Integrable and Superintegrable Systems

(y-Ti)+A+(y)!P(y+T)) = T(y)
*>(y)=(P0 TT (y _v m ) •

(1-11)

m=l

Substituting the polynomial *>(y) (1.11) into Baxter's equation (1.9), dividing both parts by
m

A (V ) _ —

m

n

n == 1

M

m

n

m

n

(y y+T,)

m-l,...,M.

(1.12)

Note that Baxter's equation (1.11) is more general than (1.12) since it covers also the case when some of v 's Bt

could coincide [9]. Coordinate BA consists in constructing the map 8fp=

13 original paper of Bethe [4] or the monograph [31]. When applying the algebraic BA one defines the map 8 by the formula [1,2]

S

(1.13)

m=l

The order of factors B(v ) in (1.13) does not matter since n

B(u)'s commute due to relations (1.1'), (1.5), (1.6). [B(u),B(v)] = 0 Vu,v. (1-14) One can show [10] that the map B coincides in fact with that constructed by means of the coordinate BA. In the context of the algebraic BA the factors A+(y) in (1.9) appear as the eigenvalues of the operators A(u) and D(u) (1.5) on u A(u)w = A_(u)u, D(u)u = A+(u)u (1-15) and are related to the guantum determinant A(u) (1.7) by the equality A_(u+|)A+(u-|) = A(u). (1.16) Both coordinate and algebraic BA encounter the same completeness problem [11,12]. The question is if the solutions

14 expression for $(u) using the so-called fusion process [6,14]: »(u) = Nlim P N (U) trQ N P N T ( 0 ) (U)T U ) (U+7))...T ( N ) (U+NT}). Here p (u) is some scalar normalization factor and P is the projector onto the (N+2)-dimensional irreducible representation of si(2) in the (N+l)th tensor power of the 2-dimensional irreducible representation. Reshetikhin has given to his improvement of Baxter's method the name analytic BA [13]. Though the analytic BA does not fit into our scheme with the operator B it leads to the same secular equation (1.9). For practical applications the analytic BA also needs the existence of the pseudovacuum and polynomial ity of the functions

2. It includes generalizations of all the variants mentioned : coordinate [15], algebraic [16] and analytic BA [13,17] (except for the functional BA for the present). 2. FUNCTIONAL BETHE ANSATZ 2.1 What is FBA Functional Bethe Ansatz is another variant of Bethe Ansatz proposed in [18,19]. The conception of FBA was influenced directly by Gutzwiller's remarkable essay [20] on the quantum periodic Toda chain. Gutzwiller's idea, which in turn was influenced by the construction of the action-angle variables for the classical Toda chain [21], is to introduce some auxiliary set of commuting operators (namely, hamiltonians of the open Toda chain) and to

15 consider the

their

basis

of

common these

eigenfunctions. Being

eigenfunctions

the

rewritten

main

in

commutative

family t(u) takes especially simple form which allows the separation

of

generalization

variables. of

The

Gutzwiller's

FBA

is

none

than

idea

to

arbitrary

the model

tractable by QISM. It turns out that the role of the auxiliary commuting family

can

be

one-dimensional separation

played

by

spectral

of variables

the

family

problem

B(u)

(1.14).

The

from

the

resulting

coincides with

Baxter's

equation

(1.9). An important distinction between FBA and other Bethe Ansatze

is

the

functional

F3

space

below

section

2.6). Another distinction: FBA does not need existence of the highest vector

(pseudovacuum) u

(1.10) which

expands

considerably the class of models accessible for study. In [18,19] FBA was applied to the Goryachev-Chaplygin top and Toda chain. In the present paper we shall analyze another example: the XXX-magnetic chain. It will allow us to illustrate more clearly the basic ideas of FBA and to make comparison with other variants of BA. We shall put our attention mainly to the general scheme of FBA. The detailed proofs will be published elsewhere. 2.2 Spectral analysis of B(u) Let the representation T(u) of the quantum group X (1.1') in the finite-dimensional finite-dimensional

space W be polynomial

degree N T(u) T(u) ==

N N

I

n * 00 n*

N N

fA B (A B 1

n=0

I n n

u"Tn= £u" CCnVDn . n n n= 0

in u of (2.1) (2.1)

In order to avoid troubles with the degenerate cases we shall impose a set of conditions on the representation T(u). The reader who is not bothered with the mathematical rigor may skip the following rather technical passage and pass immediately to the equality The Condition 0. The

senior senior

(2.4).

coefficient coefficient

T N

is

IS

number aa number

16 matrix. The quantum determinant A(u) number-valued function. Condition 1. B * 0, det T * 0. N

v

'

T(u)

is

a

N

Introducing the operators b = (-l)nB /B , n

of

N-n

n = 1,2,. . . ,N

N'

instead of B one can rewrite B(u) in the form n

B(u) = BH(uNN-biuN"1+b2u''"2-...) (2.2) The operators {b ) commute and their common spectrum n 1

B is by definition a zero-dimensional algebraic submanifold in CN. It means that B is a finite union of discrete points which can in principle be multiple. The situation is adequately described using the language of jets [22]. Condition 2. The spectrum B is simple. It means that the representation space W is isomorphic to the space Fun(B) of functions on the manifold B. The operators b are then realized as the operators of n

multiplication by the corresponding coordinates b b=(bi bN)eCN, feFun(B),

in C .

[£ f](b) = b f(b). In what follows we shall not make difference between A b and b . n

n

Condition 3. The spectrum B contains only simple points is BeC is just a finite subset of C . Conditions 2 and 3 together mean that the commuting operators b have a complete set of common eigenfunctions that

n

and to every point b€B there exists only one eigenfunction. N Let by us the consider »CN:yi—>b described formula now the mapping ®:C described by the formula y) YyN}) b (2.3) Vp(y)=^(y, •%.<*» N where cr (y) is the elementary symmetric polynomial of degree ne{l,...,N) of N variables {y } . Note that the p 1

pre-image ©" (B) is symmetric under the- permutations usS of coordinates (y } . Condition 4. Pre-image 0" (B) does not contain

17

multiple

points

that

is each beB has exactly

pre-images.

N!

The map © induces isomorphism of W and the space of symmetric functions on © _ 1 (B). Let us introduce the operators {E~} in C : E* S C"-^C»: (Yi yn y j *-+&,*. !. ry ±1, y> > . Condition 5. There

exists

such VcCN

that:

a; e'^B) = y o-v. b ) V A crV = 0

N

VaeS ,
cj Let K" = V u E ^ . Let n6{l,...,N}, ee{+,-} be fixed. Then (x n*x => x *crx )n for all x ,x eX . creS 1 2

1

1

2

2

n

N.

The parts a) and b) of the Condition 5 allow to identify the space W with the space Fun(V) of functions on the set VcC . Part c) will be used later in Lemma 1. Now we can consider the multiplication operators y in the space W which like b we shall denote by the same n

symbol as the corresponding coordinate in C . Recalling the definition (2.3) of the map ® and the formula (2.2) for B(u) we obtain the formula B(u) = B Nft(u-y ) (2.4) n=l

which describes the action of the operator B(u) in the space Fun(V)sw. To sum up, we have defined the "operator roots" y of the polynomial B(u). The operators {y } commute and have simple common spectrum VcC . 2.3 Operators Y

+ ± N

Let us define now the operators conjugated in a sense to {y } . Let Y r-. = A,

D

p

p

H

T ynp Ap » A ( y r) ,

Li p=0

. Yn =

(Yn>1

canomcally

w P V yn Dp " D ( y P) , U

(2.5)

p=0

being defined in (2.1). Using the notation A(y ) , p

D(y ) one must always remember the operator ordering in n

(2.5) .

18 Theorem

The

1.

operators

{Y ,y >" n

following

satisfy

relations

[y.yJ = [X",Y-] = 0, m

n

m

(2.6a)

P

Y*y =(y±T)5 JY*, m n

Proof-

n

the

n 1

mn

(2.6b)

m

Y V = A(y ±2). n n n Z The commutativity of y

(2.6c) was established in

p

subsection 2.2. In order to prove the relation (2.6b) for Y consider the relation n

(U-V+T})B(u)A(v)

= (U-V)A(V)B(U)

+ 7)B(V)A(U)

which can be derived from (1.1'), (1.5) and (1.6). Substitute y for v and recall the operator ordering (2.5). n

Using B(y )=0, see (2.4), one obtains n

(u-y )Y~B(u) = (u-y+TJ)B(U)Y~. n n n Using the formula (2.4) for B(u) none obtains the eguality Y~cr(y) = cr(E~y)Y~ n n for every symmetric polynomial cr(y) .n Lemma 1. Let ne{l,...,N) and ee{ + ,-) be fixed. Then for every polynomial p(y) there exists such symmetric polynomial cr(y) that for all yeV

°"(y) = P(y)

and

IT(EJ) = p(E^y) . n n

n n

The proof is simple. Note that it uses the part c) of Condition 5. Using the Lemma 1 we can replace symmetric polynomial cr(y) in (2.7) with arbitrary polynomial p(y) and, in particular, with p(y)=y which proves the relation (2.6b) for Y . The relation (2.6b) for Y is derived guite analogously from the relation (u-v-7))B(u)D(v) = (u-v)D(v)B(u) - T?B(V)D(U) . The commutativity (2.6a) of Y follows

from

n

commutativity of the families A(u) and D(u). The relations (2.6c) follow from the definition (1.7) of the guantum determinant A(u), see [18,19]. + Applying the operators Y to the function weFun(V),

19 u ( y ) = l and

putting A ; ( y ) == [Y±w](y)

(2 8)

±

one obtains the following realization of the operators Yn . Theorem 2. The action of the operators Y± on the functions feFun(V)sW is given by the formula [Y"f](y)

.

Functions

A"(y)

n

satisfy

= A"(y)f(E-y). n

the

following

tf~ = {yeV 1 iTy Then

n

: 3 )

A;(y)

b)

A

c)

= 0 A

> >

A

n

'

(2.9)

n

conditions. Let

t V).

n

VyeV*. n

VyeV.

< *y> = < ( y ) A * ( E ; y ) E

>K (E±y)

= A(yn±|)

VyeV.

The equality a) is obvious, b) follows from (2.6a) and c) follows from (2.6c). Note that the isomorphism W=Fun(V) is not defined uniquely but the gauge transformation f i—>f is possible f(y)

=p(y) f(y)

where p(y) is arbitrary function on V having no zeroes. The + functions A (y) are then transformed as follows n

, A" (y) n

_"

p(y) ± + — A"(y) P ( E "n y ) "

f o r y e YT

0

for y e V

+

(2 11)

n

±

One can consider the definition

(2.5) of Y" as the n

interpolation problem for the polynomials A(u) and D(u) . Combining this observation with the formulas (2.4) for B(u) and (1.7) for A(u) one obtains the following expressions + for the matrix elements of T(u) in terms of Y~y. Theorem 3. The operator polynomials A(u), B(u), C(u) and D(u) are given by the formulas: B(u)

— BNN

ft (u- -yj> ■ ■

"

A(U)

= 1

(2 12a)

n

n=l

u-y

N

= AH ft (u-y p ) + I

n=l

,

17 m =1

<>

y -y n

m

(2 12b)

20 D(u) = DN ft ( u - y j + I n=l

r

u-y

n

n=l

y

<•

_y

(2.12c)

m =1 r m (2.12d) a) - A ( u + | ) ] Note that the denominators (y -y ) in (2.12bc) do not

C(u) = B _ 1 (u+i)) [A(u+T))D(

n

m

vanish due to part b) of the Condition 5. 2.4 Construction of Representations The formulas (2.9) and (2.12) can be used for constructing representations of the quantum group I (1.1'). Proposition 1. Let VcCN be an arbitrary finite set and the functions A e Fun(V) satisfy the relations a)b)c) of n n

± ±

Theorem some polynomial polynomial A(u). Then Then the the operators operators Y Y Theorem 2 2 for for some A(u). given by the formula (2.9) satisfy the relations (2.6). given Proposition by the formula the relationsalgebra (2.6). 2. (2.9) Let satisfy the associative V be ± H Proposition 2. Let the associative algebra V be defined by the generators {Yn± ,yo }1H and relations (2.6) for defined by the generators {Yn ,yo )1 and relations (2.6) for some polynomial A(u) of degree 2N. Let B *0. Then the some polynomial A(u) of degree 2N. Let B *0. Then the elements A(u), B(u), C(u), D(u) belonging to the rational elements A(u), B(u), C(u), D(u) belonging to the rational extension of V satisfy commutation relations (1.1') for extension of V satisfy commutation relations (1.1') for T(u) giren by (1.5) and R(u) given by (1.6). They satisfy T(u) giren by (1.5) and R(u) given by (1.6). They satisfy also the relation (1.7). also the relation (1.7). Both propositions are proved by direct calculation. Combining propositions 1 and 2 one can obtain representations of the quantum group S in the space Fun(V) N + taking some finite set VcC and functions A~(y) satisfying n the conditions of the Proposition 1. In order to avoid vanishing of the denominators in (2.12) one should require also that V have no intersections with the diagonal hyperplanes r ={yeC I y =y ), m*n. mn

'

m

n

Let us consider now some examples. The case N=0 is trivial. It leads to a constant number matrix T . The simplest nontrivial case is N=l, T(u)=Tu+T . o Let T C/Y «"> simplest nontrivial case is N=l, T(u)=Tu+T . Let1 0T! = I* ? ] , (3*0, det ^=£"0. There should be only one variable y and respectively Y". The quantum determinant A(u) should n

+

be quadratic polynomial. Let \~ be two numbers such that

21 21=(A -A )/TJ is positive integer. Let the spectrum V of y be V = A = {A ,A +T),...,A }, dim Fun(V) = 21+1. (2.13) Define functions A+(y) on V as

€"€+=€-

M y ) = rfy-A*),

(2.14)

It is easy to verify that A (y) satisfy the conditions of the Proposition 1 for A(u) = £(u-A+-|)(u-A~+|), (2.15) therefore, the operators Y~ (2.9) constructed with A+(u) given by (2.14) satisfy the relations (2.6). The operators A(u) , B(u) , C(u) , D(u) are then constructed by the formulas (2.12) B(u) = /3(u-y), A(u) = a(u-y) + Y~, D(u) = 5(u-y) + Y + , (2.16) C(u) = 2|(u-y) - |(u-y-A+-A") + |y" + | Y + and by virtue of Proposition 2 satisfy (1.1'). The representation constructed can easily be identified with the well-known L-operator of XXX-magnetic chain [1,2] L(u) =

(2 17)

( T a) ( ^ i v i )

-

where
1=1

of the spin 1

JsJ = Ki+i). 2.5 XXX-Magnetic Chain Let us take N samples of the representation L(u) of % constructed in the previous subsection. The parameters of n-th sample will be labelled with the index n: a ,(3 ,? ,S , n

^ ,A~,1 .

Consider

now

the

tensor

product

n

n

(1.4)

n

of

22 L-operators L (u). The question arise how to realize the n

FBA procedure described in subsections 2.2-2.3 for this particular representation TN (u) . The answer is simple. - + Theorem 4. Let the sets A ={A ,X +TJ,...,A. } have no n

n

n

n

intersections Am n An = 0, m

m * n.

(2.18)

n

the conditions 1-5 of subsecion 2.2 are satisfied and the spectrum V of operators {Yn)1 can be chosen in the form V = A x A x ... x A . (2.19) Then

1

The functions

2

N

A~(yJ specifying n

± Y (2.9} can be chosen in the A±(y) = A+(y ), P

Z

the

representation

form

of (2.20a)

P

Mm) = f r ? + ( u - ^ * ) '

€+ = fU*-

n=l

(2.20b)

n=l

The quantum determinant A(u) is given by the formula (1.16). Sketch of the proof. The spectrum of B(u) is determined by induction. The case N=l was examined above. Suppose that the theorem is true for some N. Let us add one more L-operator to the product (1.4) T

or A

(u)

B

(u)

D

N+l *

C

N+l

So,

v

'

'

N+l

(u) = T ( u ) L (u) x v

v

'

N

A (u)

(u)

C (U) D (U)

v

N + 1

v

N+l

N

(u)]

N + 1

'

'

t h e o p e r a t o r I N+l B

'

v

i) i s

'

B (u)]

e x p r e s seci

A(u)

B(u)

C(u)

D(u)

as

(u) = A N ( u ) B ( u ) + B N ( U ) D ( U ) .

V

'

By virtue of the inductive assumption the operators A N (U) and D N (U) are given by the formulas (2.9) in the space Fun(AiX. . .xAN) . The operators A(u) and D(u) are in turn given by the formulas (2.16) in the space Fun(A ) N+l'

The resulting expression for B (u) is a first order finite-difference operator in the space Fun(Ax...xA ) 1

The

corresponding

eigenvalue

problem

is

then

N+l'

solved

23 explicitely and yields the required result. 2.6 Separation of Variables Now we are in a position to reach the final goal of FBA: to determine the spectrum of the commuting family of integrals of motion t(u) (1.3). Let feFun(V) be a common eigenfunction of t(u) Vu corresponding to the eigenvalue T(U)

[A(u)+D(u)]f = r(u)f.

(2.21)

Let us substitute into (2.21) consecutively u=y , n=l, n

2,...,N. By virtue of (2.5),(2.9) and (2.20) one obtains A_(y )f(E~y) + A (y)f(E+y) = r(y )f(y). (2.22) n

n

T

n

n

The set of equations (2.22) separation of variables. Let f(y) =

allows

fr*> n (y n ).

obviously

the

yn^n-

n=l

Then

the

functions

ip eFun(A ) n

of

one

variable

n

n

satisfy the multiparameter spectral problem A_(y )if> (y -T))+A (y )

p

n

T

n

p

p

y

p

p

(2.23)

n

One can put the equation (2.23) into more familiar Baxter's form (1.9) uniting the functions (p into one n

N

function
n

in the general formulation of Bethe Ansatz 1.2) is then described as S:ipi—> fj
(see Section

n=l

It is instructive to compare this result and our description of the spectrum V with the analogous results obtained in [34] in the spirit of algebraic BA. So, we have performed our program and reduced the multidimensional spectral problem to the one-dimensional one which has the familiar form. Note that in contrast to

24 other variants of BA our construction does not need highest vector (pseudovacuum). Another distinction is that the spectral problem (1.19) is not considered in the polynomial class of functions

ansatz for
25 trouble the procedure of subsection 2.3 can be generalized from the finite-dimensional case to the infinitedimensional one provided that the spectrum of B(u) is discrete. For the continuous spectrum case the rigorous derivation is unknown. c) Realization of Y~ and separation of variables. The results of the Theorem 2 can be generalized to the case of the infinite discrete spectrum in quite a straightforward way. The continuous spectrum case presents serious difficulties. For the separation of variables it is crucial that the + functions A (y) (2.9) be of the form (2.20a). In some n

situations one can prove that allthe representations of the algebra Y"y are equivalent that is connected via the transformation (2.10-11). It can be considered as an analog of Stone-von Neuman theorem for representations of CCR. Let us pass now to the examples. 3.2 Goryachev-Chaplygin Top [18,24] The model is described in terms of operators J ,x (a=l,2,3) satisfying relations of the Lie algebra e(3)

[J

«"V = - i j X e r V

[J

«'V

the self-adjoint the commutation

= _i

j>ae y V

and the Casimir operator relations

x2 + x2 + x2 =1, The

1

2

3

operators

'

x J + x J + x J = 0 . 11

2 2

33

xa and <x J can be realized on the functions on the unit sphere in R as, respectively, multiplication operators and infinitesimal rotations. In order to include the model in the range of QISM one needs to expand the algebra e(3) with the canonical operators p,q: [p,q]=-i commuting with Ja,xaThe matrix T(u) of the form (1-5) defined by A(u) = b(x+u-J{x3,J+>).

26 B(u) = e'21q K(u) , C(u) = be21q[(x+u-J{x3,J+})(u+p+2J3)-bx2], D(u) = (u+p2J3)K(u)+b(x_u-|{X3,J_}); K(u) = u2-2J3u-(J2-J3+J); {X,Y} = XY+YX, x ± = xt±ix2, J± = Jj±iJ2 satisfies the relation (1.1') with R(u)=u-2?>. The quantum determinant is A(u) = b2(u2-j) . The Condition 0 is obviously not satisfied since T(u) is cubic polynomial in u while B(u) and A(u) are only quadratic. It leads to a slight modification of the scheme presented in Section 2. The senior coefficient e" q of B(u) is now an operator having continuous spectrum and it must be added to the zeroes y ,y of B(u) in order to form a complete set of variables. The spectrum of operators J and J2 being well known the spectral analysis of B(u) presents no difficulty. The spectrum VcR is discrete, simple and consists of two parts V=V uV_ V = {yeV I y^l+i+2^,

y2=-l±|-2na),

n i2 =0,1,2, . . .

The space W is isomorphic to the space of functions f (<3' y rY ) o n KaxV which are square integrable with respect to the scalar product -

=

r +0 °

J

dq J M(y)f (q;y)g(q;y), -» yWv

u(y) = y-y_ ' 2*

This choice of measure n(y) leads to the ± . representation of Y which allow separation of variables fCq.-y^y.,) = eip(y-2) = x(y) V (y), y = 1/2, 3/2,... should be supplemented with the square integrability condition Eyl*>(y)l <»• The resulting one-dimensional spectral problem can be solved numerically.

