if it is 0{z~2) when z —> 00, then sum of both residues vanishes. This transformation will be very useful later. The Lagrangian must be varied with respect to all gk, hi and t. Denote: t~\
= U0 ,
t-%
= V0 ,
(9kAkg^);k
= Uk ,
(hiBth;%
We will prove the following proposition: 20.2.13. Proposition. The variational identity holds: /JVi SC
= tr ( S
res
° * lUk> dr> + *&]tofefffcX + t~1St)
- Y^ res6i [Vi, % + U0]{5hih;1 +
t'Ht)
1
+
(reSa^Uk^^Sgkg^1+iesbl[Uk,Vi]5hih~1)
]P k,l,ak^bi
+ J2 ak=bi
ref
W[ u k,Vi]5gkg k x
\ J
Ad^Adr)
= V{.
Further Examples and
373
Applications
+ dtr j 53^sakUk(5gkg^x + t^St) A d£ w2
\
+ 53res b l Vi(5/i(^- 1 + t _ 1 5 t ) Ad??) . Notice that the third row of this formula can be written as
since res0k[f/fc, VJ] + rest, [t/fc, Vj] = 0. The same about the fourth row: it can be replaced by + J2
re
+ i" 1 **)) A d£ A dq
Sa fe [Uk, Vi]{5gkg^
ak=bt
since iesak[Uk, V{\ = 0 when ak = bi. Now the formula becomes more symmetric, the variations of the independent variables enter only in combinations Sgkg^1 +t~15t and Shih^1 -\-t~18t. This remark has an important implication: varying the variable t we do not obtain a new equation, we just have a sum of all equations obtained by variation of gk's and his. Proof of Proposition 20.2.13. (i) We start computation with the first two terms. 5ti'^2vesakgk1(dri+t-1tri)(gkAk) I
= tT^res^i-g^Sgkg^idr,
+ V0)(gkAk) + gk 1{dv + V0){5gk • Ak)
I
- g^H^St
• t~\{gkAk)
+flfc4 - 1 Jt,,(sfci4fc)}
Ni
= tr]>^res 0 f c {-((d n + V0)(gkAkgk1)
• Sgkgkl
l
+ (-v0uk - uKri + ukv0)rHt} + d^ti^ieSa^SgkAkg^1 I
+
U^St)
+
SgkAkg^lV0
374
Soliton Equations and Hamiltonian
resQfc [Uk,d„ + V0]{5gkg^1 +
= tr ^
Systems
rx5t)
1
+ dv^2resafc UkiSgkg^1 + t'^St). l
Similarly, J t r ^ r e s 6 i h^1 {d^ +
t~\){hiBi)
l
= tr53res 6 l [V r j,a c + I / o ^ M f 1 + * -1 **) l N2
+ <% 2 res6i Vi{5hihlx + t^St). l
(ii) The next term (ak Ni
i^h)-
N2
^^Y^^ieSa^9kAkg^)~k{hiBihl\ l
I
= ^J212TeSa*{(59kAk9k1)akvi
- {OkAkg^Sgkg-1)-^
+uk(shlBlh^ix[ - Uk^Bih^shihrw = -tT^^iiesa^U^Sgkgk1}^ = tr ^
+ Tesh
Uk[Shth^\Vi}^}
^{res a f c [[/ f c , V{\5gkg^1 + res6l [£/fc, Vi]5hihl1} .
(iii) Finally, consider the last term involving the poles common to U and V, i.e. ak = &/• We want to prove: -Sti
SireSa.UgkAkgk^a^ihiBih^^gkogk1)
Y^ a,k=bi
= tr J2
ies
ak[uk,Vi]Sgkgkl•
ak=bi
(Recall that hi = gk since ak = h.) We shall prove more: each term of the left-hand side sum equals that of the right-hand side sum with the same k.
Further Examples and
375
Applications
The operator S can be written analytically (cf. Sec. 19.4.11). Let 9k(p) = go + P X a ° 9kr(z — Ofc)r and let U(p) and V(p) be U and V where gk(p) is substituted for gk- Then for any differential polynomial /(
-Str
/ p'1 Tes[U(p),V(p)}g0g-1(p)dp Jo
tTTeS[U(p),V(p)]5g(p)g-1{p).
=
The needed formula is obtained by letting p = 1. For p = 0 the left- and right-hand sides vanish. Therefore it suffices to prove the formula obtained by differentiation with respect to p and multiplication by p: -5trves[U(p),V(p)}g0g-1(p)
p(d/dp)tTTes([U(p),V(p)}8g(p)g-1(p)).
=
Obvious relations hold (here g means g(p)): p{d/dp)g{p) = g(p) -
5 0 , P(d/dp)U(p)
= ((g - 9o)Ag-1 - gAg-^g = -goiAg-1)-
+
= -{gog-'igAg-1)-)Similarly, p(d/dp)V(p)
p(d/dp)(gAg-l)~
=
g0)g~1)'
-
(gAg^gog'1)+ ((gAg-1)-gog'1)-
= [U,gQg~x\- .
= [V(p),g0g~1]~ • Further,
SU(p) = (SgAg^-gAg-'Sgg-1)-
= [6gg-\U(p)],
6V(p) =
[5gg-\V(p)}.
Thus, we are to prove t r r e s { - [ [ ^ - 1 ) ^ - ) V ] f l 0 f l - 1 - [U, [5gg'\ - [U, V]5g0g-1 + [U, = trres{{[U,g0g-1}-,V}Sgg-1
V^gog-1
Vjgog-Hgg'1} +
+ [U, V]5(g - gQ)g-1 - [U, V]5gg-\g
[U^gog-^Sgg-1 - g0)g-1} .