27 This example shows that no difficulties arise in performing the FBA scheme for the operator B(u) having discrete spectrum. In comparison with the finitedimensional case the new moment is the necessity to study the scalar product in the space W in order to describe the space Fun(V). For generalizations of the GoryachevChaplygin top see [24]. 3.3 Toda Chain [19,20] The model is described self-adjoint operators p ,q

[P.'PJ - CVU

in

= 0

'

terms

of

[

K>%1

=

The m a t r i x T(U) i s t h e p r o d u c t (1.4) of L r (u) =

u-p

n

expfgj

-exp(-q) 0

the

canonical

- ^ n ^ L-operators

1

(3.1)

n

The R-matrix is R(u)=u-iT)?>. The quantum determinant is A(u)=l. The operator B(u) coincides with the generating function for hamiltonians of the open Toda chain whose spectrum is well known. The resulting spectrum V=Rx...xR is continuous. It presents a serious obstacle for rigorous mathematical derivation of the algebra (2.6) and its representation (2.9) which is not overcome at present. The ± . . . problem is that the shift En in (2.9) should be imaginary: y ±ii). On the physical level of rigor one can simply n

+

postulate the necessary Y y algebra by analogy with (2.6). The algebra possesses the involution y - y ±iT) + + * n ID (IT) = Y" y n - *«> TT y _ y n

which follows from the involution T a g ( u ) = T a g ( u ) • T n e representations of the algebra can be constructed by analogy with the Propositions 1 and 2. The unsolved problem is to choose the factorization (1.16) of the quantum

28 determinant which could produce via (2.20a), (2.9) and (2.12) the representation T(u) isomorphic to the original one descibed in terms of the L-operators (3.1). In the paper [19] the conjecture was made that the +u

right choice equation

is

A+=i~

which

leads

to

the

separated

i"(P(Y+iTJ) + i~V(y-i-n) = r(y)
f(y) = \]
<

n(y) = n P(ym-yn), m
n(y) = y sinh|y.

It was shown in [19] that the conjecture made leads to reasonable results in the infinite volume limit N—*» which can be tested with independent methods. 3.4 Nonlinear Schroedinger Equation [25] The quantum nonlinear Schrodinger equation is described in terms of the pair of canonical Bose fields ¥(x), * (x), xeR [*(x),*(y)] = [**(x),**(y)] =

0,

[*(x),**(y)] = 5(x-y)

in the Fock space *(x)|0>=0. The periodic problem on the interval [x_,x ] is described by the matrix T(u) defined as an exponential ordered in x and normally ordered in * * [33] rx+ T(u) = :exp *(u,x) dx: (3.2) J x_ where JE(u,x) is the matrix i c 1 / 2 ( ¥ - ¥ ) -u+c 1 / a (¥+¥*) u+c 1 / 2 (*+**) -ic1/2(*-¥*) where c is the coupling constant. The operator-valued function B(u) is not polynomial now as in the previous examples but it is a hholomorphic function of exponential type. Intuitively, it is clear that *(u,x) =

|

29 its

operator

zeroes

y

must

constitute

an

infinite

n

countable set. The theory of infinite families of commuting operators exists [23], so there is a chance to succeed in performing the FBA scenario. The problem remains open. Interesting effects occur if the length x+-x_ of the interval tends to infinity. Analysis of the classical case shows that the real zeroes y of B(u) condense in a n

continuous distribution. The analytic function B(u) acquires the cut along the real axis. In the quantum case the direct spectral analysis of B(u) is a hard task. Nevertheless, acting in the spirit of Propositions 1 and 2 one can construct some representation of T(u) choosing B(u) in the form

B(u) = -+|exp{-£" ^ y C X ) } .

As in the case of the Toda chain the crucial problem is identification of the a priori constructed repre­ sentation with the original representation (3.2). In [25] some tests are discussed which indicate definitely in favor of such identification. The same approach was applied in [26] to the quantum sinh-Gordon equation. FINAL REMARKS The examples discussed above demonstrate the problems, sometimes very serious ones, which arise while applying the FBA scheme to concrete models. The majority of problems are connected with the study of various real forms of the ± algebra Y y (2.6) and its representations. A great number of concrete cases depending on the character of the spectrum V of y's present a lot of unsolved problems. Though application of FBA is as a rule more complicated then that of other variants of BA, at present FBA has no alternative in study of models having no pseudovacuum state. Even in the cases which are tractable via algebraic BA the FBA can produce new results such as

30 the new approach to the completeness problem for the XXX-magnet. Some more words about possible generalizations of FBA. The most straightforward is the generalization to XXZ-model: all rational functions are simply replaced by the trigonometric ones. The attempt of direct generalization of FBA to the XYZ-model fails since the operators B(u) do not commute. However, there is hope to overcome the difficulty using some gauge transformation, see [27], where the classical case is studied. One can also work with the algebras other than (1.1'), for example, with the algebra [28] R(u-v)T(1,(u)R(u+v)T(2> (v) = T(2)(v)R(u+v)TU>(u)R(u-v) describing the open integrable chains in contrast with (1.1') describing the closed periodic chains. For examples see [29]. In the paper [30] the Gaudin model [31,32] was studied which is obtained from the XXX-magnet in the limit Tj—>0. The secular equation (1.9) becomes then a second-order differential equation. Another possible direction is to generalize FBA to the case of Lie algebras of higher rank>l or, in other words, to search for the nested (hierarchic) FBA. This problem is especially interesting since its solution could throw light on the algebraic nature of the FBA recipe : "take the spectrum of B(u)", which at present looks like deus ex machina. ACKNOWLEDGEMENT I am grateful to L.D.Faddeev, I.V.Komarov and V.B.Kuznetsov for encouragement and useful discussions. REFERENCES [1] Faddeev,L.D.,Sov.Sci.Rev.Math.Phis. Cl,107-155(1980). [2] Faddeev,L.D. and Takhtajan,L.A.,Russian Math.Surveys,

31 34,186(1979). [3] Kulish,P.P. and Sklyanin,E.K., in "Integrable Quantum Field Theories",eds.J.Hietarinta and C.Montonen, Lecture Notes in Physics 151,61-119(1982). [4] Bethe,H.,Z.Phys.71,205-226(1931). [5] Drinfeld,V.G.,J.Sov.Math. 41,898-915(1988). Reshetikhin,N.Yu., Takhtajan,L.A. and Faddeev,L.D., Algebra i Analiz 1:1,178-206(1989). [6] Kulish,P.P. and Sklyanin,E.K.,J.Sov.Math.19,1596-1620 (1982). Jimbo,M.,Int.J.Mod.Phys.(1989). [7] Izergin,A.G. and Korepin,V.E.,Lett.Math.Phys.5,199-205 (1981). [8] Baxter,R.J.,Stud.Appl.Math.50,51-69(1971). Baxter,R.J.,Ann.Phys.70,193-228(1972). [9] Izergin,A.G. and Korepin,V.E.,Lett.Math.Phys.6,283-288 (1981). [10] Wiesler,A.,J.Math.Phys.22,750-753(1984). Creamer,D.B., Thacker,H.B. and Wilkinson,D.,Phys.Rev. D21,1523-1588(1982). [11] Faddeev,L.D. and Takhtajan,L.A.,Phys.Lett.85A,375-377 (1981). Kirillov,A.N. and Reshetikhin,N.Yu., Lett.Math.Phys. 12,199-208(1986). [12] Avdeev,A.V. and Vladimirov,A.A., Teor.Mat.Fiz.69:2, 163-174(1986). Koma,T. and Ezawa,H., Progr.Theor.Phys.78,1009-1021 (1987). [13] Reshetikhin,N.Yu.Lett.Math.Phys.7,205-213(1983). Reshetikhin,N.Yu.,Sov.Phys.JETP 57,691,696(1983). Reshetikhin,N.Yu. and Wiegmann,P.B., Phys.Lett.189B, 125-131(1987). [14] Kulish,P.P., Reshetikhin,N.Yu. and Sklyanin,E.K., Lett. Math.Phys.5,393-403(1981) . [15] Sutherland,B.,Phys.Rev.B12,3795-3805(1975) .

32 Yang,C.N.,Phys.Rev.Lett.19,1312-1314(1967). Sutherland,B.,in

"Exactly

Condensed Matterand

Solvable

Relativistic

B.S.Shastry,S.S.Jha,V.Singh,

Problems

Field Theory",

in eds.

Lecture Notes in Physics

242,1-95(1985). [16] Kulish,P.P. and Reshetikhin,N.Yu.,

J.Phys.A:Math.Gen.

16,L591-L596(1983) . Tarasov,V.O.,Teor.Mat.Fiz.76:2,184-198(1988). Reshetikhin,N.Yu.,Zapiski

nauch.

semin.

LOMI

169,

122-140(1988). [17] Reshetikhin,N.Yu.,Teor.Mat.Fiz.63,347-366(1985). Reshetikhin,N.Yu.,Lett.Math.Phys.14,235-246(1987). [18] Sklyanin,E.K.,J.Sov.Math.31,3417-3431(1985). [19]

Sklyanin,E.K.,in and

Quantum

"Non-Linear

Field

Theory",

Equations ed.

in

N.Sanchez,

Classical Lecture

Notes in Physics 226,196-233(1985). [20] Gutzwiller,M.,Ann.Phys.133,304-331(1981). [21] Kac,M.

and Van Moerbeke,P.,Proc.Nat.Acad.Sci.USA

72,

1627,2879(1975). Flashka,H. and McLaughlin,D.W.,

Progr.Theor.Phys. 55,

438-456(1976). [22] Griffiths,Ph. and Harris J.,"Principles of Algebraic Geometry" (Wiley,N.Y.,1978). [23] Berezansky,Yu.M.,"Self-adjoint

operators in spaces of

functions of infinite number of variables"(Kiev,1978). [24] Komarov,I.V. and Zalipaev,V.V., J.Phys.A:Math.Gen. 17, 1479-1488(1984). Komarov,I.V. and Kuznetsov V.B., Zapiski nauch. semin. LOMI 164,134-141(1987). Kuznetsov,V.B.

and

Tsiganov,A.V.,

J.Phys.ArMath.Gen.

22,L73-L79(1989). [25] Sklyanin,E.K.,Algebra i Analiz 1:2,189-206(1989). Sklyanin,E.K.,J.Phys.A:Math.Gen. 22(1989). [26] Sklyanin,E.K.,Nucl.Phys.B

(1989).

[27] Sklyanin,E.K.,Zapiski nauch. semin. LOMI 150, 154-180

33 (1986). [28] Sklyanin,E.K.,J.Phys.A:Math.Gen. 21,2375-2389(1988). [29] Kuznetsov V.B.,J.Math.Phys (to be published). [30] Sklyanin,E.K., Zapiski nauch. semin. LOMI 164, 151 (1987). [31] Gaudin,M., "La fonction d'onde de Bethe" (Paris, Masson, 1983). [32] Gaudin,M., J. de Physique 37,1087-1098(1976). [33] Sklyanin,E.K.,J.Sov.Math.19,1546-1595(1982). [34] Tarasov,V.O.,Theor.Math.Phys.63,440-454(1985).

34

INTEGRABILITY IN MODELS OF TWO-DIMENSIONAL TURBULENCE Yitzhak Murometz and Solomon Razboynick The University of Tennessee Space Institute Tullahoma, Tennessee 37388, USA

Abstract.

A series of finite truncations of arbitrary large size is con­

structed for the infinite Fourier representation of the Euler equations for two-dimensional ideal incompressible inviscid fluid. These truncations are proved to be Hamiltonian integrable systems. § 1. Introduction The Euler equations for incompressible, inviscid ideal fluid in two di­ mensions are: u + uu,x + vu,y = — P x , v + uv<x + vvyV = -P,y, W,l+U,y=0

,

(1.1a) (1-1&) (1.2)

where u = u(x, y, t) and v = v(x, y, t) are the x- and y-components of the velocity, P = P(x,y,t)

is the pressure, and overdot denotes the time

35

derivative. Assume that there exists a global stream function $ = ^!(x, y, t) such that u = * , s , i> = - $ , „ .

(1.3)

Recall that for the vorticity £ := u,w -

= A(*)

WfI

(1.4)

one has the motion equation

(1.5)

f + < * + < „ = 0.

If the flow is periodic in x- and y-directions, say with period 2n, and if the flow has no average drift: / / udxdy =

vdxdy = 0,

(1.6)

then for the Fourier components of the vorticity £: t = £>(*y(k*'+k*,'),

k

•= ( k i ' k 2 ) G Z 2 \{0},

(1.7)

k

one has the following well known motion equations (see, e.g., [1]): uk =

^

A(p,q)u*u)*,

w_k=wj,

(1.8)

p+q+k=0

where A(p,q):= q2:=q\

g (^2 - ^ ) ( P i 9 2 - P 2 9 i ) , + ql

for

«=(ft,fc),

and star denotes complex conjugation.

(1.9) (1.10)

36 The dynamical system (1.8) has an infinite number of components, and the motion equation for each component has an infinite number of summands. In many theoretical and experimental studies of two-dimensional turbulence, one often uses spectral models which are truncated (Galerkin) approximations to the full infinite system (1.8) [2-4]; these truncated sys­ tems sometimes possess properties that the full system (1.8) does not [5], such as additional quadratic constants of motion. Below, inspired by the beautiful Hald's paper [5], we exhibit a series of truncations of arbitrary large size each of which is a Hamiltonian integrable system. This presents a contrast to the full system (1.8) which is believed to be nonintegrable [6]. § 2. Models Let X ^Y

(2.1a)

be a pair of positive integers, and let *i < ■ ■ ■ < / »

(2.16)

be an arbitrary sequence of increasing positive integers subject to the con­ ditions I2 + X2 ^ (X + Yf ^l2 + Y2,

1 < s < n.

(2.2)

We consider the following set of 4n + 1 vectors in Z 2 \{0} : k:=(-X-Y,0),

(2.3)

a(s) := (X,ls),

(2.4)

b(s) := (Y,/ s ),

(2.5)

37

c{s):=(X,-ls),

(2.6)

d(s):=(Y,ls).

(2.7)

These vectors (and their opposite) are the ones over which we are going to allow the summation in formula (1.8) to take place. Lemma 2.8. The only relations of the form ±p ± q ± r = 0 among the vectors (2.3)-(2.7) are k + a(s) + d(s) = 0,

(2.9)

k + b(s) + c(s) = 0.

(2.10)

Proof is obvious.

Lemma 2.11. Denote As := A(d(s),k),

(2.12)

Bs := A(c(s),k),

(2.13)

As := A(b(s),k),

(2.14)

Bs := A(a(s),k),

(2.15)

D, := A(a(s),d(s)),

(2.16)

Ds := A(b(s),c(s))-

(2-17)

Then: A, = -As,

(2.18a)

Bs = -Bs,

(2-186)

38

Da = -Ds, DS = BS-

(2.18c) AM.

(2.19)

Proof. Formulae (2.18) follow from the definitions (1.9), (1.10) and (2.3)(2.7). Formula (2.19) follows when the property {p + q + r = 0} => {A(p, q) + A(q,r) + A(r, p) = 0} is applied to the vectors k, a(s),d(s),

(2.20)

and formulae (2.9), (2.186), and

A(p,q) = A(q,p)

(2.21)

are used: 0 = A(k,a(s))

+ A(a(s),d(s)

+ A(d(s),k)

= -Bs

+ Ds + As.

|

As a Corollary, we get the desired motion equations: d>a(s) = Asuj*d{s)uZ,

(2.22a)

wfc(a) = B.u^ui,

(2.226)

wc(s) = -Atu^ul,

(2.22c)

w « . ) = -Bsuj*a{s)ujl

(2.22d)

n

^ = Yl D° (<(s)uks) - uk')uk>)) ■

( 2 - 22e )

It remains to notice that formulae (2.1)-(2.7) and (1.8) imply the inequali­ ties: As?0,

BS^Q,

Ds^0.

(2.23)

39

Remark 2.24. Our system (2.22) has An + 1 complex variables consisting of 2n (complex) triads. It has the following invariant subsystem consisting of n triads: <*>6(s) = W c ( a ) = 0 ,

V

1 < s < n,

(2.26a)

(2. 22a ,<*),

1 ww k k

'

(2.25)

n

2-;A

a )w 2w 2w

(2.26b)

s=1

Remark 2.27. When n increases, the systems (2.22) change selfconsistently: letting in (2.22) Ua(n) = ^6(„) = Vc(n) = Vd(n) = 0

(2.28)

amounts to replacing n by n — 1. Remark 2.29. We can put n = oo in (2.22) [or in (2.26)] and in what follows. This will provide an infinite Hamiltonian subsystem of (1.8) which possesses an infinite number of nontrivial independent constants of motion. Remark 2.30. We chose vector k to have zero vertical component only for notational simplicity. The same system (2.22) results if formulae (2.3)-(2.7) are understood in the coordinate system of the sublattice (ZJV", ZTV'), with arbitrary

Af= ( M , M ) , ^ = M k M ) G z2\{o}.

(2.31)

What is important is that the vectors -c(s) and d(s) are reflections in k of the vectors a(s) and b(s).

40

Remark 2.32. When n = 1, the system (2.22) becomes a particular case of general 5-vector models studied by Bejnood [7]; in addition, when X = 1, Y = 2, /i = 1, the system (2.22) reduces to the Example 3 of Hald [5]. Remark 2.33. Nothing is special about the purely hydrodynamical system (1.1)—(1.2). Similar spectral analysis can be applied to two-dimensional incompressible: Yang-Mills Magnetohydrodynamics [8]; Hall Magnetohydrodynamics [9], [10, § 15.9]; Multisuperfluids [11]; etc. § 3. Constants of Motion

Define H := Im wk ^Ds

(w a ( i ) w d ( s ) - ub(s)Vc(s))

,

(3-1)

s

fcH^-E^K^

+ kwl2),

(3.2)

Us:=Bs\uja{s)\2

+ As\ud{s)\\

(3.3)

Vs:=As\ub{s)\2

+ Ba\u;c{a)\2,

(3.4)

Fs := wa(,)Wb\t) + wc(s)Wd(s) =: fs + igs, $ s := BswaMujZis)

- AtLJns)LJqa) =: <pa + iips,

(3.5) (3.6)

Theorem 3.7. These quantities are constants of motion for the system (2.22). Proof: by straightforward calculation. Thus, we have 2 + 2n + 2 ■ 2n = 6n + 2 real constants of motion for 8n + 2 real variables.

41

Remark 3.8. For n = 1 these constants can be found in [7], and for n = 1, X = 1,Y = 2, h = l—in

[5].

Remark 3.9. On the n-triad invariant subsystem (2.25)-(2.26), the constants of motion VS,FS, and Ga vanish. This leaves us with n + 2 real constants of motion for 4n + 2 real variables. i 4. Hamiltonian Structure For each (complex) variable u entering our system (2.22), set a :=

(3 := Im(u))

Re(cj),

(4- 1)

Theorem 4 . 2 . With the Hamiltonian H given by formula (3.1), the motion equation (2.22) can be cast into the following canonical form: = A,D, ' UH/i>pa(3),

<*a(*

&(*

i =--

-A,D?1 dH/da»(*)> 1

= -B.DJ

"6(5

dH/dp^,

PM

=■-jf.if.

«c(*

= A.DT> dH/d0
• au/ooH.),

& M = -A,D7> dH/da <•»>' -B.J>7

«d(»

& M = BSDJ

1

1

dH/dpd{s), dH/dad{s).

(4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) (4.10)

d k =--dH/d(3k,

(4.11)

A=

(4.12)

■-

-dH/dak. —

42

Proof: by direct calculation. Remark 4.13. We do not use the relations (2.19) in Sections 3-5. Remark 4.14. If we remove formulae (4.5)-(4.8) from the set of equa­ tions (4.3)-(4.12), the remaining formulae provide a canonical Hamiltonian structure for the n-triads system (2.26). For the case n = 1 this canonical structure is well known (see, e.g., {12]).

,,

§ 5. Integrability

Set ws:=Us Xs

+ Vs,

(5-1)

:= Us-Vs.

Theorem 5.3.

(i) The non-zero Poisson brackets between the integrals

H,ws,xs,fs,gs,^>s, {fs,9s}

(5.2)

and V>s are given by the formulae:

= -ws/Ds,

{fs,ws}

= 2\sgs,

{ws,gs}

= 2\sfs,

A, := -A.B.DJ1, {s} = -KXs,

{fs,Xs}

(5.4s) (5.5)

= ^K^s,

{Xs,ips} = 2Xs
(5.6s)

(ii) The non-zero Poisson brackets between h (3.2) and the remaining inte­ grals are given by the formulae {h,fs}

= -9s,

{h,ga} = fs-

Proof: by direct calculation.

(5.7s)

43

The Poisson brackets (5.4s) and (5.6s), after appropriate rescaling, are isomorphic to the Lie-Poisson brackets on the dual to the Lie algebra Q = so(3, R ) . We now appeal to the theory of noncommutative integrability [13, § 3.3]. Let Q be the direct sum of 2n copies of Q, understood as the linear span of the integrals entering formulae (5.4s) and (5.6s). Thus, Q is semisimple of rk = 2n. Adding H to these integrals amounts to changing ^ into ^ : = a e R x . Now, ind{Q) = ind(g) + 1 = rk(Q) + 1 - 2n + 1,

(5.8)

dim(g) = dim(Q) + 1 = 6n + 1,

(5.9)

so that ind(g) + dim(Q) = 8n + 2,

(5.10)

and 8n + 2 = 2(4n + 1 ) is precisely the number of real independent variables entering the Hamiltonian motion equations.