376
Soliton Equations and Hamiltonian
The terms with the variation 5go cancel. Denoting S = gog Sgg^1, rewrite the equality that we are going to prove as
l
Systems
and T =
trres{[t/, V)TS + [[T, U]~, V}S + [U, [T, V}~]S - [U, V]ST + [[U,S\-,V]T+[U,[V,S\-]T\}
= 0.
The second and the sixth terms: trres([[T, U]',V]S
+ [U, [V, S]-]T}) = trres([V, S][T, U]~ + [V, S}~{T, U}) =
tiTes[V,S}[T,U}.
The third and the fifth terms: trres([C7, [T, V}~}S + [[U, S]~,V}T) = trres([S, V][T, V}~[S, U]~[T, V}) =
tvies[S,U][T,V}.
All the terms together are tr res([U, V] [T, S] + [V, S] [T, U\ + [S, U] [T, V] = [{U,V},T}S+[[T,U],V}S+[[V,T},U)U}S
=0
by virtue of the Jacobi identity. Summing up all the obtained expressions, we obtain the required formula. • We already noticed that the equation 5C/5t = 0 does not give anything new, just a sum of all preceding equations. As usual, denote by Sf/Sg^r a matrix with entries {Sf/5gkr)ap = <5//^(5fer)/3« and by 5f/Sgk the series T,(sf/S9kr)(z - afc) _ r _ 1 . Then resak(Sf/dgk)6gk = Y,r(sf/S9kr)$gkr20.2.14. Corollary. S£/6gk
9ll
uk,dn + Yvi\ J a.
if there is no / such that ak = h, 8C/5hi
K1 dt +
^U^Vt
Further Examples and
377
Applications
if there is no k such that ak = bi, and
SC/Sgk=\g^
Uk,dn + \ V
h
+ {uk,vl}
+
iiak = b(. 20.2.15. Corollary. The system SC/Sgk = 0, SC/Shi = 0 is equivalent to Eq. (20.2.2). Proof. Equating expressions of all variational derivatives of the Corollary 20.2.11 to zero we obtain that all principal (singular) parts of Uv — V% — [U, V] vanish which implies that the left-hand side of Eq. (20.2.2) is identically zero. (If F = ^ £ ° Fr(z—ak)r, G = ^ 0 Gs(z—ak)~s~1, F0 is anon-singular matrix and (FG)~k = 0 then, as it is easy to show, G = 0.) 20.2.16. Two remarks. (1) It seems that we have not used the condition (20.2.10) when we derived the proposition and its corollaries. This is true. The situation here is the following. If one has a system of equations and then declares a part of unknowns to be given (as we did here when we considered diagonal matrices Ak, Bi as being given), he gets an overdetermined system for the rest of unknowns. He must impose on the "given" quantities some constraints which follow from the system of equations. Thus, Eq. (20.2.10) are nothing but the compatibility conditions for the obtained equations. (2) In fact, nothing would change, if we variated also Ak and Bi as independent variables. It is easy to check that in this case the equations 9k,it+Vgk = 0 appear if there is no bi = ak (correspondingly, —hi£ — [U, Vj]) and the sum of these equations if ak = bi, i.e. the same system as before. 20.2.17. One can obtain the symplectic form from the formula for the variation of a Lagrangian (see Eq. (19.5.2)). In our case this gives the following. The one-form is /JVi
"
(1)
=~
tr
I 5Z res«* U^9k9kl
+ *-1**) A d£
\fe=i N2
+ '^2resb,Vl(5hihfx
\
+i - 1
(20.2.18)
378
Soliton Equations
and Hamiltonian
Systems
and il = and) = - t r 2 r e s a f c \6gkg^,Uk}-k
A (6gkg^
+ t~Ht) A df
\fc=i w2
+ 5 3 resb« \Shihr\
v
i]b,A o^r 1 + t ~ l s t ) A rf??
+ 5 3 res W Uk{5gkg-1 A fy^1 - t _ 1 5t A t _ 1ft)A d£ fe=i W2
+ 53
\ res
l
1
1
_1
h Vi{5hihl A Shihf - t~ St A i ft) A di) j .
J=i
/
(20.2.19) The field Hamiltonian can be calculated using Lemma 19.5.9: % = —C + d£ A t ^ J f l W + drj A t ^ n * 1 * . This yields
{
JVi
N2
-Y,Y.res^9kAkg^)-k{hlBlhi\ fc=i(=i
+ 5 ( 5 3 ^^KdkAkg^-^ihiBih^-jgkog^1] \a,k=bi
\d£Adr). /
)
(20.2.20)
We go now to the first integrals. 20.2.21. Definition. A resolvent at a pole ak is a series oo
R
ak = 5 1 R«*AZ - ak)r = gkCkg^1, fc=0
where gk is the same as 20.2.4 and Ck is a constant diagonal matrix. 20.2.22. Proposition. Resolvents satisfy the equations Rak,S = [Rak,U]
and Rak,v
= [Rak, V]
(20.2.23)
by virtue of Eq. (20.2.2). The equality must be understood as equality of series in z — ak.
Further Examples and
379
Applications
Proof. The equations can be written as [d^+U, Rak] = 0 and [dr,+V, Rak] — 0. Undressing these equations with the aid of the matrix gk and using Proposition 20.2.8, we get it. • 20.2.24. Proposition. Let Fak = tr Rah(Uzd£ + VzdTi). The relation dFak = 0 holds by virtue of Eq. (20.2.2). Proof. Using (20.2.2) and (20.2.23), we have dFak = tr((RaktVUz
+ RakUz,n)dr) A ^ + {Rak,tVz + RakVz^)dS, A drj)
= tr(~[Rak,V}Uz
- RakUz,n + [Rak,U]Vz + RakVz,^
A 6rj
= tr(Rak [UZ,V] + Rak [U, Vz) - Rak [U, V]z)d^ AdV = 0. 20.2.25. Corollary. Expanding Fak into a series oo
*"«* = £
Faktr(z-aky
- n i t —1
we get dFaktr = 0 by virtue of Eq. (20.2.2), i.e. FaktT are first integrals of the equation. They belong to A. In the same manner one can construct resolvents R^ and first integrals at the poles 6; which are not poles of U. 20.2.26. Proposition. The following variational relations hold:
where d^( and 5V and Proposition use (20.2.6)
5tTUzRak=tT{SU-Rak)x+di(
),
6trVzRak=tr(6V-Rak)z+dv(
),
) and dv( ) are derivatives of one-forms in variations SU their derivatives. Proof does not differ very much from that of 9.3.6. We leave it to the reader. Instead of (9.3.5) one must and (20.2.7). •
20.2.27. Corollary. SFak =5trRak(Uzd^
+ Vzdr,)
= tr dzRak{5U A d£ + 5V A drj) + d{
).
(20.2.28)
380
Soliton Equations and Hamiltonian Systems
We go now to a construction of vector fields (symmetries) corresponding to the first integrals. They have a general form
a
(fe=i (i)
9k
fc=i
an
i=i
l
(i)
J
(20.2.29) where a, ft and 7 are matrices, (i) a multi-index, g£' = J^g* g^:(z — ak)r, 9/dg^ = Zn0k(d/dg£)(z - ak)-*-\ a£> = £o°° ak% - a , ) - etc. At the common poles of U and V: afe = bi and, therefore, #& = hi, and one must keep only one, either the first or the second, term. 20.2.30. Proposition. The vector field £fc0,p-i corresponding to F afco)P _i is given by Eq. (20.2.29) with the following parameters: a
k9kl = ~P^2l(z ~ ak)~r~1Rak0]P(z - ak)r , r=0
mi
fr hr1 = -pJ2l(z
- ky^RaJpiz
- hY ,
7 = 0,
(20.2.31)
r=0
where the subscript p denotes the coefficient in (z — ako)p in the expansion in powers of z — ako. Proof. The equality of Proposition 19.6.17: SFak0
)
(by virtue of Eq. (20.2.2)) must be verified. We have SUk = [5gkgk\Uk]-k SVi = [ShihJ-\Vi]^ ,
,
SU0 = r
1
^ - t^StUo ,
SV0 = t~l5t^ - r^tVo
•
According to Eq. (20.2.28), SFakQ = tr dzRako {6U Ad£ + 6VA dq) + d(
)
J2i59k9k\Uk}ak A di + ^ [ ^ ' V 1 . yl]b, 1
+ ( r Hie - t-x8tU0) Ad£ + (rl5tn
A d
V
1
- t^StVo) A drj \ + d{ ).
Further Examples and Applications
381
Using integration by parts, one can replace Rak t~18t^ by Rak t~1t^t~15t — Rako,£t~15t = RakQUot~l5t — [Rako,U]t~16t and, similarly, for RakQt~l8tv. The result is:
SFako = dz tvRako I J2i69k9k\Uk}:k A dt + YfiWT1, Ife=i
V
iK A dq
(=1
-[U-U0,rl5t]/\dZ-[V-V0,rl5t]Adr)\+d(
).
(20.2.32)
All the terms are understood as their expansions in powers of z — ako • We find the coefficient in (z — ako)p~1, «JF o , 0 , p _ 1 =pres a i t 0 (z-a f c 0 )-''- 1 trJ2 O f c 0 {.--} + d(
),
(20.2.33)
in the braces there is the same expression as in the preceding formula. Now we find ^res afc ([^fc3fc 1 ,C/ fc ] + [i~ 1 ^,i[/ f e ])a f c g^ 1 A^ fc=i N2
-\
fc=i
J
+ J2Tesbl([^ih^1,Vi] + [t-15t,Vi})pihl1Adi1\.
(20.2.34)
The vector field (20.2.29), (20.2.31) should be substituted into this formula, and after that one has to prove that the result coincides with Eq. (20.2.33). To this end both Eqs. (20.2.33) and (20.2.34) will be transformed since Eq. (20.2.34) involves residues at all the points ak,bi while Eq. (20.2.33) that at only one point ako and it is difficult to compare them. We transform them using the property of all meromorphic functions, sum of all residues, including that at infinity, is zero. Consider the term with d£ on the right-hand side of Eq. (20.2.33): pres afco (z - a f c o ) - p _ 1 tr Raico [Sgkg^ + rl5t,
Uk\~k A d£.
Let k ^ fc0. Then Rak can be replaced by the segment of the series Rak (P) = Y%Rak ,AZ ~ ak0Y• The above expression becomes a rational function which is 0{z~2) at infinity. We can rewrite it as - p r e S o J z - O f c J - P - H r f l a ^ f o J f a f c ^ 1 + t-16t,Uk}~k
A d£.