To assure the existence of

action-angle coordinates, we need only to make sure that the common level surfaces of integrals are compact, and for this we bring in the last remaining integral h: as is clear from the formulae (3.2)-(3.4), an arbitrary common level surface of h, Us's, and V^'s, is compact.

Acknowledgement

We thank Harvey Segur for a comment on his paper [12], and David Nickitich, Alwyn Popovich, and Samuel Svyatogor for helpful suggestions. This work was partially supported by the NSF and the PBSBD.

44 References

[1] R. H. Kraichnan, "Direct-Interaction Approximation for a System of Several Interacting Simple Shear Waves", Phys. Fluids 6 (1963) 16031609. [2] D. G. Fox and S. A. Orszag, "Invisicid Dynamics of Two-Dimensional Turbulence", Phys. Fluids 16 (1973) 169-171. [3] C. Basdevant and R. Sadourny, "Ergodic Properties of Inviscid Trun­ cated Models of Two-Dimensional Incompressible Flows", J. Fluid Mech. 69 (1975) 673-688. [4] J. Lee, "The Triad-Interaction Representation of Homogeneous Tur­ bulence", J. Math. Phys. 16 (1975) 1359-1373. [5] O. H. Hald,"Constants of Motion in Models of Two-Dimensional Tur­ bulence", Phys. Fluids 19 (1976) 914-915. [6] L. C. Kells and S. A. Orszag, "Randomness of Low-Order Models of Two-Dimensional Inviscid Dynamics", Phys. Fluids 21 (1978) 162-168. [7] M. S. Bejnood, "Constants of Motion in Five-Component Models of Two-Dimensional Turbulence", M. S. Thesis (UTSI, 1988). [8] D. D. Holm and B. A. Kupershmidt, "Planar Incompressible YangMills Magnetohydrodynamics", Lett. Nuovo Cim. 40 (1984) 70-72.

45

[9] D. D. Holm, "Hall Magnetohydrodynamics: Conservation Laws and Lyapunov Stability", Phys. Fluids 30 (1987) 1310-1322. [10] B. A. Kupershmidt, "The Variational Principles of Dynamics" (to ap­ pear). [11] D. D. Holm and B. A. Kupershmidt, "Superfluid Plasmas: Multivelocity Nonlinear Hydrodynamics of Superfluid Solutions with Charged Condensates Coupled Electromagnetically", Phys. Rev. A 36 (1987) 3947-3956. [12] H. Segur, "Toward a New'Kinetic Theory for Resonant Triads", Contemp. Math. 28 (1984) 281-313. [13] A. T. Fomenko and V. V. Trofimov, "Integrable Systems on Lie Al­ gebras and Symmetric Spaces", Gordon and Breach (New York and London, 1988).

46

SOLITONS, NUMERICAL CHAOS AND CELLULAR AUTOMATA

Mark J. Ablowitz

B.M. Herbstf

J.M. Reiser

Program in Applied Mathematics University of Colorado Boulder, CO 80309

Abstract In this article we briefly discuss solitons and associated coherent struc­ tures of certain integrable nonlinear evolution equations. The apparently unrelated concepts of solitons, numerical chaos and cellular automata are relevant. We indicate some of the links developing between these areas and describe our own recent findings.

1

INTRODUCTION

The concepts of solitons and completely integrable nonlinear evolution equations, chaos eminating from nonintegrable equations, and fully discrete systems such as cellular automata have been intensively studied during the past twenty or so years. At first glance these rather disparate ideas might seem to be so different as to prevent insights and techniques from one area to apply to another. Nevertheless, not only can these fields be useful in terms of general points of view, but more concretely, significant progress can be made by employing methods of one area in another.

47 One of the most significant early contributions spawning the current widespread interest in chaotic dynamical systems, was the research of E.N. Lorenz in 1963 [1] who investigated a three mode Galerkin truncation of the Oberbeck-Boussinesq equations for two dimensional Rayleigh-Benard convection. Since that time, chaos has been observed in many different physical situations. The underlying nonlinear dynamical systems which possess chaotic solutions have been studied extensively, see for example [2,3]. A thorough understanding of chaos requires a detailed knowl­ edge of the geometry of phase space. One of the few situations where a relatively complete description of the mechanism responsible for the chaos has been given, is in those cases where the chaos results from the homoclinic structure of an underlying unperturbed equation, [2,3]. Detailed knowledge about the geometry of the infinite dimensional phase space associated with nonlinear partial differential equations, is not (yet) generally available. However, significant progress has been made recently, with the integrable equations of soliton theory. The paradigm integrable nonlinear evolution equation is the Korteweg-de Vries (KdV) equation,

ut + 6uux + uxxx = 0.

(1)

Historically it was the first equation discovered with the soliton solution and many other important properties. There is an extensive literature on this subject and we refer the reader to the monograph and survey papers [4,5,6] for a review of this field. It should be noted that in 1965 Zabusky and Kruskal [7] discovered the soliton property of KdV, from numerical simulations. Shortly thereafter Gardner, Greene, Kruskal and Muira in 1967 [8] found that the Cauchy problem for the KdV equation corresponding to initial values u(x,0) = f(x),

vanishing sufficiently

rapidly as | i | —» oo, could be linearized by employing methods of direct and inverse scattering. Indeed, Lax showed in 1968 [9] that there is a rather general formulation by which the KdV equation could be viewed as a compatibility condition between two linear operators.

He also investigated in considerable detail the interaction

properties of the KdV solitons. In Figure 1 we show a typical two-soliton interaction. Note that the fast soliton is pushed forward and the slower one retarded during the

48

Figure 1: Soliton interaction for the KdV equation. interaction. Subsequent studies have established the wide ranging significance and occurrence of soliton solutions. In 1972 Zakharov and Shabat [10] found that the physically significant cubic nonlinear Schrodinger equation,

iut + uxx + 2u2u* = 0,

(2)

(here i2 — —1 and u* is the complex conjugate of u), admitted soliton solutions and that the Cauchy problem (for decaying data on |x| —> oo) could be linearized by inverse scattering methods.

In fact, Ablowitz, -Kaup, Newell and Segur [11]

demonstrated that these ideas apply to a class of nonlinear evolution equations, including the physically interesting sine-Gordon,

49 «K

— u « + sin « = 0.

(3)

Relying on the integrability properties of the solition equations, it is possible to develop the geometry of their infinite dimensional phase space, see for example, [12] and the references therein. In particular, Ercolani, Forest and McLaughlin [12] show how the homoclinic orbits of the sine-Gordon equation, (3), derive from its breather solutions. In the case of the NLS equation, (2), the homoclinic orbits derive from the dark-hole soliton solutions of the defocusing NLS equation,

iut + uxx - 2 u V = 0.

(4)

Homoclinic orbits are generically related to soliton solutions and other examples where one can establish a correspondence between soliton solutions and homoclinic orbits, include the modified KdV equation, u, + 6u2ux + uxxx = 0,

(5)

and the complex modified KdV equation,

tt( + 6uu'ux + uxxx = 0.

(6)

As in the case of nonlinear dynamical systems, the homoclinic orbits of the soliton equations are also structurally unstable and are 'broken' under small perturbations. In fact, it is found that small perturbations of some of the soliton equations, most notably the NLS and sine-Gordon equations, may also lead to temporal chaos, see for example, [13,12,14] and the references therein. In these studies, the homoclinic structure of the underlying unperturbed partial differential equation has proven to be the key in understanding the mechanism resonsible for the chaos. It has been observed that certain discretizations of the NLS equation such as the Fourier spectral scheme, may induce" irregular (even with a positive Lyapunov exponent) temporal behavior at intermediate levels of grid refinement, see, for ex­ ample, [15,16]. In sections 2 and 3 we argue, concentrating on a finite difference scheme, that the underlying homoclinic structure of the NLS equation provides the

50 mechanism responsible for the destabilization of the standard discretizations of the NLS equation. For this reason, the instabihty is refered to as Numerical Homoclinic Instabihty (NHI), see [17,18,19,20]. An important ingredient in our line of argu­ ment is the availability of the integrable discretization of the NLS equation, due to Ablowitz and Ladik [22],

iUn + ^ ( t f » + i + Vn-i - 2Vn) + UnUZ(Un+1 + Un-i) = 0,

(7)

which tends to the cubic NLS in the continuous limit, h —» 0. Due to its integrability, it has its own homoclinic orbits and we find quasi-periodic solutions at all levels of discretization. Theoretical studies of this system may be found in, [22,23], among others. The aforementioned discussion centered around problems in 1+1 dimension. However there are natural extensions to problems in higher dimensions, e.g., the Kadomtsev-Petviashvili (KP) equation (a 2+1 dimensional generalization of KdV), = -3a2Uyy,

(ut + 6uux + uxxx)x

(8)

and the so-called Davey-Stewartson (DS) equations (a 2+1 dimensional generaliza­ tion of the NLS equation),

iut +
=

( - |u| 2 )u

xx - m = ± 2 ( \ u \ 2 ) x x .

(9)

where a1 = ±1 in both cases. Soliton solutions in multi-dimensions have been obtained in isolated cases and is a topic of current research. For instance, weakly decaying solitons have been found for the KP and DS equations (2+1» dimensions), see for example [24,25]. Subsequently, strongly decaying soliton solutions have been found for the DSI equation (a 2 = 1) [26] and these have been related to the 1ST in 2+1 dimensions ([27]). We are not aware of any significant results in 3+1 dimensions. Both solitons and chaos at first were associated with solutions of partial or or­ dinary differential equations. However they also exist for discrete systems. In fact,

51 a cellular automaton may be viewed as an extreme form of discretization where the solution is not only defined at discrete locations in space and time, but the solu­ tion itself exists in a small number of states, often only two. In some cases it has been shown that the discretization can be associated with a differential equation, see for example [28,29]. Chaotic solutions have been associated with certain cellular automata (see [30] for a collection of papers discussing various aspects of cellular au­ tomata), and it was only recently that Park, Steiglitz and Thurston [31] introduced their Parity Rule Filter Automata (PRFA) admitting soliton solutions. Considering the history of soliton solutions of partial differential equations in multi-dimensions, it is remarkable that the PRFA rule may easily be generalized to yield solitary waves and sometimes even soliton interactions in 2+1 and 3+1 dimensions. The PRFA rule will be discussed in more detail in section 4 and soliton interactions in 1+1 and 2+1 dimensions will be illustrated.

2

H O M O CLINIC S T R U C T U R E O F T H E NLS EQUATION

In order to describe the homoclinic structure associated with the NLS equation, we study the NLS equation, (2), with periodic boundary conditions, u(x+L, t) = u(x, t). The starting point for developing the homoclinic structure is to find a suitable 'fixed point', in this case given by u(x,t) = oexp(i2|a| 2 <),

(10)

where a is a complex constant. Note that it is possible to get rid of the time depen­ dence by a simple transformation, justifying our use of the term 'fixed point'. Next we investigate the stability of the 'fixed point' by considering small perturbations of the form (see, for example, [32])

u{x,t) = u(x,t)(l

+ e(x,t))

(11)

where |e| < < 1. Substituting (11) into (2) and keeping linear terms in e lead to et = itxx + i2\a\2{e + e').

52 Assuming

e(x, t) = e_„(t) exp(-i/inx)

+ e„(r) exp(t/t„x)

(12)

where fin = 2irn/L, it follows easily that the growth rate, an, of the n-th mode, £„, is given by

°n± = ±pny/*\a\* -

ft.

(13)

It follows that the 'fixed point' is hyperbolic, provided, 0 < fi < 4|«| 2 .

(14)

The dimension of the (linearized, complex) unstable eigenspace is determined by the number of unstable modes and is given by 2n where n is the largest positive integer satifying (14). It also forms a lower bound for the dimension of the solution space, see, for example, [33]. Note that the dimension increases with increasing values of L\a\. These results mean that all Fourier modes of arbitrary small perturbations, sat­ isfying (14), will grow away from the 'fixed point' at an exponential rate. It is well-known that these perturbations will eventually decay and return to their initial configuration - a phenomenon known as recurrence [34,35] which is intimately con­ nected with the fact that bounded solutions of integrable Hamiltonian systems dis­ play quasi-periodic behavior. Moreover, it follows that the linear unstable eigenspace at the 'fixed point' is spanned (in Fourier space) by,

(L)-M—>(-#$)•

<15>

where <j> £ [0,2ir). This translates into an initial condition,

u(x, 0) = a + e0(/i* + icrn+) sin(p„x + ),

(16)

in the limit as to —* 0. Numerical experiments using (16) as initial condition and decreasing values of Co already indicate the existence of a homoclinic orbit whenever the instability con-

53 dition, (14), is satisfied. However, starting from the JV-dark-hole soliton solution of the defocusing NLS equation, (4), of Hirota [36], the expressions for the homodinic orbits of the NLS equation, (2), are obtained through the transformation,

x -+ ix, t -> -t, provided the evenness condition, u(x,t)

= u(—x,t),

(17) is satisfied, see [20] for more

details. It turns out that the homodinic orbits align themselves along the linear unstable eigenspaces at the 'fixed point', (10). In the case of a single unstable mode, the homoclinic orbit is given by (a similar formula has been found by Ercolani, Forest and McLaughlin [12], using Backlund transformations),

u(x,t)

= aexp(2ia

2

. l + 2cos(//x)exp(fii + 2i> + 7) + Ai 2 exp(2n< + 4i(j> + 2f) t) 1 + 2 cos(fix) exp(Hi + 7) + An exp(2fii + 27) (18)

where

\x = 2it/L, 2a sin = p,

fl = ±nyjia2

- /i 2 ,

A12 = ——-. cos^
Note that the value = TT/2 is excluded from (18)by the condition 0 < p < 2a. Hence, provided that fi satisfies the linearized instability condition, (14), the so­ lution, (18), has the characteristics of a homoclinic orbit, leaving the ring of 'fixed points' at oexp(2ia 2 <) as t —* —00 and returning at aexp(2ia 2 + 4i<^) as t —* 00 (note the phase shift from t —> —00 to t —> 00). Also note that the linearized initial con­ dition, (16), may be recovered from the homoclinic orbit, (18), if eo = exp(Jlt + 7 ) is chosen to be small and second order terms in eo neglected. Accordingly, if we take the initial values, (16), with e0 -* 0, we obtain a sequence of periodic orbits approaching the homoclinic orbit described by (18). For completeness we show numerically com­ puted examples of both single- (Figure 2a) and combination (Figure 2b) homoclinic orbits. The homoclinic structure is highly degenerate and is 'broken' under suitable perturbations of the NLS equation. This may lead to a weak form of temporal chaos,

54

(a)

(b)

Figure 2: Single- and combination homoclinic orbits.

55 as was shown for certain forced damped perturbations in [13]. In the next section we briefly discuss how the temporally irregular solutions obtained from a standard difference approximation of the NLS, is related to the homoclinic structure.

3

N U M E R I C A L HOMOCLINIC INSTABILITY (NHI)

In this section we wish to point out some of the qualitative differences between the integrable discretization, (7), of the NLS equation derived by Ablowitz and Ladik [22] and the standard difference approximation given by

ilfj + ^ ( [ / , _ , - 2Uj + Uj+l) + 2UfU; = 0.

(19)

Periodic boundary conditions are used throughout; given by UJ+N = Uj< The two schemes, (7) and (19), are both of second order accuracy and the 2

0(h ) difference, a purely nonlinear phenomenon, between the two schemes becomes transparant if the standard scheme, (19), is rewritten as

iifj + ^ ( £ / ; _ i - Wt + Ui+l) + |CA,f(CVi + U]+l) = WiWV^

- 2Uj + U]+l). (20)

It is therefore quite remarkable that the two schemes display completely different behavior at intermediate levels of discretizations - the standard difference

scheme,

(19), suffers from NHI, whereas the integrable scheme (7) has quasi-periodic solu­ tions at all levels of discretization, by virtue of its integrability. In order to illustrate the instability, we consider the initial condition,

u(x,0) = \(l+0.lcosnx),

(21)

where \t = 27r/L, with increasing values of the spatial period, L = 2\/2mx, m = 1,2,3. According to the expression, (13), for the growth rates of the unstable modes, these choices correspond to maximal growth for the m-th mode. The semi-discrete systems, (7) and (19), are solved using the Runge-Kutta-Merson code, D02BBF, in the NAG library, with the relative error specified as 1 0 - 6 (experiments with smaller tolerances did not give significantly different results).

56

(a)

(b)

Figure 3: Standard scheme, m = 1, N = 32.

57 For m = 1, i.e., only one unstable mode and a single homoclinic orbit, both schemes perform satisfactorily. This is illustrated in Figure 3 showing the solution obtained from the standard scheme, (19), using N = 32. The time evolution of the modulus and the time evolution of the spectral decomposition of the spatial structure are shown. It is clear that the spatial structure consists of only a few low order modes. It should be pointed out that the phase instability reported elsewhere [20], is avoided by exploiting the symmetry u(x,t)'=

u(—x,t) in all calculations reported

here. If the value of m is increased to two, two modes are unstable according to the linearized instability condition, (14). This implies a three homoclinic orbit structure - two single ones, each corresponding to one of the unstable modes, and a combi­ nation orbit. Because of the more complicated homoclinic structure, this example is not easily solved by the standard finite difference scheme (N = 32), as shown in Figure 4. The time evolution has become completely irregular. This solution should be compared with the solution obtained from the integrable scheme, using the same parameter values (note that this situation differs slightly from the experi­ ments reported in [17] where the second mode was just on the edge of the instability region, (14)), shown in Figure 5. If the number of elements of the standard scheme is increased to N = 64, the irregularity in the time evolution improves dramati­ cally (Figure 6). It is of interest to note that the higher order modes in the Fourier decomposition of the spatial structure are apparently negligible according to Fig­ ure 6b. Yet, the time evolution still becomes increasingly irregular with increasing time. Figure 7a shows the time evolution of the modulus of the solution at x = 0 over 256 time units, using N = 64. After about 150 time units, the time evolution becomes quite irregular. The solution obtained from the integrable scheme, also for N = 64, is shown in Figure 7b. This solution is well-behaved over the entire interval of integration. Finally, m is increased to three, resulting in four unstable modes according to the instability condition, (14). Now the time evolution becomes very complicated, but not chaotic. Figure 8 shows the results from the from the standard scheme, using

58

(a)

(b)

Figure 4: Standard scheme, m = 2, N = 32.

59

(a)

(b)

Figure 5: Integrable scheme, m = 2, N = 32.

60

(a)

(b)

Figure 6: Standard scheme, m = 2, N = 64.

61

(a)

(b)

Figure 7: a) Standard scheme, b) Integrable scheme, m = 2, TV = 64.

62

(a)

(b)

Figure 8: Standard scheme, m = 3, N = 64.

33

(a)

(to)

Figure 9: Integrable scheme, m = 3, N - 32

64 N = 64. Although the time evoluton is apparently completely irregular, the higher order modes remain relatively unaffected, as seen in Figure 8b. Again a remarkable improvement is obtained when we resort to the integrable scheme, shown in Figure 9 for N = 32. From this figure it is quite clear that the time evolution is not simple! The instability described above has also been observed in connection with Fourier spectral and pseudo-spectral methods (see, for example [18]) and in all cases inves­ tigated, the instability disappears as the spatial resolution is refined, as it should. Numerical evidence strongly suggest a close relationship between the instability and the homoclinic structure of the NLS equation. The standard difference approxi­ mation, (19), is not integrable, but may be viewed as a perturbation of the integrable problem. The perturbation apparently allows the phase space to be 'opened up' in the sense that the solution is free to cross the homoclinic orbits of the underlying integrable equation. As the spatial resolution is refined and the numerical scheme converges, the homoclinic structure is restored and the anomalous behavior disap­ pears. The numerical experiments show that this may only happen at a level of refinement much higher than one would normally expect. On the other hand, the integrable scheme has its own homoclinic structure, approximating the one associ­ ated with the NLS equation, and does not show any of the ill effects suffered by the standard discretizations at intermediate (and not so coarse!) discretizations.

4

CELLULAR A U T O M A T A A N D S O L I T O N S

In this section we turn to the question of investigating soliton and particle-like solutions in Cellular Automata (CA). It turns out that there is a class of CA that admits soliton interactions, bearing a rather close resemblence to the KdV solitons shown in Figure 1. The prototype model which shall be discussed is called the Parity Rule Filter Automata (PRFA), introduced by Park, Steiglitz and Thurston [31] in 1986. First we turn out attentions to alternate formulations of the PRFA.

4.1

Formulations of PRFA

The PRFA is given by the following implicit rule. Define the sum

65

Figure 10: Solitonic interaction of two particles, r = 3.