(20.2.35)
382
Soliton Equations and Hamiltonian Systems
The term with drj transforms similarly. If k = ho, then the corresponding term is pTesako (z - akJ-P-1
tiRako
(p + nk + l)[5gkg^
+ t^St,
Uk}~ko A d£. (20.2.36)
If there is / = Zo such that b/0 = ako, then among the terms with drj there will be a term pres afco (z - akJ-P-1
tr RakQ (p + nk + l)[
Now let us transform Eq. (20.2.34) taking into account Eq. (20.2.31). If k ^ ko, then \{z - ak)-T-lRako]p
= xesako{z - ako)~p~1[{z
-
= resafco(z - ako)-p-xRako{p){z = - r e s a t ( z - akoyp~lRako{p){z =
a^^R^J - afc)_r_1 -
ak)~T^
-{(z-ako)-p-1Rako(p))r,
where the subscript r denotes here the coefficient in the expansion in the powers of z — ak (and not of z — ako). Then (20.2.31) becomes ctfcffjT1 = P{z ~
ako)'p~lRakQ{p),
where the right-hand side is understood as an expansion at the point ak. Substituting this into (20.2.34), we obtain exactly the expression (20.2.35). In the same way we discuss the term with dr). If k = ko, then a
ko9ko
nk = -P^2Rak0,p+r+l{z r=0
-
dko)r
= -p[Rak0(P + nk + l)(z - a f e j - ' - 1 ] ^ • Substituting this into (20.2.34), we obtain the term (20.2.36). The same occurs with the term containing dr]. • 20.2.37. Proposition. The Poisson bracket of the first two integrals FakiT (Sec. 19.7.1) (they can also be Fbur) is zero. Proof. We have to prove that for any two first integrals Faki iPl and Fak iP2 the form {Fakl,Pl,Fa is an exact differential. We P2] = ^(Ukl,P1^ak2,p2)
383
Further Examples and Applications
have (see (20.2.29) and (20.2.34)):
U=i
fc=i where ak'
'
J
and others are determined by (20.2.31), e.g., a{ ]
k 9k
1
= -Pi Y2((z
_ a
fc)" r _ l f i « f c l )PI {* - o-kY ,
r=0
and the subscript pi means the coefficient in the term (z — a^ ) P l in the expansion in powers of z — akl • Further,
{
Ni
nk
E
[("fc^Sfc^n.f/fe.n+^Kaif^fc1)^^
E
fe=l ri,r2=0
iV2
mi
+E E
>
K/fVVi.^n+^K/fVM.?}
i=l ri,r 2 =0
{
JVi
E
J rife
E
i((Z~ak)~ri"lRakl)PiUk,r1+r2]
fe=l r!,r2=0
dots mean a similar term with drj. Let us write the generator A{z\,z2) of the right-hand sides with different pi and P2 multiplying the right-hand side by {z\ — ak1)Pl{z2 — afc2)P2 a n d summing over all pi and P2. It suffices to show that the generator is an exact differential. We get
i
Ni
nk
E
E
KZl
-akyri~lRakl(zi)Uktri+r2]
fe=l r i , r 2 = 0
x (z2 - a f e ) - ^ - 1 - ^ . (z 2 )Ade + - - - i -
384
Soliton Equations
and Hamiltonian
Systems
Using the formula
(^-^rri-l(z2-akr^-1
(z2-*i) 5Z = (zi - ak)-r~l
- (z2 - afc) - " 2 - 1
we obtain Pi V 2 ^ 2 - zi)A(zi,z 2 ) = t r ^ 3 ^ ( i? afcl («i), t—1
Rak,{Zl),
uk.
(z3-ofc)r+1.
r
^ -
^fc,r
rt.i.V+1 (zi -- afc)
^fc2fe)
RakAz2))dz +
= tr([i?0fci (zj), [ / ( z i ) ] ^ (z2) + [Rak2 (z 2 ), £/(z2)]JRafci ( Zl ))de + • • • = ti(d^Raki
(Zl) • Rak2 (z2) + dsRak2 (z 2 ) • Raki {Zl))di + •••
= ditv{Raki{Zl)Rak2{z2))di
+ --- = d{
).
a
A construction of solution to Eq. (20.2.2) was suggested by Harnad et al. [HSS]. 20.3
Principal Chiral Field
20.3.1. The theory of the principal chiral field (see 10.4) will be presented here from the point of view of multi-time formalism. This is, in fact, a special case of the preceding system, with two poles, but we treat it independently. Equation (10.4.6) can be written as Mv-Nt
= [N, M],
Mv + Ni = 0.
The first of these two equations is equivalent to the existence of a matrix g(£,r]) such that M = g^g-1,
N =
gng-1.
Now take A = t r 0 ? 0 _ 1 5 „ g - 1 d f A dr\ as a Lagrangian. We have SA = -ti(Mr, + N^Sg • g'1 Ad£Adr} + dtr(-Md£
+ Ndrf) A 5g • g~x
Further Examples and
385
Applications
whence 6A/5g = -g-1(Mri
+ Ns),
where SA/Sg is as usual, the matrix (SA/Sg)^ — SA/Sgpa. Therefore, the equation SA/Sg = 0 is just the remaining equation Mv + N^ = 0. According to the general rule we find ilW = - t r ( - M d £ + Ndr)) A Sg • g'1, n = SSlW = -tr{(-<JM A d£ + SN A drf) A Sg • g'1 - (-Md£ + Ndrj) Ag-g'1
ASg- g'1} = tx{~8gv Adn + Sg^ A d£)g~l A Sg • g~l.
The energy-momentum tensor is the set of two forms T4 = - t r M2d£,
T„ = tr N2dn.
The field Hamiltonian is H =A =
trMNd£Adr}.
The Hamiltonian form of the chiral equation is SU = {di A i(d() + dr]A i(0„))ft (check directly that this is equivalent to the desired equation!). In this case the coordinate—momentum variables can be used. We choose the elements of the matrix g as coordinate, and the elements of the matrices P£ = SA/Sg^ = g~xN and pv = SA/Sgv = g~lM as momenta. Then fi'1) = —ti{p^drj — pvd£)Sg ,
0 = —tr Sg(Sp^drj — Spnd£),
H = tr gpngpzd£ A dn, and the Hamiltonian equations have the form dig = S7i/5pi,
dr,g = S'H/Spri,
d^
+dvpv
=-STi/Sg.
20.3.2. Resolvents and First Integrals We have seen in Sec. 10.4 that Eq. (10.4.6) are equivalent to the zero curvature equation [d^ + U, dv + V) = 0, i.e. Uv-Vt
= [U,V\
(20.3.3)
386
Soliton Equations and Hamiltonian
Systems
where (see (10.4.7), A is replaced by z) U = -M/(z + 1),
V = N/(z - 1).
The equation must be satisfied identically in the parameter z. According to the presence of two poles, there are two kinds of the resolvents which are series in z + 1 or in z — 1: oo
oo
(a) fl = £fl fc (z + l)fc, (b) fl = £ i 4 ( z - l ) \ fc=0 fc=0
they are common solution of the equations Re = [R, U],
Rn = [R, V]
(20.3.4)
and the matrices U and V are assumed to satisfy Eq. (20.3.3) which is the compatibility condition for the system (20.3.4). Let F = tr(Uzd£ + Vzdri)R,
Uz = dU/dz,
Vz = dV/dz.
(20.3.5)
20.3.6. Proposition. By virtue of Eq. (20.3.3) we have dF = 0. Proof. Indeed, dF = ti(-Uzv
+ Vzt)Rd£ Adr] + tr(-f/ z i? r ? + VzRi)d$, A drj
i tr(-[C7, V]z - UZ[R, V] + VZ[R,U])d£ Adr] = 0.
•
The expression F is a generator of first integrals, i.e. the coefficients of its expansion in powers of z (or z~x) are first integrals. 20.3.7. Proposition. The relation 5F = dz ti(5U A d£ + 5V A drf)R + d{ ) holds. Proof. It can be proved as 20.2.27.
•
20.3.8. Proposition. The characteristic of the first integral (20.3.5) is a = -dz(z(z2-iy1Rg)
(20.3.9)
(this is understood in the sense that the coefficients in the expansion of (20.3.9) correspond to those of (20.3.5)).
Further Examples and Applications
387
Proof. The following equality should be verified: 5F i i(da)U + d{
)
(as usual, = means equality by virtue of the equation, this time of Eq. (20.3.3)). Let b = Rg,a= -dz(z(z2 - lyH. We have i(db)Sl = -txiiRg^g-Hgg-1 The expression ti(Rg)^g~18gg"1 one hand,
A d£ - Sg^R
A d£ - (£ o r,)} .
can be transformed in two ways. On the
triRg^g-^gg-1 = trid^RSgg-1)
+ Rg^Sgg'1
= txid^RSgg-1)
- g^lSgg-^R]
- RSg^1
RSgg^g^g'1)
+
+ 25gg-1Rgig-1
RSg^g'1).
-
On the other hand, triRg^g^Sgg-1
§ tr([fl, tfjfor1 +
Rg^Hgg^)
= tr((z + iy1[gig-1,R]6g
• g~x +
= tr((z + l ) - 1 ^ , Sgg-^g-'
Rg^Sgg-1) Rg^Sgg-1).
+
Equating one expression to the other we have tr^z+l)-1^-1^,^-1] S
tvi-d^RSgg-^-Rg^g-Hgg-'+RSg^g-1).
Substituting this into the expression for i(db)Q we obtain
W
= -trUiRSgg-1) + ^{-d^gg-^-Rg^g'Hgg-1
+R6gig-1)
+ 28gg-1Rgig-1 - 2R6gig-1\ A d ? - ^ ! ) , ^ -z) = -tr {-htiMgg-1) -(^i),ze-z).
- ^(-Rg^Sgg'1
+ Rdg^1)} dt;
388
Soliton Equations and Hamiltonian
Systems
The first term, together with the corresponding term with dv make a complete differential. We have
i(da)n = -dzz(z2 - \yxi{dh)si = -dztA-^—(-Rg!:g-Hgg-l
+ R8gig-1)/\dZ
\ z+1
+ j^ji-Rgvg^Sgg'1
+ RSg^'1)
\ A dq + d(
).
Now, according to (20.3.7), we have SF = dz tr(6U A d£ + 5V A dq)R + d{
)
= dztr!-j^jS(g^1)RAd^+^-[S(gr,g-1)RAdi1\+d(
= dz trj—J— (-RSg^g-1 + Rg^dgg'1) \z
+1
+ — - { - R 8 g „ g - 1 + Rg^dgg'1) Thus, i(da)ft = 5F + d(
)
A d£ Ad-q\+d{
).
) as required.