S(«|+1) = I > ! 3 + £«!+; 3=1

(22)

j=0

on the interval - c o < i < oo where a{ denotes values at site i and time t with a' taking on values 0,1 only. The new state at level t + 1 is given by t+1 1

_ \ 0 1 1

if S(ati+1) is odd or zero if S(a* +1 ) is even, nonzero

(23)

The computation is carried out by sweeping from left to right, assuming that at the initial time there is a finite number of nonzero sites, a', and that to the left we always have an infinite number of zeros. We shall refer to r as the radius. Figure 10 shows two particles, initially completely separated by a large number of zeros, emerging intact after an interaction. Note that a ' 1 ' is represented by

66

Figure 11: Solitonic interaction, represented by its sum profile.

a black box and a '0' by a blank space. The resemblence with the KdV soliton interaction of Figure 1 is striking. Note, for instance, a similar behavior in the phase shift. Changing the representation of the CA to plot the 'sum profile' allows the resemblence to be even more pronounced, as shown in Figure 11 (note that the CA rule is not changed). The sum profile at level t is simply given by S{at),

-co <

i < oo, in (22). It should be noted that the value of S{a$ is always in the range zero to 2r + 1. The aforementioned formulation is the one introduced in [31]. We now consider alternative formulations. An alternate formulation of the PRFA, referred to as the Fast Rule Theorem (FRT), is given by [38],

67

{ r

- ~ \ Sf if i £ B(t) '

(24)

where a{ = 1 — a{, i.e., the complement of a\, and £(<) is a certain set described as follows: (a) The site of the first nonzero value is an element of B(t). (b) We place all subsequent sites in steps of r + 1 bits in B(t) so long as there is at least one nonzero value in any of the intervening r + 1 bits. (c) If there are at least r + 1 zeros after a site in B(t), then we go to the next available nonzero value and repeat step (b). Steps (a) to (c) are repeated until all nonzero values are exhausted. The FRT has been used to study many of the analytical features of the aforementioned CA, including, stability, soliton interactions and periodic particles, see for example, [38,40,41]. In the next section we briefly discuss some of its consequenses. In order to formulate the PRFA in terms of a difference scheme, we first rewrite the sum formulation, (23), in the general form,

a

i

= / ( a ; i • • * > a ;+r> a ; - r > • • • - a ; - i ) -

(25)

Since a\+1 depends on the arguments of / only through their sum, it is immaterial in which order they are listed. In formulating the rule it is only important that / is a function of n = 2r + 1 arguments and we rewrite the rule as a +1

'

=/(ai,...,a„).

(26)

Following [39], we now write / as the most general polynomial function of its arguments. For example, if n = 3,

/ ( a i , 02,03) = 7o + +

7i«i + 72
68 In general / will contain 2 n free parameters, ij,j mined by specifying the value of a'

+1

= 1 , . . . , 2", which can be deter­

for each of the 2 n different values assumed

by the arguments of / . Specifying a' + 1 according to the sum formula, it is easily seen that the coefficients of the constant and linear terms are zero. The difference scheme for general n is given by,

«;+i = E ( - i M E ■*«*• ■ ■ *=2

where Pk = I

a

i*

(27)

ii,-.i*

— 1 and LrA —«fc denotes the sum over all possible combinations

of the products of k different terms. The proof is by induction on n. It is easily established for n = 3(r = 1) and we assume that the formula holds for n — 1. This amounts to assuming that the formula holds for all cases where some of the dj, j = 1 , . . . , n are zero and all that remains, is to show the validity of the formula in the case where all a, = 1, j = 1 , . . . , n. Making use of the identities, ( - 1 ) " = £ q ? ( - 2 ) * and 0 = £ C J ( - l ) - \ t=o

(28)

it=o

where C£ = , _^!.,t!, it follows that

5 = Ec?(-i)*(2*- l -i)=|[(-ir+ii.

(29)

k=2

Since S = 0 for n odd and S = 1 for n, the difference formula, (27), is established. The fact that the value of o| + 1 depends only on the sum of values of neighboring sites is reflected in the symmetry of the difference formulation - the formula is invariant under the interchange of any of the variables.

4.2

Basic Strings and Soliton Interactions.

A number of basic properties of the PRFA are easily derived from the FRT formu­ lation. First we observe that PRFA is not reversible. If we define the null string a' = Nt := 0 . . . 0, then by FRT, a'+1 = Nb =

0 ^ . r+l

69 Now consider the prenull string, PNb, a' = PNh := .LO^J), r+l

then by FRT, flj+l

—Nk

= o...o. r+l

Clearly a state may have many predecessors, demonstrating that the CA is not reversible; information is lost in the evolution of the CA. Using the FRT, one can show that the PRFA is stable in the sense that a finite number of ones at any given time, can evolve into only a finite number of ones at a subsequent time. This follows from the fact that if the box set at time t, B(t), is finite (there is a finite number of ones at time t), the number of ones at time t + 1 is finite since the state at time t + 1 difffers from the state at time t only over the set

Bit). Periodic particles repeat their spatial structures after a certain number of time

steps, i.e., they have the property that,

«2S = «J.

(30)

where p is the period and d is the displacement. The speed of the particle is nat­ urally defined as, c = djp.

In the next section we briefly address the problem of

constructing periodic particles; here we note a few of their basic properties. An interesting group of periodic particles is referred to as basic strings. These are particles consisting of r + 1 bits. Let B' be a basic string which is neither the null string nor a prenull string, written in the form Bt=.S.S...SJ

(31)

a

where a is the number of identical substrings, S. If z, is the number of zeros and /, the number of ones in 5 , then a(l, + z,) = r + 1 and the total number of ones in Bi is given by / = al,. The FRT shows that Bi evolves as a solitary wave with (a) period, p = I, =

l/a.

70 (b) speed, c =

itiilzl.

Recalling Figures 10 and 11 and our remarks concerning soli ton interactions of particles, it is natural to consider what happens when one solitary wave is fired into another one. It turns out that the interaction is solitonic. Let B\ and Bi represent two basic strings and at t = 0 form the state, B\ZaBii where Za is a string of zeros of width r + l+za>r

+ l. If the number of zeros

in Bj is denoted by /,-, j = 1,2, then the speeds, Cj, satisfy Ci < c2 if h < h (if C\ > c 2 , the two basic strings never interact). The interaction itself may be quite complicated, but eventually the two particles separate and the final state is found to be solitonic, i.e., BiZpB\, where Zp = 1 + r + zp and 2,3 —* oo as 2 —► oo. During the interaction the faster soliton is pushed forward by an amount 6+ and the slower particle is retarded, by an amount S_. The phase shifts, S± can be calculated explicitly and for period one particles (p = 1) are given by

8+ =

2/1(r-c2),

6. = -2(r + l), and the total phase shift is given by, A = <5+ — «5_. For general p the situation is a little more complicated with the phase shift of the faster soliton depending on the relative distribution of the nonzero bits and the number of times the particles oscillate during their interaction. In the case of more general particles, not all interactions are solitonic. However, algorithms have been found which allow the computation of periodic particles. This is discussed in the next subsection.

71

4.3

Construction of Periodic Particles

In constructing periodic particles the following parameters in (30) have to be spec­ ified, 1. The radius, r. 2. The period of the particle, p. 3. The spatial displacement, d, of the particle over each period. According to the FRT, 0 < d < r — 1 for the period one particle. For particles of period higher than one, the intermediate displacements during the course of one period, must also be specified. For example, for the period two particle, d = d\ + di where

0
1, j = 1,2.

The period one particle is the easiest to construct as it repeats itself at every subsequent time level. Let X be the particle with sites at I , at t = 0 and Y denote the particle with sites j/,- at t = 1, displaced d units from X. The FRT gives,

Vi T

~

_[

Xi

~ \

Xi

for i £ 5(0) for i e 5(0) '

B(0) being the box sites at t = 0. We will take x0 = 1, {5(0)} = {0, (r + 1), 2(r + 1 ) , . . . } (if the box sites are separated by more than (r + 1) sites, then the particle could have been truncated earlier). Periodicity implies,

2/i = xi+d

(32)

whereupon we have

Xi

-"

=

/ ti \Xi

for i $ 5(0) for ie 5 ( 0 ) ,

,,,* (33)

with p = r — d. This equation may be written as a linear difference equation,

* 4 = »«_,+ ( l - 2 * i - , ) * ( ^ r j ) , where

(34)

72 _,,. | 1 > = \Q

S{k

if k is a nonnegative integer otherwise.

,,_» (35)

Since (34) is a pth order scheme we need p initial conditions which we choose as Xj = 0, t = — 1 , . . . , —p. Notice that this choice of initial conditions imply that XQ = 1. This difference formula yields a spatially periodic sequence of the form,

•

POPOPO

•

(36)

where each O is a string of at least r +1 zeros separating the last 1 in one P from the first 1 in the next P . The procedure is then to compute Xi from the linear difference equation, (34), stopping after the first P has been computed; the details may be found in [41]. The formula for period two particles is obtained in a similar spirit and is given by,

,-*. + (1 -*,-,) [(,(;£) + ,(i = £ = ii)

(37)

where p = 2r—d, d\ and d2 denote the displacements during'the first and second time steps respectively, d = d\ + d2 and 6(k) is defined as before. The inital conditions are given as for the period one particle, again leading to a spatially periodic particle that can be truncated to the desired period two particle. For example, if we choose the radius r = 3, with intermediate displacements, di = 1 and d2 = 4, then Jthe period two particle is constructed, using

Xi

= Xi_5 + (1 - 2xj_5) [S ( i ) + S ( i z i ) ] ,

(38)

where the initial conditions are Xi = 0, — 5 < i < — 1. The resulting particle is given by 1000111001011010011100010 where members of the set of box sites are indicated by a bar. Particles with higher periodicity can be constructed in a similar way. However, for these higher periodic particles one has to deal with a phenomenon called splitting.

73 This happens when the particle splits into two or more separate particles, loosing its periodicity. We have recently obtained a characterizaton of situations where this will- happen, in terms of solutions of a diophantine equation [42].

4.4

Soliton Interactions in Higher Dimensions

Many of the ideas of the preceeding subsections can be extended to higher dimensions and here we demonstrate soliton interactions in 2+1 dimensions. Consider a two-dimensional grid and let o{- denotes the value at site (i, j) at time level t. In order to advance to the next time level the one-dimensional PRFA is applied to each of the successive horizontal levels, sweeping from left to right. Next the PRFA is successively applied to each of the vertical levels, sweeping from top to bottom (these sweeping directions are clearly not important). The two-dimensional automaton is represented by its sum profile by adding the horizontal and vertical sums. The main observation is that we find that in multi-dimensions it is easy to construct localized solitary waves. Indeed, interactions of these waves can be solitonic. Figures 12 and 13 show the interaction of two different pairs of particles in 2+1 dimensions (r — 3). We start initially with two localized particles. The time evolutions of the two individual particles, without any interaction, are shown in Figures 12a,b and 13a,b. As the time evolves, the images of the particles at the old positions are frozen. In Figures 12c and 13c the particles are allowed to interact. Again the particles emerge virtually intact from the interaction, apart from a phase shift - the particles behave like solitons. These ideas are generalized in a straightforward manner to three spatial dimen­ sions, or, for that matter, n + 1 dimensions, for any n > 1. We apply the one dimensional PRFA successively in each of the spatial directions in order to advance to the next time level. Again the order in which the rule is applied is not impor­ tant. It is hard to illustrate soliton interactions in dimensions higher than 2+1 in a visually attractive way and the interested reader is referred to [21] for illustrations of soliton interactions in 3+1 dimensions. Finally we note that the way in which the PRFA is generalized here, increases dissipation. By this we mean that the 'energy' (see, [43] for a discussion of the

74

(a)

(b)

(c)

Figure 12: Soliton interaction in 2+1 dimension.

75

(a)

(b)

(c)

Figure 13: Soliton interaction in 2+1 dimension.

76 'energy'associated with the PRFA) decrease is greater in higher dimensions and that a larger number of particles will loose their character under propagation by this rule. Nevertheless, the fact that localized solitary waves exist and the fact that some particles maintain their character, even under interactions as we have seen, merit our interest in these straightforward generalizations. Also, these generalizations are by no means the only possibilities. In fact, 2+1 dimensional soli ton interactions have been observed for rules based on the FRT mentioned above. Although not quite as straightforward as the ones described above, preliminary studies show that they may be less dissipative. Acknowledgements

This work (MJA) is partially supported by the TCSF, Grants

No. DMS-8803471, the Office of Naval Research, Grant No. N00014-88-K-0447 and the Air Force Office of Scientific Research, Grant No. AFOSR-88-0073. One of us (BMH) would like to express his appreciation for the hospitality of the University of Colorado and the support of the University of the Orange Free State and colleagues in the department of Applied Mathematics. fPermanent address: Department of Applied Mathematics, University of the Orange Free State, Bloemfontein 9300, South Africa.

References [1] E.N. Lorenz. J. Atmos. Sci., 20, pl30 (1963). [2] J. Guckenheimer and P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York (1983). [3] S. Wiggins. Global bifurcations and chaos. Springer, New York (1988). [4] M.J. Ablowitz and H. Segur. Solitons and the Inverse Scattering Transform. SIAM, Philadelphia, 1981. [5] M.J.Ablowitz and A.S. Fokas. In Nonlinear Phenomena,

ed. K.B. Wolf, p3,

Springer (1983). [6] A.S. Fokas and M.J. Ablowitz. In Nonlinear Phenomena, ed. K.B. Wolf, pl37, Springer (1983).

77 [7] N.J. Zabusky and M.D. Kruskal. Phys. Rev. Lett., 15, p240 (1965). [8] C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M Muira. Phys. Rev. Lett., 19, pl095 (1967). [9] P.D. Lax. Comm. Pure Appl. Math., 2 1 , p467 (1968). [10] V.E. Zakharov and A.B. Shabat. Sov. Phys. JETP, 34, p62 (1972). [11] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur. Stud. Appl. Math., 53, p249 (1974). [12] N. Ercolani, M.G. Forest and D.W. McLaughlin. Geometry of the modulational instability Part III: Homoclinic orbits for the periodic sine-Gordon equation. To appear Physica D (1989). [13] A.R. Bishop, D.W. McLaughlin, M.G. Forest and E.A. Overman II. Phys. Lett., 127A, p335 (1988). [14] G. Terrones, D.W. McLaughlin E.A. Overman II and A.J. Pearlstein. Stabil­ ity and bifurcation of spatially coherent solutions of the damped-driven NLS equation. Preprint (1989). [15] J.A.C. Weideman. Computation of instability and recurrence phenomena in the nonlinear Schrodinger equation. Ph.D. Thesis, University of the O.F.S. (1986). [16] Mei-Mei Shen and D.R. Nicholson. Phys. Fluids, 30, p3150 (1987). [17] B.M. Herbst and M.J. Ablowitz. Phys. Rev. Lett., 62, p2065 (1989). [18] B.M. Herbst and M.J. Ablowitz. On numerical chaos in the nonlinear Schrodinger equation. Report INS # 120, Clarkson University (1989). Also to be published, Proceedings of the workshop on integrability, Oleron, France. Springer (1989).

78 [19] B.M. Herbst and M.J. Ablowitz. Rounding error and the loss of spatial sym­ metry associated with discretizations of the nonlinear Schrodinger equation. INS Report # 121, Clarkson University (1989). [20] M.J. Ablowitz and B.M. Herbst. On homoclinic structure and numerically induced chaos for the nonlinear-Schrodinger equation. Program in Applied Mathematics Report 1, University of Colorado, Boulder. To appear in SIAM J. Appl. Math., April (1990). [21] M.J. Ablowitz, B.M. Herbst and J.M. Keiser. Nonlinear evolution equations, solitons, chaos and cellular automata. INS Report # 123, Clarkson University (1989). To appear Proceedings of the International Conference on Nonlinear Physics, China. Springer-Verlag. [22] M.J. Ablowitz and J.F. Ladik. Stud. Appl. Math., 55, p213 (1976). [23] N.N. Bogolyubov and A.K. Prikarpat-skii. Sov. Phys. Dokl., 27, p l l 3 (1982). [24] A.S. Fokas and M.J. Ablowitz. Stud. Appl. Math., 69, p211 (1983). [25] A.S. Fokas and M.J. Ablowitz. J. Math. Phys., 25, p2494 (1984). [26] M. Boiti, J.JP. Leon, L. Martina and F. Pempinelli. Solitons in two dimensions. To be published in Integrable systems and applications, Eds. M. Balabane, P. Lochak, D.W. McLaughlin, C. Sulem. Lecture Notes, Springer. [27] A.S. Fokas and P.M. Santini. Solitons in multidimensions. INS report # 106, Clarkson University (1988). [28] B.M. Boghosian and C D . Levermore. Complex Systems, 1, pl7 (1987). [29] J.L. Lebowitz. E. Orlandi and E. Presutti. Physica, 3 3 D , p 165 (1988). [30] S. Wolfram. Theory and applications of cellular automata. World Scientific, Singapore (1986). [31] J. Park, K. Steiglitz and W. Thurston. Physica, 19D, p423 (1986).

79 [32] J.T. Stuart and R.C. DiPrima. Proc. R. Soc. London, A362, p27 (1978). [33] C.R. Doering, J.D. Gibbon, D.D. Holm and B.Nicolaenko. Phys. Rev. Lett., 59, p2911 (1987) and Nonlinearity, 1, p279 (1988). [34] H.C. Yuen and B.M. Lake. Phys. Fluids, 18, p956 (1975). [35] H.C. Yuen and W.E. Ferguson. Phys, Fluids, 2 1 , pl275 (1978). [36] R. Hirota. Direct methods of finding exact solutions of nonlinear evolution equations, in Backlund Transformations, R.M. Muira, ed., Lecture Notes in Mathematics 515, Springer -Verlag, New York (1976). [37] N. Ercolani, M.G. Forest and D.W. McLaughlin. Notes on Melnikov integrals for models of the driven pendulum chain. Preprint (1989). [38] T.S. Papatheodorou, M.J. Ablowitz and Y.G. Saridakis. Stud. Appl. Math., 79, pl73 (1988). [39] M.F. Maritz. Soliton behavior in cellular automata and difference equations related to cellular automata. Technical Report 2/88, Department of Applied Mathematics, University of the Orange Free State (1988). [40] A.S. Fokas, E. Papadopoulou, Y. G. Saridakis and M.J. Ablowitz. Interaction of simple particles in soliton cellular automata. Stud. Appl. Math., 8 1 , pl53180 (1989). [41] J. M. Keiser. On the computation of periodic particles for cellular automata. Masters Thesis, Clarkson University (1989). [42] J.M. Keiser and M.J. Ablowitz. Private Communication. [43] C.H. Goldberg. Parity filter automata. Preprint, Department of Computer Science, Princeton University (1987).

80

The Unstable Nonlinear Schrodinger Equation TetsuYAJIMA and Miki WADATI Institute of Physics, College of Arts and Sciences, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153, JAPAN.

ABSTRACT A nonlinear evolution equation iqx + qit + 2 | q \2q = 0 , which we term unstable nonlinear Schrodinger equation, is introduced to investigate soliton phenomena in unstable systems. The equation describes a competition between instability and nonlinearity. Initial value problem is solved by the inverse scattering method. Based on the exact results, properties and roles of soli tons in unstable nonlin­ ear systems are discussed.

1. INTRODUCTION In this paper we study a nonlinear evolution equation iqx + qu + 2\q\2q

= 0.

(1.1)

We refer to eq. (1.1) as unstable nonlinear Schrodinger (UNS) equa­ tion [1]. The UNS equation may be considered as a prototype am­ plitude equation for the study of soliton phenomena in unstable sys­ tems. It is easy to notice that interchange of space i and time t in eq. (1.1) leads to the conventional nonlinear Schrodinger equation ift + qXx + 2 | q | 2 q=0.

(1.2)

81

We refer to eq. (1.2) as stable nonlinear Schrodinger (SNS) equa­ tion. Since the SNS equation has been proved to be a soliton system (completely integrable system) [2,3], readers might wonder that the solvability of the UNS equation is trivial. However, eq. (1.1) is a second order partial differential equation in time and initial value problem is very different from that of eq. (1.2). To have some insight into the physical significance of the UNS equation (1.1), we proceed from simple analysis on the evolution of a small disturbance. Linearized equation of (1.1) is iqo,x + qo,u = 0 . Substitution of qo(x,t) = Aexp(ikx

(1.3)

— iuit) into (1.3) yields a dis­

persion relation u2 = -k.

(1.4)

Therefore, a small disturbance with positive k components, however small it is, might exponentially grow. Because of this instability, numerical analysis of eq. (1.1) is rather delicate. We set k = 4ri2, q0(x,t)

u = 2irj,

T) > 0 ,

= Ae^2x+2r>i.

(1.5) (1.6)

As time goes on, we have to consider a nonlinear effect due to 2 | q \2 q. To see this, we expand q(x,t) as q{x,t) = e^'iAe2"* + eB^t)

+ e2B2(t) + •••)•

(1-7)

The constant e is a "smallness" parameter for perturbative calcula­ tion and we finally take e = 1. Substituting (1.7) into eq. (1.1), we

82

have n-l

- A W +^ r

+ 2 E 5;(t)5m(t)5B_,_m_1(*) = o, 7,m=0

n = l,2,3,---, ,

(1.8) BQ(t) = Ae2*.

(1.9)

We solve (1.8) with (1.9) iteratively and the solution is

Bn(t) = (-l-0r)nAe^^^.

(1.10)

Thus, summing up contributions of all orders, we arrive at «(x,t) = e 4i " 2 * e 2 ' 1 A ( l + E

A?

(-if^r

e4

"'")

n=l

1ir\ e*«V*+»>

(1.11)

cosh(2?7i + p) where with real constants, p and , we have set A = -title*4*

.