•
20.3.10. Remark. One can ask: do we really need all these intricacies, why, in order to find the characteristic of a first integral, just not to use its definition? For example, why the following reasoning is not good? F = tr(Uzd£ + Vtdq)R. dF = tr{(-UZT) + Vzi)R + (-UzRr, + VzR^)}d^ A drj. Using R% = [R, U] and Rn = [R, V] after obvious transformations, taking into account that Mv — N^ = [N, M] and Mn + N( = —g5A/6g, one can get dF = tr{(d/dz)(-Un
+ V^ + [U, V\)}Rd£ A Sri
= -(z2 + l)(z 2 - l ) - 2 t r ( M , + Nf:)RdZ A 5-q = -[dzz(z2
- l)'1} • trRg(5A/5g)d{ A Sr)
from where the characteristic is —\dzz{z2 — l)}Rg. This expression differs from the one given by (20.3.9): now, the operator d/dz acts only on the numerical coefficient z(z2 — l ) - 1 . What is wrong? The matter is that the
Further Examples and Applications
389
equations R$ = [R, U] and Rv = [R, V] themselves are true only by virtue of the equation and the above expression of dF brings no more information than the fact that dF = 0 modulo the equation. 20.3.11. Proposition. Let Fk and Fj be two first integrals constructed with the aid of resolvents R and R(x\ Then {Ffc,F/ '} = 0, i.e. the first integrals are in involution. Proof. The generators of first integrals are F = tx{Uzdi + Vzdv)R,
F « = tr(Uzld£ + VZldi})R^
(in the expression for F^ the parameter z\ is substituted for z). It is convenient to consider two other generators, F and F^\ instead of F and F^\ such that F = dzF and F^ = dzPW and to prove their involutiveness. Their characteristics are & = —z(l — z2)"lRg and a,W = - Z l ( l - z\)-lR^g. For them SF = tr(5U A d£ + 5V A drj)R + d{ ) , 6PW = tr(<5l/(1> A d£ + 5VW
A dt))R,W
+ d( ) .
Now we calculate {F, F ( 1 ) } = d a F ( 1 ) = i(da)$F{1) = i(0 a ) tr(<5t/(1) Arf? + dV (1) A d»j)iJ(1) = i(d&) tT{-(zi
=
1
+l ) -
^ -
1
- gtg^Sgg-1)
+ (Zl - ly^Sg^g-1
- g^g'^gg-1)
_ z ( z 2 _ 1 } -i
+
tr{
_(zi
+ (Zl - lyHiRg^g-1
+ (Zl - ly'dR^R^ = -z(z2 - iyHr{((Zl +
i)-i((^ 5 )^-i _ gig-iR)d£
+ ly^lR^U} +
= -z(z2 - lyitri-izi
A dr,}R^
- gT1g-1R)dr,}R^
= -z(z2 - l)-1tv{-(z1 + (z! - ly'dR^}
A d£
+
[Kgtg-1])^
[R^^-^d^R^ + ly'dR^R^ _
-
[RW^zg-'jRW
[R^^g-^dv}
+ l)~\z + ly^geg-^RW
((z1-iy1(z-iy1[R,grig-1}R^~R^R)dr}}
- R^R)^
Soliton Equations and Hamiltonian Systems
390
(all equalities are modulo the equation and modulo the exact differentials d( )). The expression within the braces is skew symmetric with respect to interchange of R and RSl\z and z\. The multiplier z(z 2 —1) _ 1 violates this symmetry; however, the Poisson bracket must be skew symmetric. This is possible only if the expression is zero. • 20.3.12. An additional set of first integrals. Besides the first integrals related to resolvents we can produce another set of first integrals. Namely, Gk = ^ / ( £ ) t r M f c + 2 d £ ,
-^f{ri)txNk+2dv,
Hk =
where / is an arbitrary function. Let us check this: dGk = - 2 / ( 0 trAf* + 1 M,de A dr) = - / ( £ ) tr Mk+1(MV
- JVj)d£ A dq
- /(£) tr Mk+1(MV + Nt=)d£ A dq = - / ( £ ) tr Mk+1 [TV, M]d£ Adn + /(£) tr =
Mk+1gSA/5g
f(t)trMk+1g5A/6g
(because Mv + N$ = -g5A/Sg).
Thus dGk = 0. Similarly
dHk = -f(v)
Nk+1g6A/6g.
tr
At the same time we have obtained characteristics of these first integrals: XGk = f(0Mk+1g,
XHk
=
-f(v)Nk+1g.
20.3.13. Proposition. The Poisson brackets are
iG^>Gk2} = — - | — { ( k 2 + l)f[f2 - (h + l)hf2} fcl + «2 + I xtrMkl+k2+2d£, {Hkl,Hk2}
= -ki
2 +
h + 2{(k2
+ l ) / { / 2 - (fci + l ) / i / 2 }
xtr7Vfei+fc2+2d7?)
{Gkl,Hk2}
= {Gk,F}
= {Hk,F}
where F are our old first integrals.
= 0,
Further Examples and Applications
391
Proof. We calculate: {Gkl,Gk2}
= dGkiGk2
= i(dGki)6Gk2
=
i(dGki)tvf2{0
• (fc2 + 2 ) M f c 2 + 1 ( % • g-1 - gzg^Sgg-1) = 2tr/2Mfe2+1{(/iMfcl+1^g-1 = 2tr f2Mk2+1{f[Mkl+1
•
9ig-\h
•
tTM
' Mv
{Gkl,Gk2}
g trM
fel+fc2+1
M^+1g)g-1}d^
+ (fci + l ^ M ^ M ^ R .
The expression 2tr/ 2 M f c l + f c 2 + 1 /i(fci + l)Mvdr) kl+k +1
-j^K
can be added since
|[AT,M] = 0. Then
= 2tx{f[hMki+k>+2}d£
+ 2trf1f2
= 2tr j/;/ 2 M f c ^+ 2 t x = lTTT-^> ^ fci + fc2 + 2
2
+
•^ A ± ^
. dMfc*+fc*+2
1 fcl+fca+2 (/i/a)e} fcl^2 +2^ W ^ 2 - (fel
+ 1
de
)/i/2}^fcl+fea+2^ •
Similarly for Hk. {F,Gk}
= dFGk = i(dF)6Gk
= dz(z -
z-1)i(dRg)5Gk
= -dzz(z2
- \)-x2txMk+lf
• {(Rg^g'1
= -dzz(z2
- l)~l2tvMk+1f{R^
= -dzz(z2
- I ) " 1 • 2 • (z{z + l)-1)trfMk+1[R,M]d^
+ RM -
- g^Rg
• f ^ R
MR}d£ = 0,
etc.