(1.12)

This illustrates that the instability in eq.(l.l) will not continue but be suppressed by the nonlinearity. The outline of this paper is the following. In §2, we derive the UNS equation (1.1) from a set of equations describing electron beam plasma. In §3, we solve initial value problem of the UNS equation through the inverse scattering method[4]. In §4, using the results of §3, we investigate properties of solitons in unstable systems. The last section is devoted to concluding remarks.

83

2.

D E R I V A T I O N OF T H E U N S E Q U A T I O N

The UNS equation (1.1) was derived in plasma physics[5,6]. We con­ sider a plasma system where an electron beam is injected under high frequency electric field. By the continuity relation and the Bernoulli equation, we have the relations among the density n and the velocity u of electrons in plasma and beam. Using subscripts p and b meaning plasma and beam, respectively, we have ^L

+ V-(npup)

= 0,

(2.1a)

2 £ + (iv V K = - - E W - - ^ Vn, , r

at drib

at

^

*

m

mnp

+ V-(n 6 u 6 ) = 0 ,

(2.1c)

+ (u,V)u l = - i # ) .

at

(2.1b)

*

(2.1d)

m

Here J&W is the electric field, Tp the plasma temperature, m the electron mass and — e the electron charge. In (2.1d), the term in­ cluding the beam temperature is neglected, because we assume that the beam velocity is sufficient large and we take only high frequency part of the field. The densities and velocities are devided into three parts: the average , the high frequency and the low frequency parts. These are distinguished by the superscripts 0, h, and I. We investigate the high frequency part of (2.1). The higher order terms, such as npup

\ of the high frequency part are considered to

be small. Eliminating up and u&, we have

+ n|")E< 1 ) = 0 ,

(2.2b)

84

where uo is the average velocity of the beam. From the Maxwell's equation, we have V-E^

-Awe{n^+n[h)).

=

(2.3)

Combining (2.2) and (2.3), we get

K

dt2

-§V 2 +a, e 2 (l + ^y))E(") K 2

m.

(1

+ uo-V)2-4*>

=

aw?

=

v

A

47re

- ^V 2 )4jren£*\ m

(2.4a)

)E<">,

(2.4b)

■0 +

«i 0)

where a =

»e

=

«i 0)

(2.4c)

n<°>' 4Tre2rrp'

(2.4d)

m

We call a and we beam constant and electron plasma frequency. The frequency of the electric field can be approximated as ue. Then we introduce new complex variables E and p as E ==

i(E + E*),

4iten\ ' -^ ( P + /»•),

(2.5)

whose time dependences are given by E = e-'^E,

p=

e-'^p.

(2.6)

85

Both E and p are slowly varying functions of time. Neglecting the higher derivatives of E, we can show that d2

d ,, E + E £■■(-*«.!+.«■. W> *<-*"'m ^ -

(2.7) (2.7)

For p, we think on the right hand side of (2.4a). When we approximate p as a traveling wave with wave number vector k, we have

d2 K

at' dt2

T

TV2

2

_^V )p--a, e V(l-^)m rnijji m mw*

(2.8)

The second term on the right hand side of (2.8) can be neglected because we/fc corresponds to the "velocity" of the high frequency part which is much larger than the thermal velocity y/2Tp/3m. Then using (2.7) and (2.8), we can rewrite (2.4a) as (0 dt

"2„r

2mue

)E

-

We

(2.9)

The term np' /np ' originates from the ponderomotive force and it is expressed as [6] 2 4° 4 0J) - _ >1EEI 2 0|) ( 2 1(2.10) 0)

44 ° >""

167r4 r/ levn^Tp-

(

0)

Then we have from (2.9) and (2.10) V(4— dt <

■

We|E|2

We

ee + ^ V ^ - ^ i | ^ -0)2)2 E = - ^ pP . 2mwe Z2irnp Tp 2 ' 32^4°> r/ " ~T

(2.11a)

In rewriting (2.4b), we can neglect the second term on the right hand side because rvb' is usually much smaller than n\\ So we have ((— — ++ u 0 V)V = aw aa>22V-E. V-E.

(2.11b)

86

From now on, we restrict our discussion to one dimensional prob­ lem. Let us take uo to be parallel to i-axis and the wave travels along this direction. We consider a longitudinal wave for p, so the non-zero element of the electric field E is E\. Then the equations (2.11) are reduced to 4 3

<«£-i + 5£+!/!•>/ =" 2 7 &

+

v

Kff

'

(2.12a) (2.12b)

& ' - ' -

The dimensionless quantities in (2.12) have been defined as follows: J2mu}e

i =\ x

f=

r = uet,

1 Tp '

-F

El

9-

\j32nn{°)Te /m

/v

V = uQ<

K.

p a\/327rn ( e 0) re

27a

3

-

(2.13)

0

The dispersion relation of the linearlized (2.12) is a cubic equation for UJ: uW

4K

L

27 (u-kV)*-1

1 I X/-k 2 ' 2 •

(2.14)

There exists always a real solution. Since we are interested in the evolution of unstable mode, we shall not consider the real solution. When V2 < 2(1 + K), (2.14) always has complex solutions. When V 2 > 2(1 + K), the condition for k which gives complex u> is

k
v(l-Jl-

2(1 + i t ) \ V2 ) '

(2.15)

87

We investigate the region near "the critical point" kc. We denote by wc the frequency at k = kc which develops into complex numbers when k ^ fcc: uc = kcV —

3

(2.16)

-K.

We express the solution of (2.12) as (2.17)

Here and hereafter we write the variables £ and r as x and t. There should be no confusion about this change of notation. In the region where the wave number and the frequency are near kc and u>c, the envelopes <$>\ and 4>2 a r e slowly varying functions of x and t. Using this approximation in (2.12), we have 2 =

,9

2 +

M/c

27i d

4/c

-£&

3(

&

d

+

0x)

(2.18) + *&-»■

From (2.17), (2.18) and (2.12), we obtain ^ 9 ^ ~\i\2i - ► ? >

x

V2nt X

~2V^ *

3

-t,

(2.19)

(2.20)

gives the UNS equation (1.1). To summarize, under the conditions that the beam velocity is sufficiently large and that the system is one dimensional, it is shown that the envelope of the high frequency electric field near the wave number kc obeys the unstabnle nonlinear Schrodinger (UNS) equa­ tion. The nonlinearity comes from the ponderomotive force.

88

3. I N I T I A L V A L U E P R O B L E M We consider the following auxiliary linear equations (an eigenvalue problem") 111:

C k*

l«l

dv dx dv

~di

=

2

-2i(2 -2Cq*

f-i<

W

iqi + 2(q -i | q | 2 +2i(-i

)

v

,

?\,

(3. 1) (3.2)

K)V>

(3.3)

v - ( : ) ■

The spectral parameter C is? i n general, a complex number. It is easy to show that consistency of (3.1), (3.2) and d = 0 gives the UNS equation (1.1). We shall investigate the direct and inverse scattering problems of (3.1). The results is used to obtain solutions for initial value problems of (1.1). We assume that q(x,t)

approaches to zero sufficiently fast as

|x|—* oo : q(x...0 - > 0

1*1--* 00 .

for

(3.4)

Let v and w be solutions of (3.1) with eigenvalues (^ and C,2 , respectively. Calculating the derivative of the Wronskian, W[v, w] = v-[W2 — v2w-± , we have

d ,

-(vlW2-

-

v2wi)

= ~2(Ci "- C2){i(viw2 + «2t«i)(Ci + C2) - q*viw2 --qv2wi) = "2(Ci "-C,2){v2iWi

-WuVi)

.

(3.5)

89

It is obvious from (3.1) that when v is a solution at £ = £ + IT/ , then v = I

1 u* is a solution at C* = f — »»7 - For real £ = £ , v and

u are independent solutions. Let us first consider scattering problem and examine properties of reflection and transmission coefficients. For real eigenvalue C = f, we define solutions u(x,f) and w(x,£), Jost functions, whose asymptotic forms are

e v{x,Z) -* ( I 7"')I ««.o

a;x->--oo, —»• —oo, (3.6) (3 6)

/ o \ The Jost functions v(x,C) a n a w(x,0 Im(C 2 ) = 2fr/ > 0 . We can show that / ee- 2 . < 2 * \

»(**0 -» «..o - (( "f")) 0

«0(X,C) «>(z,C)

-- **

/

o2 \

2.C2 x) xj ((ee2.C

x —» +oo. are

'

analytic in the region

|I C C HHO oo, O,

Kl-oo. ICHOO.

((3.7) 37)

Since the set of w and w forms a fundamental system of solutions, we can express v(x, f) as a linear combination of w and w : v(x,00 = v(x, = a(Ow(x, a(£)W(x, 0 + K 6(f0M » (**. 00..

(3.8) (3-8)

in which a(f) and 6(f) are transmission and reflection coefficients, respectively. It is easy to show, using the relation (3.5), that 2 2 I aa(0 ( 0 I2I++I 1 6 (6(0 0 | 2 =| = I- 1. 1

(3.9) (3.9)

Also from (3.5), it is shown that a

a(0 wi»2)(x,0( 0 = (»iw («l w 22 "--U>lV2)(x,0-

(3.10) (3-10)

90

The function a ( 0 is analytically continued into the region Im(C ) = 2£r/ > 0. As is seen from (3.7) and (3.10), a(() tends to unity in the limit of ICI —^ oo:

4 0 - •» I

for

IChoo.

(3- 11)

We denote zeroes of a(Q in the region £ij > 0 by (j (j = 1,2, • • ■»N). A bound state occurs at £ = (j since v(x,Q and w ( x , 0 become lin­ early dependent. The proportional coefficient at £ = Q is written as bj . The set of quantities, a(0> b(£), bj and Q, is called scatter­ ing data. Let us take the limit \x\—► oo in (3.2) to evaluate time developments of a(0> H O

an(

«(£,*) = e " -'^ao ,

l fy- Then, we have ,

KM=■- e^b

0

l *>(*) = e ^bj0

,

(3.12)

i = 1,2,-- ,N, where the subscript 0 means the value at t = 0.

We next consider inverse problem of the scattering: we construct q(x) from the scattering data. The Jost functions v(x, Q, w(x, 0 and the transmission coefficient a ( 0 can be analytically continued into the region £rj > 0 in the complex £-plane. We introduce the following function;

(

1 v(x 0e2i?*

Im(C2)>0;

$(0 = MO

Integration of a function $ ( 0 • (z — 0 N

*(C) + ^ R e s ( $ ( z ) ; 2 : = i-1

(l\

\o)

f

1 27H

r

J —oo

1

in complex 2-plane yields

0)

00 >(0

-c

(3.13)

Im(C2) < 0 .

2i x kuJ(x, C)e ?

# -

1

z"00 #■«;(„)

91

Here, ^ 1 ) ( £ ) and (j^2\vi) are the jumps of $(£) across £ and 77 axes, respectively. We assume that a(() has N simple zeroes at C = Cj, 3 — 1,2, • • ■, N. Then, (3.14) becomes

*(O=(;)+E^,C*) r*mvd(_

+1

1 /-*<■%)„,

(3.15)

where 7* is defined by

(3.16) Ik . 7fc = -~r a'(C fc) • a (Ob) ■ e - 2 ' ^ s in powers of £ - 1 , we obtain an ex­

By expanding w(xX)

pression of q{x) from (3.13) and (3.15). In particular, the asymptotic form of w(x, £) in the limit of ( —► 00 is

^ * = ( i ) + E r+f c ( ; ; ) . u;(x,C)e"2'c2a;

-(S) S^(2)-

(3.17) (3-i7)

We compare the expansions on both sides of (3.15) to get q(x) =2iai

= -2i^7fc*6-2'^U;2*(x,a) fc=l fc-1

1

- - /

f°°

v J-00 J-00

0 it0

di^H)

- - /

00

* J-00 J-00

dT,
.

(3.18)

This formula can be rewritten as follows. We introduce functions K^\x,y)

(j = 1,2) and express w(x,()

w(x,C) = f

as

;)^.+f*^(.,.)^. /•oo

+ /

2i 2a 2 , t ds(KW(x,s)e < . '^ . dsC^ 1 >(x,a)e

(3.19)

92

From the explicit form of $(C) in (3.13), the jump functions $l>(£) and 4^2\r}) are expressed in terms of a and b as

S^wfrfle*"-

e>o,

-mv,{x,ty*'' 1-ji^.Oe^-

*
(1)

0 (O =

lr\

<^%)=

-

ri/ >| >0 ,0 ,

^|«,(x,ii?)e-«*'-

<

(3(3.20 20)

1,V l ^ « . ^ We define functions F (x) (j = 1,2,3) by

77 < 0 .

;

-i a^« * , ( « ) := «.) 2.- £ i f1 W V«*' "- i(jf -f/ >> K a(O S^-v*e N

e

4
*

Jo

J — <x

" / )<*!? a l r a(in) ^-oo ( 7)

i-U

-2tV:c

(3.21

We substitute (3.13) and (3.20) into (3.15) to obtain formulae for the following three quantities: ,iV. i) $(C)atC #(C) at C =■C*,i l,2,---,iV. q, j= 1,2,--

k)-#(C-ie))/2. ii) (#(e + k)- * ( C - - w))/2. ( e - --i»7))/2. iii) ($(e (*(e + 177) --- #$(« "?))/ 2 -

A calculation shows that these quantities can be written in terms of functions K^\x,y) (j = 1,2) and F ; (x) (j = 1,2,3). From them, we arrive at j) #P(a: y)++ J°° f K[ (x, ,v)+*7(» y)+F*(x : + y) /•OO

+ /

ds dsKl K^(x,2)*(x,s)F*{s s)F*(s + y)

dsK^*{x. dsIQiy*(x,s)F? s)F? (s(s + y) = 00,, +1+1

(3.22a)

JX

,yh r

( i} dsl<[ K I<[22\x, \x, S)FJ( j #2>*(x, ( * y)- j " ds s)Fj(s S + y)

-r J J XX

OO

/

dsK^ix, + y) y) = = 00.. j+1(s + dsK\1\x,s)Fs)F j+1(s

(3.22b) (3.22b)

93

The set of equations (3.22) is the Gelfand-Levitan equation[8] for the system (3.1). We have from (3.18), (3.16) and (3.21) 1 1} q{x) = g(ar) = -K\ \x,x). -jr{ (x,x).

(3.23)

The initial value problem for the UNS equation (1.1) is thus solved. That is, for given q(x,0) and qt(x,0) we calculate the scat­ tering data at t = 0. The time dependence of the scattering data {a(£),b(£),bj,Cj}

is given by (3.12). The sought function q(x,t) is

obtained by solving the Gelfand-Levitan equation (3.22) and by using (3.23). 4. SOLITON S O L U T I O N S 4.1 Solitons In the following, we assume that 6(f) = 0 and look for soliton solu­ tions. From (3.18), we see that q(x,t) is given by iV JV

2i x g(x,*) q(x,t) == -2iJ2Mt)w -2i^(7 2f(x,C k)e ^ yc w^(x,a)^

x

)*.

(4-1)

fc=i

Equation (3.15) at £ = £*, j — 1,2, ■ • •, TV, are now 7

*

e

W i ( x , C i ) e - 2 ^ E ==-j^^-V Wi(*,Ci)e-«$* k=\

u>J(*,C,>««''

= 1

«

+ Zw* fc=i V fc=l ^'

Defining ur-

x

^

w

*^x^ '

/-u "WOOu

and /J.J, respectively, as

™f i = = V^W \lk(t)wj(x,(k), ^Cfe), w fc)

itQk

2 CW x c Mi ==\fW)e ' >*, , W A-(*)e

(4-3)

94

we have from (4.2) a set of linear equations; N

fc) 4 * =0, C; - Cl

«#> W

«£>

*

(4.4) fc=l ^

s

*

We write the coefficient matrix of eq.(4.4) as A. The determinant of this 2JV x 2N matrix, \A\, satisfies a relation:

a(iog | A |) 0(log|A|) dx dx

v ^ / , ( fc) * fe) (*)*>> =«„ . f; (a^ i (Jfc) - /■* cfc»i *) jb=i

= I m ( f ; --fito-i*). a//rf)*)= 88Im(E"

(4.5) (4-5)

In terms of a* and /?& in (3.17), the right hand side of (4.5) is ex­ pressed as 8Im(/?2). Using (3.15), (3.17) and (4.5) we get | A 1) ^0(log |M J1 = - ( U9 ||2!)«) „++4|
(4.6) (4.6)

One soliton solution is obtained when 6(f) = 0 and a(£) has a simple zero (a bound state). Denoting a zero of a(£) by ( = £ + irj, we obtain q(x t) = q(x »*) q(x,t) =

2 22 - eeXp( x p (-4i(f - 4 i ( e -2 - rj / 7)x ) x- -2i(t 2^ 2l ->m

Zirj

++ i(j>) ^) cosh(8^r;a ^x ;++ 2T1 277* t ++p)p) ' cosH8

((4.7) 4J)

where = 0)), = -2ugfci(t = 0)), 0<(> = -2arg(/x(<

-p loc( = log( |

P -- l o s ( |

2T)

^

M(t = 0

) ).

)p)-

(4.8) (4.8)

This solution, when x and £ are interchanged, is the same as that of the stable (conventional) nonlinear Schrodinger equation. We find

95

that e q . ( l . l l ) is a special case, £ —* 0, of the one soliton solution (4.7). It is known that solitons experience the position shifts due to their mutual collisions[9]. In the stable media, during the collisions, the faster soliton accelerates and the slower one decelerates. In or­ der to investigate the position shift in the unstable media, we shall examine the asymptotic behaviours of soliton solutions. We assume that a(Q has distinct N simple zeroes: £;- = £ ; -f iijj, j = 1,2, • • •, N. Each sohton has different velocity since the velocity of j-ih soliton is — ( 4 £ ; ) - 1 . Without loss of generality we assume that £i < 62 < • • ■ < £N- It means that a soliton with a smaller number has a larger velocity. From (3.2), the time and spatial dependences of \nj(x,t)\

are

N (x,t) | = | ^ 7 ^ 0 ) I .e-*ivd*+*/*ti).

(4.9)

We observe the iV-soliton solution in a coordinate moving with the m-th soliton, i.e. in the coordinate such as x + t/4£m is constant. Using Hj(x,i)

in (4.9), we have asymptotic forms:

ftj -» 0 fij —* 0

for j < m , for j > m,

as t -> +00 , as t —> —00.

^

1Q.

'

Then, we obtain the asymptotic forms of (4.4) in the limit of \t\—» 00, which gives the relation between w\m' and tfig

. For t —> 00, we

have

u;(m) + fJ'm

L^_Pu;(m)* = 0) 1

(m)

-%—~w\ 2«»?m

;

,

(m)*

+ ^2

(+1*

= /4T .

I

A

11

-V

(4.11a)

96

where TT 11

/4+) - M m

Cm C; , _ ,* ■

S m

>=m+l

S

(4.11b)

J

In the limit of t —* — oo, we have a set of the equations which has the same form as (4.11a) but fim' is replaced with fAn ■

r*m

= Mm

n

l>m V>m

(4- l i e )

-c;'

Equations (4.11) clarify properties of the soliton scatterings. As in the stable media, collisions occur pairwise. Each soliton experiences the shifts of position and phase due to the collisions. Recalling the expression of one soliton solution, we see that the center and phase of the soliton in the asymptotic regions are given respectively by •Em

=

1 Mm(< =o)l

1

-1 Pm 8^m^?m

2»?m

2

>

m = -2arg(^ m (< = 0)).

(4.12;

We shall write xm and (j>m in the limit of t —* oo (t —► — oo) as i „ and 4>~m

(a™ and
find that the position shift and the phase shift of m-th soliton are given by AT

— ■rl't") -

1 f]m

^m

(E

log|

j'=m+l

S>m _ t i Sm

-c;

l1

n-l

w ~4' n E l o s i S>m ^7 7= 1

A ^ m = *<+> -- 4"> m_1

/-

L

r-

=2<Er -^

N

r

r

j=m+l

S m

S

;

(4.13)

97

We consider two solitons; one with a larger velocity — ( 4 £ i ) - 1 and the other with a smaller velocity — ( 4 & ) - 1 • According to the formula (4.13), the faster soliton has the negative position shift A*, = i - ] o g | £ ^ £ | < 0 , m Ci - C2

(4.14a)

and the slower one undergoes the positive position shift Ax 2 = — l o g I £ ^ £ I > 0 . m C2 - Ci

(4.14b)

That is, during the collision, the faster soliton decelerates and the slower one accelerates. This property is common to the unstable sine-Gordon (USG) equation, (j>u — xx — m 2 sin0 = 0 [10]. We con­ clude that the interactions between solitons in the unstable cases are attractive in the sense that solitons form virtual bound states during the collisions. 4.2 Initial Value Problem for sech-Type Potential We consider an example of initial value problem.

For the initial

condition, we take q(x, 0) = 0 ,

qi(x, 0) = Asechx.

(4.15)

This corresponds to the situation that an "impulsive force" is applied to the system.

The constant A represents the magnitude of the

impulsive force. From (3.1) we eliminate the function i>2 and use a new independent variable z = (1 — tanhx)/2. Then we have Z { 1

-

z )

^-

+

i

2-z)7z-

fl

+

, ,,

<'- 4 ' +

4C4 + 2 i C 2 ( l - 2 z ) 1

4,(1-,)

>"'=°- <4'16>

98

This equation is the same as the one appeared in the initial value problem of the conventional nonlinear Schrodinger equaiton[ll], when C is replaced by 2£ 2 . We have two linearly independent so­ lutions, v^

and i>(2);

v\V = z<\\

- z)-*? F(-A, A;l-+ 2*C2; z),

vP = z-«\l

- z)«* F(-A, A; - - 2i?; z),

,W = z*A-< \l-z)

-

■

<

"

x F ( i - 2iC2 + il, i - 2zC2 - A; | - 2iC2; z), vi2) = **+«* ( 1 - z ) *C x F ( i + 2iC2 + A i + 2iC2 - A; | + 2iC 2 ;*) . (4.17) The function .F(a,6;c;z) is the hypergeometric function[12]. The Jost functions, and the transmission and reflection coefficients are given by /

Af

(

2

w =

V

2(e-ri )+i/2 42)

w =

J

,(!)' -Av

(2)«

h 2)-i/2j \2(e-rj

{ r ( - 2 ^ 2 - r ? 2 ) + l/2)}2 a = r ( - 2 i ( £ 2 - 772) + A + 1/2) T(-2i(e -V2)~A+ 6 =

\

i I r(2i(^ 2 - Ty2) + 1/2) | 2

r(A)r(i-A)

1/2) ' (4.18)

The function a(C) can be analytically continued into the region £77 > 0:

{r(-2ic 2 +1/2)} 2 «(C) = r(-2t'C2 + A + 1/2) r(-2*C2 - A + 1/2)

(4.19)

99

Fig.l Zeroes of a(() for the initial condition (4.15) with A = 3.75. Then the zeroes of a(£) are located on the line £ = 77 in the £-plane, and their values are

£ ^ = 4vA-n+i, n = 1,2,---,N

< A+

-.

(4.20)

We see from (4.20) that the number of solitons increases as the impulsive force becomes larger. This suggests that the energy injected into the system is transported in the form of solitons. In a separate paper, we shall report some other initial value problems [13]. 5. C O N C L U D I N G R E M A R K S The unstable nonlinear Schrodinger (UNS) equation (1.1) is a typical amplitude equation for nonlinear unstable systems. In fact,we have shown that it describes weak instability in electron beam plasmas. By using the inverse scattering method, we have solved initial value problem of the equation. As shown in §3, initial value problem of the UNS equation is much more involved than that of the conventional

100

stable nonlinlear Schrodinger (SNS) equation since the UNS equation is a second order partial differential equation in time. It is interesting to notice that as a by-product of this work we have solved boundary value problem, q(x, t) = f(t) for x = 0, of the SNS equation. From the exact soliton solutions, properties of the soli tons in unstable systems have been clarified. During the mutual collisions, a faster one decelerates while a slower one accelrates. Then, the position shift has an opposite sign to that of the stable case. This suggests that soliton interactions in unstable systems are attractive. We have shown in §1 that soli tons in the UNS equation arise as the competition between instability and nonlinearity. We may say that solitons carry away the energy pumped into the system and that instability occurred in the system is relaxed by the creation of solitons. Those properties and roles of solitons in unstable systems are to be confirmed experimentally. In conclusion we point out that the soliton concept, which has been so successful for dispersive nonlinear systems, will again play an important role for the studies of nonlinear unstable systems. Acknowledgments The authors would like to express their sincere thanks to Dr. M. Tanaka and Professor K. Yamagiwa for stimulating discussions on electron-beam plasma systems. This work is partially supported by Grant-in-Aids for Scientific Research Fund from the Ministry of Education, Science and Culture (63302062).

101

REFERENCES [1] Yajima, T. and Wadati, M., J. Phys. Soc. Jpn. 59 41 (1990). [2] Zakharov, V.E. and Shabat, A.B., Sov. Phys. -JETP 34 62 (1972). [3] Zakharov, V.E. and Manakov, S.V., Theor. Math. Phys. 19 551 (1974). [4] Gardner, C.S., Greene, J.M., Kruskal, M.D. and Miura, R.M., Phys. Rev. Lett. 19 1095 (1967). [5] Tanaka, M. and Yajima, N., Suppl. Prog. Theor. Phys. 94 138 (1988). [6] Tanaka, M. and Yajima, N., "Solitons and instability of electric beam plasma", Reports of Institute of Applied Mechanics, Kyushu University, 63 281 (1987). (in Japanese ) . [7] Zakharov, V.E., Sov. Phys. -JETP 35 908 (1972). [8] Gelfand, I.M. and Levitan, B.M., Amer. Math. Soc. Transl. ser.2,1 253 (1955). [9] Wadati, M. and Toda, M., J. Phys. Soc. Jpn. 32 1403 (1972). [10] Yajima, T. and Wadati, M., J. Phys. Soc. Jpn. 56 3069 (1987). [11] Satsuma, J. and Yajima, N., Suppl. Prog. Theor. Phys. 55 284 (1974). [12] Abromobitz, M. and Stegun, A. (eds.), "Handbook of Mathematical Functions", p.555-566, Dover, (1964). [13] Yajima, T. and Wadati, M., in preparation.

102

CLASSIFICATION OF INTEGRABLE EQUATIONS R. K. Dodd Department of Mathematics, & Computation San Jose State University, San Jose, California December 21, 1989

1

ABSTRACT

On of the outstanding problems in the theory of integrable p.d.e.'s is the classification problem: Is it possible to algebraically classify the integrable equations so that some order can be imposed on the vast zoo of known equations? In this article I present some results on this problem.

2

INTRODUCTION

The invention of the inverse scattering method in 1967 by Gardner, Greene, Kruskal and Miura [1] led to an intensive investigation of completely inte­ grable systems which continued throughout the seventies. All the physically important equations discovered and solved through these endeavours essen­ tially involve only two independent variables. There are exceptions to this such as the Kadomtsev-Petviashvili equation which involves two space and one time variable. The problem here however is that the exact solutions are non-local. This is appropriate for shallow water waves, but neither this nor most of the higher dimensional integrable models can be interpreted as field equations since this requires the existence of exact local solutions which can represent particles. Thus by the beginning of the eighties the basic properties of integrable systems in two independent variables were understood and a vast zoo of such equations had been introduced. In particular we mention that integrable sys­ tems are infinite Hamiltonian systems and that the exact solutions of the

103

equations are special functions defined by a remarkable family of equations, the Hirota equations [2]. Both these properties are discussed in other chap­ ters in the book. In the present decade only a few new physical integrable p.d.e.'s have been discovered and undergone intensive investigation. The most important of these are the stationary axisymmetric space-times and their Einstein-Maxwell generalisations [3]. The current situation can therefore be summarised by saying that we have a cookery book of equations but no systematic way of classifying them. It is the classification problem which I want to pricipally discuss in this article. The classification of the equations would have two advantages: (i) simplification of the existing jungle (ii) a possibly deeper understanding of integrable equations This last possibility might indicate a fruitful direction for research into in­ tegrable equations in higher dimensions. The most succesful attempt so far at classifying integrable systems is due to Mikhailov and Shabat [4],[5]. They studied systems of a specific type, namely two component systems of second order. In Hamiltonian systems with finite (n say) degrees of freedom complete integrability is indicated by Louiville's theorem: the existence of n constants of the motion which are in involution with one another. Mikhailov and Shabat were able to show that the existence of an infinite hierarchy of local conservation laws for their system reduced to the requirement that two conservation laws of sufficiently high order existed. These conditions produced a system of contraints on the functional defining the equations which were exhaustively analysed in their papers, ultimately leading to the classification of integrable two component systems of this type. There are several problems with this approach: (i) different systems of equations have to be investigated separately from scratch (ii) the method does not give any simple relation between the integrable equations of the system - they are defined by complicated conditions on the functionals defining the equations (iii) the method does not relate the p.d.e.'s to the other integrable sys­ tems. The last point refers to the fact that besides completely integrable p.d.e.'s there are also completely integrable o.d.e.'s, differential-difference equations and difference equations.

104

3

AFFINE LIE ALGEBRAS

A detailed exposition of the theory of Kac-Moody algebras is given in Kac's book [6]. The particular algebras which arise in soliton theory are a subclass of the Kac-Moody algebras called affine Lie algebras. These can be realised as central extensions of twisted loop algebras associated with the simple Lie algebras over C. Let g be a simple Lie algebra over C and let a be a dth order automorphism of g {ad = Id). There is a decomposition of g ihto the eigenspaces of
C1)

g(
fc(,+j)

where s,eg(,)

If a — Id then the loop algebra is untwisted. If the group of automorphisms of finite order are considered for a simple Lie algebra g then it is found that the twisted loop algebras fall into equiv­ alence classes the members of a particular class being equivalent up to an automorphism of the twisted loop algebra. The automorphisms of g which determine different equivalent classes are the outer automorphisms defined by the Dynkin diagram symmetries. Thus for a given untwisted loop algebra g(Id) we have the non-equivalent twisted loop algebras determined by graph automorphisms of g and for each Lie algebra of this type there are a family of twisted loop algebra realisations. In this article we shall only deal with the twisted loop algebras connected with the simple Lie algebra sl(n,C), the Lie algebra of traceless matrices which is a subalgebra of the Lie algebra of n x n matrices gl(n, C). It is convenient to also consider twisted loop algebras associated with gl(n,C). For these algebras the theory outlined above is easily developed. Let Eij := (5,\»6 ri j) 1
105 and the simple coroots a,y := [ei,fi]o where [., .]„ is the matrix commutator. Then we have [°n,ej]o = Oijet

[at, fj]0 - -an

fi[a, fj]0 = Sijaij

where (OJJ) is the Cartan matrix for the Lie algebra a„_i S sl(n, C) in the usual classification. For this algebra we find that oij = 26i,j - 6ij+i -

6ij-i

Observe that the algebra a„_i admits the triangular decomposition a„_i = n+ © h © n_ where h is the Cartan subalgebra of diagonal matrices and n± are respec­ tively the nilpotent upper (+) and lower (-) triangular matrices. Let {({} be the basis of h*, the dual of h, such that (i{Ejj) = fkj- Then the simple roots of an-i, II = {a,- : »" = 1, ...,n - 1} are defined by a,- = e,- — e,+i, i = 1, ...,n — 1. We have [h,Ei,j)0 = a(h)Eij and a = Ci — ej = a,- + a,+i... -I- c*j_i Then Ea := CEij is the root space associated with a and the basis element Eij is generated by [[■■■[ei,ei+i]0,...ej]0 if j > i. The root space E-a is generated by a similar expression in which the /; 's replace the' e, 's. All the root spaces can be generated in this fashion so that the complete set of roots are therefore A = A+ U A_ where A+ = {a = a, + ... + a, : i < j} U H and A_ = — A+. It follows that the triangular decomposition is defined by the root space decomposition of a„_i. In the case of gl(n, C) we have to include an additional basis element in the Cartan subalgebra which we conveniently take to be 7 the identity element. The highest and lowest roots are ±(o,- + ... -I- <*n-i)- The corresponding basis elements are Co : = En-1,1

/o •'= 7?l,n-l

The algebras a„_i and gl(n, C) are generated as we have seen by the ChevalIey generators and the simple coroots n v (together with I in the case of

106

gl(n, C)) which satisfy the relations given in (3.1). Since all the simple Lie algebras over C can be defined in this manner the particular algebra a„-i is also characterised by the Dynkin diagram fl

n-l:

"~^~^

°

which consists of n — 1 vertices. The Cartan matrix is recovered from the diagram by defining (i) an = 2, (ii) atj < 0, » ^ j with a,j determined from the constraints, | Oij \\ aJX |= (the number of lines connecting the tth and jth vertices of the Dynkin diagram) and | o,j |<| aji | if the diagram contains an arrow pointing from the »th to the j t h vertex or a,j = aji otherwise. Let v be a graph automorphism of the Dynkin diagram for a„_i depicted above. The set of such automorphisms is equivalent to the permutation group 52 = {^0, v\) where VQ is the identity element and i\ is the transpo­ sition v\ : j —» n — j . It is easy to check that the Chevalley generators of an-i can be generated from {e 0 ,...,e„_i} alone and so consequently they generate the algebra itself. Let s := (so, ■••,sn-i)eZ+, where Z+ is the set of non-negative integers, then it is a theorem due to Kac [6] that any order

d-h^2

Si «'eZ£

automorphism a of an_i where /»e{l, 2} labels the order of the elements comprising 52, can be defined up to conjugation, by setting a.ti = As'e,-

t = 0,..., n — 1

where as before A is a rfth root of unity and AJ is the eigenvalue for the eigenspace of a which consists of elements of degree j in the Zj- gradation of a„_i defined by cr. Kac's calls this the ft — s gradation of a„_i. In particular the 1-principal gradation of a„_i is defined by putting h = 1 and Si = 1 for i = 0,..., n — 1 and the homogeneous gradation or "root space" gradation is given by h = 1 and SQ = 1, s, = 0, » = l,...,n — 1. The loop algebra an-i^c) in these two cases is called respectively the principal realisation or the homogeneous (untwisted) realisation of an-i- Define an order 2 automorphism v of am by v.ti = e„(,)

t = 0, ...,m

Denote the fixed point set of am by a£,. Then we find that a^ is a simple Lie algebra generated by m = 2 n - l : e° = 2ene? = e 2 „-i+e,

f° = jnft

= fm-i+fi

» = 1,.-, n - 1

107

m = 2n : e? = e2„+i_,- + e;

/? = f2n+i-i

+ fi

i = 1,..., n

The simple Lie algebras generated by these Chevalley generators can be obtained as before. The Dynkin diagrams are found to be

m = 2n-l :

bn

o=>o—o

o

(n vertices)

m = 2n

d„

o—^—o

o

(n vertices)

The afline Lie algebras which correspond to the loop algebras introduced above are obtained by a central extension 0 —* C —* g —► g(cr). We have g = g(
fi = f?®k~ai

i= 0

n-1

(2)

The generators eg, /Q are respectively the basis vectors for the lowest and highest root space of g° chosen so that [eg, ffl = dt

[«oV°. eg] = 2eg

[a0Vo, /0°] = -2%

108

From the Chevalley generators (2) we obtain

«,y = h,/.'] = ctf + d- 1 * Tr (e?tf) * The element z is the canonical central element since n-l

From the Chevalley generators we can construct a Cartan matrix in the usual way. In this case however, in contrast to the simple Lie algebras, the matrix is singular. This is a characteristic feature of Kac-Moody algebras. It is the number h which defines the particular type of affine Lie algebra. Changing the numbers *, defines a twisted Lie algebra which is isomorphic to the given affine Lie algebra. The algebras which we obtain from the three cases which we consider are called ajjij, a ^ and a^„-i- They have the Dynkin diagrams

o _ ,' ' :

o

o

•

o

o

D-l

*2n(2)

*2n-,(2):

Each graph has n vertices. The Dynkin diagram for gl^1', g 4 n a n c ' S^2n-i a r e t>ne s a m e > but as explained above since the Cartan subalgebra contains an additional element I basis elements / ® Wd jtZ have to be included in the root space decomposition. In this paper we only study the integrable equations associated with aj,2j and gl!£'. The representation theory of these algebras is based upon the existence of a maximal Heisenberg subalgebra. Rather than develop this

109

theory in the abstract we shall in the next section show how the fundamen­ tal representations of these algebras can be obtained for a canonical set of twisted realisations. For each such representation a distinct hierarchy of integrable equations exists.

4

AFFINE LIE ALGEBRAS AND INTEGRABLE SYSTEMS

In the introduction we mentioned that the representation theory of affine Lie algebras will be used to classify the integrable equations. The fact that the completely integrable equations were related to the representation theory of affine Lie algebras is comparatively recent and first occurs in the pioneering work of Dxinfel'd and Sokolov [7] and the Kyoto school of mathematicians (Sato and Sato, Date, Jimbo, Kashiwara and Miwa [8]-[10]) published in the early eighties. Previous studies of finite dimensional integrable systems had exploited the theory of simple Lie algebras over C such as the work of Bogoyavlensky [11] but the recognition that the representation theory of affine Lie algebras is associated with integrable equations belongs to these later authors. In fact this work is far from complete and a "grand classification scheme'* based on the algebras does not yet exist. Early in the development of soliton theory Hirota [2] had developed a method for finding special classes of solutions to integrable equations which could be obtained in closed form, the n-soliton solutions. This technique originated from the well-known Cole-Hopf transformation which transforms the nonlinear Burger's equation into the linear heat equation. It is precisely this special class of solutions which the representation theory of affine Lie algebras produces. The Hirota equations which these solutions satisfy are also obtained. The representation theory of infinite rank affine Lie algebras (those with an infinite number of Chevalley generators), is easy to develop. The ad­ vantage of studying these algebras is that besides the class of integrable equations which they define it is also possible to obtain the integrable equa­ tions for ak without recourse to a detailed analysis of the representation theory for this algebra as we outline below. Let gl(moo) be the Lie algebra of complex matrices, with the matrix commutator [., .]<> as the Lie bracket, whose elements have the form g — 0?'?£?>*egl(moo), t.jeZ, 1 < a,b < m. A repeated index is summed and there are only a finite number of non-zero elements. The standard basis

110

element is defined by £ # := (S^^)u,vcZ,i
■—■—'

The Chevalley generators {e,, /,• : ieZ} and the simple coroots H v := {oti : ieZ} are given by e»n(r-l)+m = -E^r'+l

^m{i—\)+a = Kf+l

<*r = [er , fr]o

0 = 1, ..., m - 1 ™Z

where fr := ej (g' := (h,j i. The commutator [.,.]„ extends to gl(moo) by linearity. The m-principal realisation of g l ^ [12] or the m-component g l ^ [10], denoted by gl(moo), is defined by gl(moo) = gl(moo) $ C z with the bracket [Si + Aiz, (ft + A2z] = [si,92I0 + iKffi, 9i)z

V C , p,egl(moo)

and where ip(.,.) is the C-valued cocycle

Klitf,Eg) = gg%m-9V)) 'W = \ o ;>o The corresponding coroots II = {a* : ieZ} are defined by <*i = [e«. /•] = «iy + hflz Consequently z is the canonical central element

* = £<*? itZ

111

The Cartan subalgebra h C gl(moo) is spanned by H v , / and z. The Lie algebra gl(moo) contains the subalgebra © m g(oo) which consists of m copies of gl(oo). Denote the ath copy by gl°(oo). The Lie algebra ©gl(moo) has the principal gradation defined by dege? = 1 = - d e g / f

o = l,...,m»eZ

where ef := E?f+1, ff := (e?) t are the Chevalley generators of gl a (oo). Put

«°(J):=£3X+>egl(™>°)

(3)

««Z

and let s denote its image under the canonical homomorphism gl(moo) —* gl(moo). Proposition 1 (a) The Lie algebra ffimgl(oo) contains the principal Heisen­ berg subalgebra s m := (&™=x s° © Cz and s° = s° © Cz can be identified with the principal Heisenberg subalgebra o/gl°(oo). The algebra s m is generated h {s"(j),z : a = 1 m, jeZ}, [stt{j),s\k)]=j6*;b_kz (b) The Lie algebra gl(moo) contains the maximal Heisenberg subalgebra s m which can be identified with the principal Heisenberg subalgebra o/© m gl(oo) C gl(moo) [12]. We call s m C gl(moo) the maximal Heisenberg subalgebra of gl(moo); its canonical projection onto gl{moo) determines a Cartan subalgebra. The principal gradation of ©gl m (oo) induces a Z-gradation of gl(moo) called the m-principal gradation which is defined on basis elements by deg £ # = (fi - r)

deg z = 0

Let d be the corresponding degree operator on gl(moo) so that d.£# = ( « - r ) £ # The generating functions for gl(moo) are given by

y/°-V, ob) := D K'hi %j+«0,ta - f r 1 * ,j e z

(4)

Pa

We now consider the representation theory for these algebras. Let g v denote either gl(moo), gl^1^ or aj,_i. Introduce the category C\ of g v -modules such

112

that Ve& [13] if (a) the canonical central element z acts as the scalar 1 (b) the degree operator d of g v has an induced action on V such that V = U. £ F V* where F is a field of characteristic zero (it may happen that F = | Z for example) and V* consists of homogeneous elements of degree i and moreover there exists toeN such that V"+,° = 0. Thus V(Ci is a level 1 g v -module. It is convenient to represent the action of
V

hZk

i-P' : =

v

D

ii...ikPi-Pk

«i,...,;'teZ

where ^...^eV. Note that the generating functions belong to EndV{p, q}. Define the delta function 6{p) and the formal function (1 — p ) - 1 by the series

5( P )=EP I icZ

a-pr^Ep* tVZ+

We can define the action of the delta function on V{p} by introducing truncation operators Tn : V{p} —> V^jp]

T„v{p}= 53

M„(Z*)

W

where <£„(i) = •! * ! f . j ~ " and | i |:=| ii | +... | ik | Then the following results are easily established by considering v\p, q]eV\p, q]. *(-) v{p, q} = 6(J) w{p,«} = 5 ( | ) w{g, g} 5(p /, )v{p} =

ft-1E5(pw"i)vV}

where w is a non-trivial hth root of unity. In terms of the module notation we get from (3) and (4) for Ve& a g(moo)-module, the following fundamental relationships in End V{p, q, r, s}. [ • " O W f o . f c ) ] = («°V. - «°,V.)fll6,c(Pl,9c) [«?/ a ' i '(Pa,?6),S/ e ' / (re,«/)] = 5 M 5 ( - ) f f / a ' / ( P a I S / ) - « a , / « ( 7 - ) ^ , 6 ( ' - e , 9 6 )