D
Thus, the first integrals Gk (and Hk) form a graded Lie algebra Q = @Qk (respectively, H). The simplest way to describe it is the following. The elements of {Gk} are pairs (k, / ) , where / are functions, and (fc, Xih + X2f2) = M(k, h) + X2(k, f2) [(fci,/i), (k2,f2)]
= (fci + fc2, (fca + l ) / i / 2 - (fci + l ) / i / 2 ) •
Commutators in H differ by sign.
392
Soliton Equations and Hamiltonian
20.4
Systems
Lagrangians of the n t h Reduced K P (GD) Hierarchy
20.4.1. This section generalizes the examples of 19.8.4. Our aim is to show that symplectic forms of the non-stationary and of the stationary theories can be combined as two components of one united form, that Hamiltonians of these theories are components of the energy-momentum tensor, and that first integrals are also components of one-form. We return to Chap. 1 and consider the GD-hierarchies. Let oo
1 n
L = (
(20.4.2)
l
be the dressing substitution expressing {u/t} in terms of <j>k, k = 1 , . . . ,n (the rest of 4>k does not play any role and
- zn) - (L - zn)(T(L
- *»))+ = 0 -
Recall that basic resolvents satisfy also the variational relation 3.5.3; we shall assume our resolvent to be a basic one. The GD-hierarchy comprises equations L = [L,5resT{r)/6L},
(20.4.3)
where L = dtL, t = tr. 20.4.4. Proposition. The form A r = (-res T( r) + res L^T14>)dx Adt = res(-T ( r ) +
+ dw*{r),
(20.4.5)
where u?-. is a form. Then SA{r)
= res{-(Jres T{r)/5L)6L
+
+ d{u)*(r) Adt) = r e s { - ( * r e s r ( r ) / < J L ) ( 0 - 1 a n « 0 - 0 - 1 ( J # ~ 1 a n ^ ) + <j>-1^8~ldn5<j) - -1S^(f>-1dn^}dx A dt
Further Examples and Applications
393
+ d(w*r) Adt + res (j)-1dn6
-
(^-10»;}0_1ty
+ d{(u)*r) + LJ**)) Adt + res <j>-1dn5(f> A d x } ,
where uJ\ should be determined from dwfo =
res[(6resT{r)/6L)
(20.4.6)
Let ujf, +
= ([L,<J res T ( r ) /«L] - L)<j>-X54>
whence the required assertion immediately follows.
•
Moreover, the one-form is fiC1) = - W ( r ) A dt - res (f>-1dn6(j) A da:
(20.4.7)
and Q, = res4>~15(j)(j>~1dn A 50 A dx — £w(r) Adt = —tl^dx + il(x)dt. The energy-momentum tensor is T (x) = - r e s (f>-1dn(t>'dx - (res 0 - 1 d n 0 + i(d)cj(r) - res T ( r ) )dt = -T(xt)dx + T(xx)dt, T (t) = - r e s T(r)da; - i(dt)uj^dt
= -T^dx
+ T(tx)dt,
and the Hamiltonian is H = (res T(r) — i(d)u>(r))dx A dt. 20.4.8. Now we compare three theories: (a) the present field theory where x and t are equal in rights, (b) the former single-time theory, and (c) the stationary theory, i.e. the theory of the stationary equation <5resT(r)/<5L = 0. 20.4.9. Proposition. The form — J fi(t)dx restricted to the algebra Au is none other than the symplectic form fi(°°) of the single-time theory (b).
394
Soliton Equations and Hamiltonian
Systems
Proof. Let X be an element of i?_/fl(_ 0 0 ] _ n _ 1 ) and X H-> H, H = [L, X}+ be the Hamiltonian mapping. • 20.4.10. Lemma. The vector field du(x) c a n be extended from Au to the whole of A$ by its action on the generator: <j>-1dH(t> =
X.
Proof of the lemma. Let the vector field 8H be defined by this formula. Then dHL = dH{rldn4>)+
= {rldndH<j>)+
- O T
1
^ •
i.e. it acts on Au exactly as required.
•
We continue proving the proposition: -n{t){dHl,dH2) = -ves
+ X 2 0 _ 1 a n 0 X i } = res[L, XX\X2 + 0(
and — / fl^-)(dfjJ, dH2)dx = Jres[L, X\\X2dx
)
(see also 19.8.1).
20.4.11. Proposition. The Hamiltonian in the theory (b) is — Proof. This is evident. 20.4.12. Proposition. Let F = —F^dx + F^dt
D jT^dx. D
be a first integral of the
theory (a), i.e. dF = 0. Then / F^dx is a first integral of the theory (b). Conversely, if J F^dx is a first integral of the theory (b) then there exists F(x) such that F is the first integral of the theory (a). Proof. The equation dF = 0 is equivalent to dF^/dt
+ dF^/dx
= 0.
The sign = means that dt acts as the vector field corresponding to the equation. Then dt J F^dx = J dtF^dx = — J dxF^dx = 0. Conversely, if dt f F^dx = 0 then dF^/dt is an exact derivative, i.e. F^ exists such that dF{t)/dt = -dF{x)/dx. • Now we compare the theories (a) and (c). 20.4.13. Proposition. The form Q^ is the symplectic form of the stationary equation (all the functions are assumed to be independent of t and are understood modulo the equation S res T^/SL = 0).