(5)

113

We now introduce an important concept which enables the Hirota equa­ tions for gl^1' and a } ^ to be obtained directly from those for gl(moo). The principal and homogeneous representations of these algebras can also be obtained from the fundamental representations of gl(moo) but for an arbitrarily twisted algebra further modifications are necessary [12]. The J-reduction of gl(mcc) is the fixed point set gP(moo) of the auto­ morphism, (P; *#.-£&.,.+,., z~z where n = (ni,...,n m )eZ+. It was shown in [13] that gl n (moo) can be identified with a realisation of the finite rank affine Lie algebra gl^1', or more correctly with a completion of it, where n = £(n) := £ n 0 - The particular realisation which is important for the representation theory is the one defined by the J-reduction of the generating functions gT' (pa,qb) of gl(moo). These are obtained by applying the canonical homomorphism gl(moo) —»• gl(moo) to the generating functions (4). Let d = [ni,...,nm], the least common multiple of {rt\,...,rim], and put n 0 := d/na. Introduce the basis elements Ffj , teZn so that 9 £f#f/egI(n,C)

where s'j'eC. The 1 - n l gradation defined on basis elements by

n

gradation of gl(r»,C) is a Zd-

deg Ff'j = jni, — ina mod d

(6)

Let gl(r»,C) = 8j e z d g(j) with the bracket [ffi(Jfc) + hz,g3{k)

+ \2z) = [gi(k),g2(k)]0 + cT1 Tr Res

where gi(k)e ®jtZ gy

mod

d) ® **'

{dk(gi(k))g2(k)}z

*ieC

Throughout the rest of the paper when we refer to gl^1' or a!^^ we always mean that it is the realisation with the Sln-gradation defined in (6). The explanation for the notation is as follows. Let a be the auto­ morphism of gl(n,C) which defines the 1 - Sln-gradation. It is given by a.Q := X~l a X, oegtfn, C) where X = diag (Xlt...,Xm)

Xa := diag (1,A n »,...,A*"'- 1 )"")

114

and A is a non-trivial dth root of unity. Then the eigenvalues of d, and the corresponding eigenspace is g(j) in the decomposition (1). Under J-reduction, © m gl(oo) C gl(moo) becomes the Lie algebra

c u gilV The Lie algebra gV^K has in Kac's notation introduced in the previous section, a Z-gradation, the 7ialn0-gradation [12]. It is defined by deg e? = na = - deg ft 4 ■= K&i ® *''"*

#

:

(?;

= ei

iiZ

on the Chevalley generators {e",/°} of glj,1'. In (7) we assume that tf := k_1 and i — i mod n„ if a is the corresponding "upstairs" index to i. This gradation is associated with a Zj-gradation of gl(n a ,C), where d = £(nalna). It is defined by the automorphism p, iPp) = •eZ„ a ,

J2

p^+^-J^^ina^-jntpaj,

0

k-(,d+ina-jnb)

jiZ„b,*tZ

where and tp are dth roots of unity. In subsequent expressions of this type we will omit the limits on the summation sign. The J-reduction of the maximal Heisenberg subalgebra s m C gl(moo) defines a maximal Heisenberg subalgebra s n C g l ^ ; if 7r : g$' —► gl(n, C) is the covering homomorphism then 7r(sn) is a Cartan subalgebra of gl(n, C) The maximal Heisenberg subalgebra s n is generated by {ra(j),z : jeZ, 1 < a < m} where

^ ( i ) ~(r>(j)y

jeZ

r°(0):=£i^a

The commutation relationships are canonical, [ra(i),rbU)] = i6°;b_jz

^ " E C ®

1

"

115

In the case of a ^ l i C gv£' the maximal Heisenberg subalgebra has to be redefined (since r°(0) jtsglj. Let pb := (Af,..., \bm), q6 := (mAj, ...,nmXbm) be the solutions to p a .q 6 = 6a'6

p6.n=0

«,»=l,„,m

(8)

Define h n C a ^ i j as the Heisenberg subalgebra generated by {ha(j),z : jeZ,a = 1 m - 1} where ha(j) := ft°(0) ® k>d and A°(0) := £ A£r6(0). Then the maximal Heisenberg subalgebra Sn C a j ^ is SQ

— © a : n a ^ l ^ a © **B

where s„ a is the Heisenberg subalgebra of a } ^ generated by {r a (j),« : jeZi,j £ 0 mod na}. The generators satisfy the canonical commutation relationships. Proposition 2 (a) A basis for gl},1', n = ^(n) with the xilQ-gradation is provided by the homogeneous components of the generating functions g£c(A'p,p) f,

i = 1,..., nc - 1 c^bifm

=l

c

gQ (A.'p,p)

i-l,...,fb,cb
c=l,...,m

and gl^° (A'p,p) where A is a non-trivial dth root of unity, d = [r»i,..., nm] and fb,c is the greatest common divisor of {nb,nc} together with the gener­ ators of the maximal Heisenberg subalgebra s n C gl£,'. (b) A basis for aj,_ a is provided by the generating functions listed in (a) to­ gether with the generators of the maximal Heisnberg subalgebra s n C a ^ i j . It is important to note that except for the case na = N, a = 1,..., m there is no general formula for the Chevalley generators of gv£> or a ^ I j . The fundamental relationships satisfied by the generating functions {gQ (p, q)} and s n for gl£^, which are analogous to those for gl(moo) given in (5) can also be derived [12]. For our puposes we only require the first of these rela­ tionships. Let A be a dth root of unity whose multiples are the eigenvalues of the automorphism cr of gl(n,C) defined above. The following operators belong to End{p,g} where VtC\, is a gl^-module.

[ra{j),lfc{*P, 4>p)] = Wn°F* ~ ¥n'^)PinaltC{
*P)

(9)

116

The proof is by straightforward calculation. Notice that gln' (up, Xp) is not a generating function if na = 1 = nj, by proposition 1. The result (9) can also be inferred from the J-reduction of the analogous expression in (5) for gl(moo). The m-principal realisation of gl^ has a vertex representation on a space which can be formally identified with the representation space for the Heisen­ berg system associated with ak introduced by Frenkel and Kac [14]. The lattice associated with the representation however is not a root lattice. In [13] Lepowsky and Wilson show that with every maximal Heisenberg subalgebra of aj121 it is possible to associate a representation of a fundamental module. A maximal Heisenberg subalgebra of a^ij is defined by an ele­ ment of the Weyl group W of an_i. The construction of representations of level 1 for arbitrary maximal Heisenberg subalgebras of a^ij was given by Lepowsky [15]. Kac and Peterson [16] show that a complete non-redundant list of Heisenberg subalgebras of a^ij is afforded by a set of representatives of the conjugacy classes of W. For a^ij there are p{n) distinct Heisenberg subalgebras (p(n) is the classical partition function; the number of partitions of n without regard to order, p(3) = 3, p(5) = 7 etc.). In fact it is easy to show that we can obtain p(n) distinct maximal Heisenberg subalgebras of aj,_i by J-reductions of gl(moo). Lemma 1 (a) Let {gl q (moo) : £(q) = n, 1 < m < oo} be the set of Jreductions which define realisations c/gl„ '(*n-i)- There are p(n) distinct realisations o/gln ( a n-i) defined by distinct maximal Heisenberg subalgebras SqCgtf'KCai1^). The proof is by a straightforward counting procedure. It follows that we have the following fundamental result. Theorem 1 A canonical set of realisations o/gl*,1' and&ni1 can be obtained from the J-reductions of {gl(moo) : 1 < m < n}. The determination of the Chevalley generators of gl^1' with the Hl n gradation is difficult except in the case when n0 = N, a = l,...,m. They are then given by e m( jv-i) =

FN-I,O

®*

em(r_1)+m = * * £ , ® k

e m(r -i) + a = F?f+1

a = l,...,m

0
1< r < N - 1

117

with ft := e}. The principal and homogeneous realisations of gffl and a^lj correspond to the J-reductions denned by: (i) m = 1, r»i = N / 1 (principal realisation) (ii) na = 1, a = 1,..., m (homogeneous realisation)

5

LEVEL 1 INTEGRABLE EQUATIONS

The integrable equations associated with the representations of gl' 1 ' and a^ii can be obtained from those associated with gl(moo) [12]. They are obtained in their Hirota form from the standard modules for this algebra. In the literature usually only level 1 standard modules the fundamental modules are considered. This is the case we shall consider in this section. In the next section we shall investigate the level 2 case for a^'. Introduce the Schur polynomials

5Z PiirW '■= exP J2 y*^ r>l

j>0

Thus the Schur polynomial Pj(y) is a polynomial of degree j in the variables {j/i,..., J/J}. The first five Schur polynomials are po(y) = l

Pi(y) = 3/i

P2(y) = y2 + 2 ^

My)

= 2/3 + 1/12/2 + ^y?

P4(y) = m + j ^ + 2/13/3 + jj2/i2/3 + 2^!/? l 2 l 2 l 5 ps{y) = 2/5 + yiv* + 3/22/3 + -2/12/5 + ^3/13/3 + y ^ i The Hirota equations are defined in terms of polynomials in m variables of this type. Let Eka[i\y) ■= (-l)h*+"+h'->

£Pj(-2yW)pi+fco+1(zW)eXp(y.z)

where y^a\ a — l,...,m are m sets of the y variables defined above and y<») : = £ > » > > )

y<°> : = ( ^ , 2 4 a )

H/W,...)

Introduce the Z-lattice M with the basis {sa : a = l,...,m} and put ^(5Da=i /*o»a) = /*i + ••• + Mm- Consider a fixed AeM such that the vec­ tor of coefficients of c, ceZ m and 1(c) = s + 2 for a given s > 0. The

118

corresponding family of Hirota equations is the s — (gl(moo), P1") family ( r + denotes the subgroup of the affine Lie algebra which defines the evo­ lution [12]). They are obtained from the coefficients of y in the following equations, with c subject to the above conditions. m

E ^ . - 2 [ft: y] (r«=+A-.. (* - *)**+- («+■)) Lo = ° o=l

The functions rc+A-»0,rA+*0) are called r-functions. The transformations A —> /i + A, £(JI) = 0, defines a new set of r-functions which satisfy the same family of equations. Define the totality of Hirota polynomials by the generating functions Hir[z:y]:=£Hir,[z:y] *>o m

Hir,[z : y] :=

£

£

ECa-2[z : y] ® e 8 " 2 "

CtZ™j*(c)=»+2 o = l

Let Par consist of finite sequences of non-increasing positive integers 7 = (7j 7i). 0 < 71 < ■ •• < 7i and put (y)"" = (i»n>0tti-.>*)■ Then Hir,[z : y] consists of sums of the form ( y ^ y V y j c - ^ z ) ® e c - 2 5 a over all possible 76 Par and c such that 1(c) = s + 2. The degree of the polynomial in this expression is deg ^ ; C ( ,-i(z) ® ec~2r' = £(7) + \ I c | | where | . |B is the Euclidean norm. The s—(gl(moo), r + ) family of equations is made up of equations of homogeneous degree. The relationship between the variables in the fundamental representa­ tions of gl(moo) and the maximal Heisenberg subalgebra s m , is given by saU)^9x(')

s0(-i)'-Jz5o)

J>0

m l

(10)

The representations of gl^1' also in part involve a similar identification to (10) in which the generators of the maximal Heisenberg subalgebra s n C g l ^ , {r°(j), z : jeZ}, replace those of s m , but in this case deg r°(j) = jn„, j £ 0. Observe that the actions of s°(0) and r°(0) are not defined; they act on the vacuum space of s+ := ( space spanned by {s°(j) : 3 > 0} or {r°(j) : j > 0}) in the representations.

119

For a^ij it is not hard to show using (8) that the corresponding Hirota equations arise from the identifications z i-> 1 r°(j) t-+ d' (a)

ra(-j)

m

i-» jxj°'

j £ mod n a , a : n a ^ 1

m

E "«*£ = °

E n-^W = 0

o=l

o=l

i >0

(11)

'""

It is straightforward to obtain the corresponding Hirota equations for a^2 x corresponding to the fundamental representation of its n l n realisation by imposing the conditions (11) on the Hirota equations for gl(moo), 1 < m < n. The rigorous proof of the correctness of this is given in [17]. We consider as an example a few of the Hirota equations associated with a^ (the complete list of non-redundant representations for a x ' is already known and consists of the principal and homogeneous representations). From the theorem the realisations associated with aj are derived from the J-reductions of {gl(moo) : m = 1,2,3}. In fact if we consider the 0 — (gl(3cc),r + ) family of Hirota equations then we find that they also contain the 0 — (gl(2oo), T+) and 0 — (gl(oo), I"*") families as subsets. The corresponding Hirota equations for a 2 arise from the reductions ne{(l, 1,1), (3), (2,1)}. The first of these defines the homogeneous representation, the second the principal and the third corresponds to a new representation. To obtain the Hirota equations for these reductions we use the correspondence given in (11). In particular if we put / := rx+(i,o,o,o). 9 '•= T-A+(O,I,O)> g~ := rx+(2,_i,o), ft := rx+(0,o,i). h := rA+(2,o,-i) then we find that the following well known equations are included in the reductions. T h e 2-component nonlinear Schrodinger equation [18]((l,l,l)-realisation) Hirota equations: dji)dj2)f.f-2g.g =0

-dxw (dxw + dxmj f.f - 2ft./. = 0

( 2 ^> " " fy) fQ = ° faw

+ B?p) f.h = 0

( 2 0 ^ ) + <£<,)) f.g = 0

120

integrable equations: 2 t r * - r l a ; + 2 r ( | g | 2 + | r |2) = 0 2»fc - qxx + 2q (| q |2 + | r | 2 ) = 0 where (*, -it) = ( x ^ , ^ ) and (r,r*,q,q*) = Boussinesq equation[19] ( (3)- realisation) Hirota equation:

(h/f,h/f,g/f,g/f).

(^)+3^1)j/./ = 0 integrable equation :

3u« + 3u2 + uix - 0

where u = 2d2 log / and {x,t) = (*i , 4 )• Redekopp's equation[20] ( (2,l)-realisation) Hirota equations: dmdmf.f + 2g.g = 0 integrable equations: vt = - 2 | q | 2 where (x,-it)

= {x\l,,x^>)

iqt + qxx = vq

and (
A detailed classification of integrable equations associated with ak' is given in [17].

6

THE PRINCIPAL LEVEL 2 REALISATION

In the literature Hirota equations and integrable equations associated with level 1 or fundamental representations of the affine Lie algebras are only considered. However the Hirota equations for the other standard modules also should be considered in the full classification problem. This turns out to be a problem whose difficulty rapidly increases with the level number. In next few sections I will derive the Hirota equations for the principal realisation of aj . Choose the following basis of s^(2, C)

/»=(! _°i) «/»=(? 5) *-o=(-°i S)

121

so that with respect to the matrix commutator [., .]o as Lie bracket we have IP, X±p)o = ±2x±0

[X0, X-f3]0 = -/?

The principal gradation of s^(2, C) can be defined by the involution cr.g = — T 9 , 9^s£(2, C) where T is the transpose operation. The eigenspace g, of IT corresponds to the eigenvalue (—1)', ieZ2 = {0,1}. Thus < 2 , C) = go © gi go = C(x0 + x-p)

gi = C/3 © C(xp -

x-p)

The loop algebra s£(2, C[fc_1,fc]) with the principal Z-gradation is then defined by s£{2, Cik-1, k]) = e,-«z & ® ** &=&i mod Z2 with the bracket [gi ® k{, g2 ® f']o = [ffi,ff2]o® ki+j For yes£(2, C) let g — go + g2 be its principal Z2-decomposition and define

9(f) ■= 9i ® k"

9{p} •= J2 9(i)& jeZ

g(k) ■= Yl 0(0 t'cZ

In the principal gradation deg 5(v) = i and g{p} is a generating function such that the coefficient of p' has degree't. The principal realisation of aj is the central extension of s£(2, C[fc_1, k]) defined by a[1} =s£(2,C[k-1,k])®Cz with the bracket bi(*) + Ai*,«a(*) + A2z] = bi(t),fl2(fc)]o + i Tr Res {^(ft(fe)) ««(*)} * (12) In equation (12) Res {£,- a>'fc'} : = a - i a n ^ Tr i s t n e t r a c e operator. Observe that {/?(,•),£ : je2Z + 1} defines a Heisenberg subalgebra s C a ) ' ,

D%)>%)] = ' V i *

(13)

122

Let a\ {p,q} denote the vector space of formal Laurent series in p and q with coefficients in a^ . Then x^{p}eai '{p, q] and (14)

\P(i),*p(.P)] = 2p~j*l3{p}

Define 6(p) := YljeZP' an<^ Dv{p) := Pj^v{p) for v{p}eaj '{p, q} so that in particular 6(p)v{p} = 6(jp)v{l}. Let < gi,g2 >:= Tr (ffijfe) for y,€si(2,C) then the commutation relation for the generating functions is

[x0{p},xy{q}] =

5.1> '*0,1 *-y]M >■ ((-• 1 ) ' P / 9 ) +5^<£r,x'3'^>z?5((-W«)

7

(15)

THE PRINCIPAL LEVEL 2 REPRESENTATIONS

Let s± := ©±i>o C0n\ and define Sym(s_) as the symmetric algebra generated by s_. Then the commutation relations (3), (4) and (5) can be used to construct standard representations of a\' as endomorphisms jr : a.\ ' —» End V of a vector space V = Sym (s_) ®c Oy where 7r(s+)£2y = 0 defines Qy [13]. The representation has level £ if n(z)v = £v

ve Sym(s_)

The action of s, using (13) can therefore be taken as (in module notation) /3(,).(D®W)

=

(d/j(l)v) ®u>

»'>0iodd

%).(« ® w) = -it{P(i)v)®w i < 0 »' odd z.(u®w) = fu®o> v®weV

(16)

The action of the generators x1{p}, j — ±/7, is given by the vertex operator 7rx7{p} = X^(p;n,£) := E-(-j,p;n,£)E+(-y,p;w,£)

® Z^(p;n,£)

where

£±(7,p;7r^):=exp(2 £

V i>Oaodd

T(7(±J))P ±J /

± jA /

(17)

123

The operator Z^(p;n,£) := £ , t Z z7(,)p' generates the space Civ through its action on any element VQiClv- The endomorphism corresponding to x 7 (,jeap is obtained from the coefficient of p' in the formal expansion of Xy(p; ir,£). The principal degree of z^ is ». Consequently the space V decomposes into elements of homogeneous degree V:

-u V$ jtZ

such that Vj = 0 for j > N where N depends on the level. Because of the structure of V, (8) can be inverted to define Zp(p; ir,£) in terms of ir{xp{p} and £*(/?,p;ir,£). In fact this is how Lepowsky and Wilson derive the properties of the Zv algebra generated by the components of Zp(p;ir,£). There are two properties common to all levels which are derived respectively from (14) and (15). [Z0(p;n,£),s] = O (i-p/qr'ii+p/q)-2" - (1 - q/pfl\\

ZP(P; *,*) Zp(q; »,*)

+ q/pT211 Zp(q; n,£) Zp(p; n,£) = --£D6(-p/q)

(18)

In equation (18) the rational functions have the formal expansions given by

(l+p)" = £

( J" )pi

j>o V

I

neZ

Equation (18) therefore defines relations between products of the operators ZM{\ which can be used to order or straighten elements of Zv ■ The simplest module constructed in this way corresponds to the fundamental modules for which £ = 1 and dim Civ = 1. In this case

^(p;,r,l) = ±±(-l)* The ± signs correspond to the two possible non-equivalent fundamental modules for a; '. The corresponding vertex operators is * , ( p ; n, 1) = ±\i

exp ( 2

£

\ j>0;jodd

p->"T(_i)) exp ( - 2 /

V

£ j>0;jodd

p> ^ / j ) /

(19)

124

where ir is the representation denned in (16). The level 2 standard modules and representations are constructed from the representions (W, 7r) = (Vi ® V2, JTI ® ^2) where (K", TT») are fundamental modules. Let vo = "o ® "o be a highest weight vector of (W, TT), (i.e. VQ eflv; ) then the standard level 2 module is = u ( a « ) vo where U(a^ ') is the universal covering algebra of a^ . The corresponding vertex operators in the representation are X 7 (p; TT, 2) = £ - ( - 7 , p ; TT, 2)E+(-j,p;

w, 2)ZY(p; TT, 2)

and Z 7 (p; TT, 2) = ZW(p; TT, 2) + Z™(F> ^, 2) Z7x'(p;ir,£)

The operator representation is

(20)

for the representation associated with the level £

Z!f>(j>;ir,£) -

£T (7, p;*-,-£)(*-! z 7 {p}® ... ® 7r;x7{p} ® ... ®Trex^{p})E+(f,p;n,£)

where 7Tjx7{p} := 1

j=l,..,£

For level 2 representations there is the anti-commutation relation [ZP(p;w,2),Z{J\q;ir,2)}+

= -±6(-p/9)

where [a,b]+ := ab + ba, which can be obtained from the definitions (15). Since we also have Z01\p;n,2) = ±Z{?p(p;n,2) where the + sign is used if Vi = V2 we can obtain from the formal expansion of (21), (22) and the definition of Z 7 (p; TT, 2) the results [13] z0(i) = < W) 1 I 0

-{

W

=

f 0 (241(i,

Vi « V2 »£2Z + 1 «'e2Z ie2Z+l

* * *

(21)