Further Examples and
395
Applications
Proof. Equation (20.4.5) shows that w?% is the one-form of the stationary equation. Equation (20.4.6) yields that the form wf\ vanishes under our assumptions. • 20.4.14. Proposition. The Hamiltonian of the stationary equation is T(xxy Proof. The equation T(xx) = - r e s T(r) + i(9)w( r ) holds for functions independent of t. If the fact that res T(r) is the Lagrangian and W(r) is the one-form of the stationary equation is taken into account, then this formula yields the usual expression for a Hamiltonian. • 20.4.15. Proposition. If F = —F^dx + F(x)dt is a first integral, then F(x) is the first integral of the stationary equation. If da is the vector field related to the first integral F then it is also the vector field related to the first integral F( x ) of the stationary theory. Proof. We have dF = 0, i.e. dF{x)/dx
+ dF(t)/dt
independent of t this reduces to dF^/dx
= 0, i.e. F(x) is the first integral.
Further, 5F = — i(da)0, + d{
= 0. For solutions
) . For the terms containing dt this means
SF(X) =-i(da)n{x)
+ dt(
).
The last term vanishes for solutions independent of t.
•
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Index
Abel differential, 137 mapping, 135, 137 action-angle variables, 308 additional symmetries, 113 Adler mapping, 19 Adler-Gelfand-Dickey (AGD) algebra, 48 algebraic-geometrical solutions, 132 Backhand transformation, 71, 73, 215 Backlund-Darboux transformation, 73 Baker, 91 Baker function from the Grassmannian, 125 Baker-Akhiezer-Krichever lemma, 135 Baker-Akhiezer, 91 bilinear identity, 92 Boussinesq equation, 13 characteristic of the first integral, 273 classical W»-algebra, 251 Clifford algebra, 106 coadjoint representations, 34 coordinate-momentum variables, 33, 348 differential Fay identity, 102 dressing, 89 Drinfeld-Sokolov reduction, 154
Euler-Lagrange equation, 30 exactness of the bi-complex, 336 Faa di Bruno polynomials, 81, 101, 111 Fay identity, 102 field theory energy-momentum tensor, 346 Lagrangian, 346 Poisson bracket, 356 symplectic form, 346 variational derivatives, 342 first integral, 14 first structure, 47 Fock representation, 106 Gardner-Zakharov-Faddeev-Poisson bracket, 41 genus of the Riemann surface, 298, 312 Grassmannian, 124 group characters, 240 Hamilton mapping, 28 Hamiltonian, 26 Hamiltonian of infinitesimal diffeomorphism, 256 Hamiltonian mapping, 47 for AKNS-D, 147 for KP, 83 Hamiltonian pair, 52 Hamiltonian structure of stationary equations, 303
408
Hamiltonians of the AKNS-D, 151 Hamiltonians of the KP hierarchy, 87 hierarchy AKNS-D, 141 GD, 13 general matrix, 195 KdV, 13 KP, 75 modified KP, 216 multi-pole, 173 q-KP, 224 single-pole, 165
Soliton Equations and Hamiltonian Systems primary field, 253 primary generators of the W„-algebra, 258 principal chiral field, 384 principal chiral field equation, 177 pseudodifferential operators (*DO), 10 "quasi primary" field, 253 resolvents, 18 rotation of the n-dimensional rigid body, 323
isomonodromic deformations, 187 Kontsevich integral, 250 Korteweg-de Vries equation (KdV), 13 KP equation, 76 KP hierarchy, 75 constrained, 218 discrete, 222 Kupershmidt and Wilson theorem, 68 Lax pair, 12 Liouville integration, 284 Magri's Poisson bracket, 42 Miura transformation, 67 Miwa transformation, 240 modified GD, 73 modified KdV equation, 71, 147
Schouten bracket, 28 Schur polynomial solutions, 237 second structure, 47 soliton-type solutions, 16 stabilizing chain, 231 stationary equations of the KdV hierarchy, 278 stationary equations of the matrix hierarchy, 295 string equation, 119 symplectic form, 25, 29, 35 r-function, 97 T-function for AKNS-D, 206 r-function from the Grassmannian, 126 ^-function, 138 Tulczyjev's operator, 337 universal property of KP, 94
nonlinear Schrodinger equation, 146 normal ordering, 103 Poisson bracket, 26, 84 for AKNS-D, 149 for KP, 84 Poisson-Lie-Berezin-KirillovKostant bracket, 36
variational bi-complex, 331 of a differential equation, 350 variational derivative, 9 vertex operator, 103 Virasoro algebra, 44 Woo-algebra, 267
Soliton Equations and Hamiltonian Systems Second Edition The theory of soliton equations and integrable systems has developed rapidly during the last 30 years with numerous applications in mechanics and physics. For a long time, books in this field have not been written but the flood of papers was overwhelming: many hundreds, maybe thousands of them. All this output followed one single work by Gardner, Green, Kruskal, and Mizura on the Korteweg-de Vries equation (KdV), which had seemed to be merely an unassuming equation of mathematical physics describing waves in shallow water. Besides its obvious practical use, this theory is attractive also because it satisfies the aesthetic need in a beautiful formula which is so inherent to mathematics.
Reviews of the first edition: 'There is a bibliography of 112 items. This book is pedagogically written and is highly recommended for its detailed description of the resolvent method for soliton equations." Mathematical Reviews "The book of LA Dickey presents one more point of view on the mathematical theory of solitons or, in other words, on the theory of nonlinear partial differential equations... The series of joint papers of IM Gelfand and L A Dickey in the middle of seventies was an important step in the development of the mathematical theory of nonlinear integrable equations... As a whole the book presents a very good exposition of the important part of the soliton theory." Mathematics Abstracts
ISBN 981-238-173-2
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