125

The anti-commutation relations for Zg(p;ir,2) derived from (21) and (22) can now be used to show that (18) is identically satisfied. If ZviVi ls t n e algebra generated by the components of Zg'(p; TT, 2) then Q\y — Zv^-vo and similarly Uv = ZV-VQ. Therefore the standard level 2 module structure for a[ originally found by Frenkel [21] is recovered. T h e o r e m 2 Let C(z) be the Clifford algebra generated by {zpij\ ■ jeZ} with lzP(i),z0(j)]+ = -2( - 1 )' 5 <+i.° and let C(z) be the exterior algebra with the basis {zg(ni)...zg(nr).vo : r > 0,ni, ...,n r eZ,m < ... < nr < 0}. Then fiy C C{z) is the exterior algebra with the basis {zgtni\...Zp{n\.vo : r > 0,n 1 ,...,n r eF,n 1 < ... < rw < 0} where F = 2Z (F = 2Z + 1) ifv[^. V2 (Vi 9* V2). The corresponding vertex operator is X , (p; TT, 2) = exp I 2

£

\ j>0;jodd

p J 7 ( _ i } J exp I -

£

/

j>0;j odd

\

P'A-V-'] /

IZ)^ZW) Sjtf

(22)

8

THE INTEGRABLE EQUATIONS

It is convenient to use C[x], x = (zi,Z3, ...X2r-i, • ••)> as a realisation of s_ by putting Xi = /?(,). On C[x] we have the contravariant Hermitian form < p(x), g(x) > Q x ] := \p(Dx)q(x)

|x=0

where •Cx := (dxi, -dxz > •••> 2_ _ f**»r-i' •■•) The action on C[x] is the natural one xi ,p(x) = z,p(x)

l.p(x) = p(x)

9^ ,p(x) = dXip(x)

The contravariant property is then defined for p(x),g(x), r(x)eC[x] by < r(x).p(x),g(x) > C m = < p ( x ) , r ( r > x ) . g ( x ) > Q x ]

(23)

126

For simplicity we write Zj := z ^ j . Let j = (h,h,—Jn) k < h < —< in < 0 and define 2j := z^ hzj% A ... A ZJ„. On C(z) we define a Hermitian form on basis elements by <*h.*k > c ( Z ) : = d e t I (zhi,Zkj)

|

with (*>*j) = 26i>>

(24)

Introduce the dual C (z) of C(z) which is the exterior algebra generated from iz; : teZ_ \ such that *«(*i) = 2 6, 'J

The following wedging operations can be defined on C(z). Zj -Zh = Zj A zv. Zj.Zb = % ( % i ) A % 2 A... -Zj(zh2)zh1 Az/i3 A... (-l/ + 1 ii(zfc,)2 h l A...*„._, A 2 i u i A... It is clear that ij is just the odd derivative operator DZj := \dZi. of the odd derivative operator

In terms

<*h,*k> / c o l c (z)>=-D* h (*k) where DZh := £>/,! A Dh2 A ... A D„„. The choice of normalisation in (24) and (25) has the following nice properties. The adjoint of Zp(p;n,2) defined in (20) and given the form in the Theorem is, if we define pt = p~x

^(^,2)* =$>-'*] However from the definition of the Clifford algebra its action restricted to C(z) is given by *fc =

{

(-l)fc+1iU Zk

k>0 with io := 2o — 0 Jb < 0

127

It follows that Zfiip) *, 2) = Z$(p;, ir, 2) + Zp(p; r, 2) where 2f(pjJr,2) t = ±2*(p;jr,2) A contravariant Hermitian form can be denned on Qy by the restriction of < •>• >c(z) to fiv-

< !»(■)./(■),*(■) >„v=< /(z),^(i).s(z) >„v where f(z),g(z)e£lvby

A Hermitian contravariant form on V is then defined

< p(x) ® / ( z ) , g(x) ® ff(z) > v :=< p(x), g(x) > Q x ] < / ( z ) , ff(z) > „ v (25) Let y W , ^ 2 ) be two copies of a level 2 module, not necessarily the same, and form the representation (V*1) ® V*2', TI-W ® x(2>). Clearly we can proceed in the same way as before with this representation to obtain an irreducible representation of aj . Let x*',zW, i = 1,2 be copies of the variables denning the representations. We can take «o ® vo = 1 a s a highest weight vector of W — V^' ® V' 2 ). An element gea[' or an element GeAj ', the group associated with a^' acts on W according to g.(vW ® t,(2>) = g.vW ® VW + „(i) ® g.vW G.(vW®vW)

=

G.vW®G.vW

Introduce the change of variables x=

I(x(D+x(2)) !/• (1) ,

y=

I(x(l)_x(2))

00\

*/ (1)

(2)\

where P = C[x] ® C(z£°), Q = C[y] ® C(z^2)). The decomposition of W with respect to the contravariant Hermitian form (restricted as necessary),
>PeiQ-=p<

9,Q>Q

is W=V®U

(26)

128

and V is a standard module of level 4, (leV). On the other hand from the theory outlined in §3 W = C[x] ® Q\v and V = C[x] ® fty. Therefore Qw = nv®ttv

<^v,^v>PSQ=0

(27)

and U = C[x]ft^. The space Hir := fly is the space of Hirota polynomials in the level 2 principal representation of a^ . We can therefore calculate the g-dimension of Hir by using the results of Lepowsky and Wilson [22]. For the decompo­ sition of the module W into subspaces of homogeneous degree we define


where dim W{ is the dimension of the Wi subspace. The decomposition (28) then gives dim,C[x] dim,ft (1) dim,ft (2) = dim, I V + dim,ft£

(28)

The precise values of the g-dimensions in (29) depend upon the fundamental modules used in the construction of the modules V*1' and V. Label any standard module by the ordered pair of non-negative integers (fc_, k+) which correspond to the number of times the ± - fundamental module, equation (19), occurs in the tensor product of the representation. In fact though for V® we have the possibilities (2,0), (1,1), (0,2) and for V, (0,4), (1,3), (2,2) etc., the formulas are invariant under cyclic permutations so that we need only consider the casesfc_< k+. The results of Lepowsky and Wilson give the following. dim9C[x] = n (

1

+
n>0

dim,ftW

rin>o(l + ?2n) * - < * + (0,2) n n >o(l + ? 2n - 1 ) * - = * + (1,1)

n„>o(i-«Br1(i-96n"5)(i-96n_1)(i-«6") dim,IV = { n„>o(l - 9 2 "- 1 )- 1 n„>o(l-?Tr1(l-93n)(l-?6n-3)

* - < * + (0,4) *- < k+ (1,3) * - = * + (2,2) (29)

Since the decomposition by degree of the Hirota polynomials is Hir = U Hiri «
129 we get a family of polynomials partially ordered by degree. The family corresponding to the case (k-,k+) in (27) we call the (k+ — k-)-family of Hirota polynomials. The dimension of Hir,- for low decrees is listed below.

degree 0-family 2-family 4-family

0 (1.0) (1.0) (1.0)

-1 (1.2) (1.1) (0,1)

-2 (2,2) (1.2) (1.2)

-4 (3,10) (2,7) (2,5)

-3 (1.6) (2,4) (1,3)

-5 (3,16) (3,11) (1,9)

-6 (5,24) (4,18) (3,14)

Table 1: The entries (a, b) correspond to the dimensions a — dim (Qy)> and b = dim Hir,- respectively for the (fc+ —fc_)-familyof Hirota polynomials

The generating function xp{p} acts on the module W as the vertex operator Xp{p} = E(x,p)E(-Dx,p)Zp(p) where Z/3(p) = M^{y,p)Z(zp,p)

+

M~(y,p)Z{w0,p)

(30)

The operators in (28) are defined by

E{x,p)

:= exp(2

J2

Xjp 3

j>o,j odd

Z(zp,p)

=

£Z?zm 7tZ

Z(wp,p) := J2JwfiV) and

M ± (y,p) = E(y,p)E(-Dy,p-1)

±

E{-y,p)E(Dy,p-1)

From the theory of §3 for the form ot a level 4 representation ot x±p\p) it follows that Q.w is stable under the action of the operators Zly.ma) Z{y,vtp)

= M+(ytp)Z(zB,p) - M (y,p)Z(vf0,p)

130

The choice of normalisation for C[x] and C(zy) given earlier shows using (22) that the operators Z±{y,zl3;p) Z±(y,vfp;p)

= =

M+(y,p)Z±(z/3,p) M-(y,p)^(w^)P)

are self-adjoint (+) and skew-adjoint (—). The basis for fly can be calculated directly from the action of

Zp{p) = Y,ZiPi icZ

on the highest weight vector 1. The calculations can be checked against the results of Lepowsky and Wilson [22], Theorem 14.4. The complementary basis for Hir can then be calculated using the orthogonality property (28). The Hirota polynomials to degree —4 are given in the following table. deg

(1,3)

-1 -2 -3

f2 - Z y . u , , y2 v

u

2 ~ 2 yi u i , yf

(0,4)

«u>y?

3 + 2&U2, 4j/fu, - u 3 , y,2")-!, y?, y3, y i ^ . , ™ . ,

UJUJ + 2j/ 3 , y?

-4

(2,2) W-I , V! y,z_!, y,2 +VJ_1W_J

»i% - K > yiu3

-ht,

y3"x - 6 u 4 , y ? u i - s u 4 . Viih - g f 3 « i . y ^ + ^ « 3 « i , yi" 2 «i + 2 u 3 u i

yi

y ^ - j , yf.ys y?^-2 - ?*-4

y.y3 - K * - 3 * - I , y?*-,, » - , ^ , y,4 + j y ^ - i Z - i - j z _ 3 Z - i ,

y?-|«'-3«'-i + ?yf «-i<»-„

y^s + ^ - j ^ - j

y* + ^ z - 2 ^ w_ 2 y 2 , u<_,

z-aW-i, y 3 ^ - i , y ^ - 3

Table 2: The basis for Hir,- i = — 1, ..,—4 for the three possible cases. For simplicity we have put ut := vQ and vt := «£2/. We now produce the Hirota equations associated with this representa­ tion. The integrable equations can be recovered from the Hirota equations upon the introduction of-some auxiliary variables. For a level 2 module (vW.irW) i = 1,2 defined in §6, with V*1) not necessarily distinct from V^2', r*'g, ge&\' is an endomorphism of V™. The automorphism exp n^g of V is well defined because it is locally finite, that is for any veV^ there exists a finite n such that (rr^g) .v = 0. If A^ is the group of such automorphisms generated by the elements of aj , then the orbit of the highest weight vector 1 is 0(A^) := {G.l : G(A^}. It follows

131

that if rWeO(AW) then rW®r<2>eV where V is the level 4 standard module of the previous section. The relation (25) now implies if p(y,z^ ,z^){ Hir then < p(y, z « , ■ z ( 2 ) ) ,rW(x + y, z(1>)® r< 2 )(x- -y> « « ) >=:0 This is just the corresponding Hirota equation defined by this element of Hir. It can be written in a more familiar form as

V(\Dy > 29z(D>2~3z(2))'rW(x + y.zW)r( 2 )(x- - y . * ( 2 ) ) 1

:0,Z( 1 )= :0,Z<2> = .

' 'y=

' (31) The r-functions can be expanded in the basis for C(z^). For the examples listed in table 3 we only require the basis elements up to degree -4. 7"(x,z) = /0 + /l2_ 1 +/2Z_ 2 +(/3Z_3+fl3.Z_ 1 2_2) + /4Z_ 4 +S4«_iZ_3 + " • (32)

Put T^> := T : ■n'' is defined similarly but we replace /i, fft, «i etc. with fi, &, hi if the representations (V®, nW) t = 1,2 are distinct. The Hirota equations and the integrable equations are easily obtained and include a linearised Korteweg-de Vries equation as well as a coupled Korteweg-deVries equation. A full discussion of the equations is given in [23].

9

CONCLUSION

In this paper I have discussed the classification problem for integrable equations. I hope that it is clear from the article that there are still many interesting problems associated with "known" integrable systems. Most of this work was done while I was a JPS fellow in 1987. I would like to thank Professors Sato, Miwa and Jimbo for their hospitality at RIMS when much of this material was first presented.

132

REFERENCES 1. Gardner,C.S. Greene,J.M, Kruskal.M.D. k Miura,R.M. 2. Hirota, R.

3.

Dodd.R.K. k Morris,H.C

4. 5.

Mikhailov.A.V.fe Shabat.A.B. Mikhailov.A.V. & Shabat,A.B.

6.

Kac.V.

7. 8. 9.

11. 12.

Drinfei'd, V. G. k Sokolov, V. V. Sato,M k Sato.Y. Date, E., Jimbo, M., Kashiwara, M. k Miwa, T. Date, E., Jimbo, M., Kashiwara, M. k Miwa, T. Bogoyavlensky, 0 . I. Dodd.R.K.

13. 14. 15. 16.

Lepowsky,J. k Wilson,R.L. Frenkel, LB. k Kac, V.G. Lepowsky,J. Kac.V.G. k Peterson, D.H.

17.

Dodd, R.K.

10.

Phys. Rev. Lett.19 1095 (1967) "Direct methods in soliton theory" article in "Solitons" eds. Bullough, R.K. k Caudrey, P.G. pub. Springer-Verlag,(1980) "Solitons in General Relativity" pub. Plenum Press (1990) T.M.F. 62 163 (1985) preprint "Integrability conditions for the system of two equations of the form ut = A{u, )uxx + F(u, ux) II" Landau Inst. (1985). "Infinite dimensional Lie algebras" 2nd edition CUP (1985). Sov. Math. Dokl 23, 457 (1981). Lect. Notes in Num. Anal. 5,259 (1982) R.I.M.I.S. Kyoto Univ. 18, 1077 (1982) R.I.M.I.S. Kyoto Univ. 19, 943 (1983) Comm. Math. Phys. 5 1 , 201 (1976). " Fundamental representations of the m-principal realisation of gl^." In press /. Math.Phys.(1990) Invent. Math. 77,199 (1984) Invent. Math. 62, 23 (1980) Proc. Natl. Sci. U.S.A. 82, 8295 (1985) article in "Symposium on anamolies, geometry and topology" ed. Bardeen, W.A. k White,A.R. pub. WorldS'cientific, Singapore (1985) "Classification of a restricted class of solvable equations associated with gloo" - in preparation

133

18. 19. 20. 21.

Manakov.S.K. Zakharov.V.E fc Shabat.A.B. Ma,Y-C. & Redekopp.L.G. Frenkel.I.B.

22. 23.

Lepowsky.J. & Wilson, R.L. Dodd.R.K.

Sov. Phys. JETP 38 248 (1974) Funct. Anal. Appl. 8 226 (1974) Phys. Fluids 22 1872 (1972) Article in Proc. 1981 "Conference on Lie algebras and related topics." S.L.M. 933 71 (1982) Invent. Math. 79 (1985) (1985) "Level 2 Hirota equations associated with the principally twisted affine Lie algebra a„ Los Alamos preprint LA-UR 89-3205 (1989)

134

LIST OF ALL INTEGRABLE HAMILTONIAN SYSTEMS OF GENERAL TYPE WITH TWO DEGREES OF FREEDOM. "PHYSICAL ZONE" IN THIS TABLE A. T. FOMENKO Department of Mechanic! and Mathematict, Moscow State University, Motcovi 119899, USSR

1. I n t r o d u c t i o n This paper contains a short review of the new theory of topological classification of integrable Hamiltonian systems of differential equations with two degrees of freedom. This theory was created by the author and then was developed in collaboration with his colleagues, in particular H. Zieschang, S. V. Matveev, A. V. Bolsinov, A. A. Oshemkov, A. V. Brailov, and V. V. Sharko. The whole text will be published separately. The theory develops some important ideas of I. M. Gel'fand, S. P. Novikov, V. I. Arnold, S. Smale, R. Bott, J. Marsden, T. Ratiu, A. Weinstein, J. Moser, V. V. Kozlov, M. P. Charlamov, and F. Waldhausen. The basis of the theory is contained in the following main publications: A. T. Fomenko, "The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability", Math. USSR Izv. 29 (1987) 629-658 (see [15]). A. T. Fomenko, Integrability and Nonintegrability in Geometry and Mechanics (Kluwer, 1988) (see [19]). A. T. Fomenko, "Topological invariants of Hamiltonian systems integrable in Liouville sense", Fund. Anal. Appl. 22 (1988) 38-51 (see [16]).

135

A. T. Fomenko, "Symplectic topology of completely integrable Hamiltonian systems", Usp. Matem. Nauk 44 (1989) 145-173 (see [17]). Farther developments of the theory can be seen in: A. V. Bolsinov, A. T. Fomenko, and S. V. Matveev, "Topological classifi­ cation and arrangement with respect to complexity of integrable Hamiltonian systems of differential equations with two degrees of freedom. The list of all integrable systems of low complexity" (in print) (see [4]). A. T. Fomenko and H. Zieschang, "Criterion of topological equivalence of integrable Hamiltonian systems with two degrees of freedom", Izv. Akad. Nauk SSSR, 1990 (in print) (see [27]). Then — the papers: A. V. Brailov and A. T. Fomenko [7]; A. T. Fomenko [14]-[21]; A. T. Fomenko and V. V. Sharko [22]; A. T. Fomenko and H. Zieschang [25]-[27]; S. V. Matveev, A. T. Fomenko, and V. V. Sharko [37]; S. V. Matveev and A. T. Fomenko [38], [39]. In the present paper we consider the Hamiltonian systems of differential equations which are integrable in Liouville sense and are the systems "of gen­ eral type", or "of general position" (see details below). We will consider the integrable systems up to topological equivalence, namely two systems are topologically equivalent iff there exists some diffeomorphism which transforms the set of Liouville tori of the first system in the set of Liouville tori of the second system. Let us remind here several basic problems: 1) Let us consider two integrable Hamiltonian systems "of general type" v\ and t>2- How to recognise: are they topologically equivalent or not? 2) How to classify integrable Hamiltonian systems "of general type" up to topological equivalence? 3) Does there exist some topological invariant which can classify the inte­ grable systems? 4) How to find some new topological obstructions to integrability? 5) What is the "complexity* of integrable system? 6) How to describe all integrable Hamiltonian systems (of general type) which have low complexity? It happens that the answer to all these questions is "yes" (in some strong sense; see below). Let us present here some rough skeleton of the whole work. We list here only the main topics of the work. 1) Liouville integrability of Hamiltonian systems. Liouville tori and Liouville foliation of isoenergy surface.

136 2) Bott integrals. Nonresonant integrable systems. The systems of ^general type" = the systems "of general position". 3) Topological equivalence of integrable systems. 4) Formal brief description of the topological classification of integrable systems. 5) Geometrical construction of new topological invariant of integrable systems. 6) The invariant I(H, Q) and the marked invariant I(H, Q)*. 7) Main theorem: the integrable Hamiltonian systems of general type are topologically equivalent iff their marked invariants coincide, i.e. I(Hi,Qi)* = /(#2,Q2)*. 8) Classification of integrable Hamiltonian systems of general type with two degrees of freedom. 9) Classification of three-dimensional isoenergy surfaces of integrable Hamiltonian systems. Canonical representation of constant-energy surfaces of integrable equations. 10) New Morse-type theory for the Bott integrals on the isoenergy surfaces. 11) Classification of all bifurcations of Liouville tori inside isoenergy sur­ faces of integrable systems. 12) The main examples from mathematical physics and mechanics. 13) The nonsingular constant-energy surfaces of integrable Hamiltonian systems possess specific properties which distinguish them among all smooth three-dimensional manifolds. Not all 3-manifolds are realized in the form of a constant-energy surface of an integrable system. 14) Multidimensional case, the theory of integrable systems with arbitrary number of degrees of freedom. 15) New topological obstacles to integrability of Hamiltonian systems in the class of Bott integrals. 16) Application to the theory of three-dimensional closed hyperbolic manifolds. Calculation of the volumes of about 1000 closed compact hyperbolic 3-manifolds. 17) The estimation of the number of periodic solutions of an integrable Hamiltonian system in terms of homology groups of the corresponding isoen­ ergy surface (three-dimensional case). 18) The notion of the complexity of integrable system with two degrees of freedom. The connection with the deep properties of three-dimensional manifolds.

137 19) The algorithm of enumeration of all integrable systems (up to topological equivalence). 20) The list of all integrable systems of low complexity. 21) The code of integrable system can be represented as some molecule which is constructed from the "atoms". In this sense the set of all integrable systems is precisely the set of all "molecules". The list of all integrable systems is similar to the table of all chemical elements. 23) Connection with the theory of Morse-Smale flows. The Hamiltonian mechanics and the theory of "round Morse functions". 24) Computers in Hamiltonian mechanics and symplectic topology. Our computer experiments and results. 25) The values of new topological invariant for the famous physical systems (Kovalevskaya case, Euler case, Lagrange case, Toda lattice and so on). 26) The "physical zone" inside the list of all integrable Hamiltonian sys­ tems. This zone is formed by all the systems which have been discovered in concrete problems of physics and mechanics. It is very interesting to "draw" this zone inside the table of all possible integrable systems (most of them of course have the formal character and never appear in the concrete physics). 2. B a s i c N o t i o n s of t h e T h e o r y 2.1. Symplectic manifolds and Hamiltonian systems D e f i n i t i o n . Let / be a smooth function on M2n and w a symplectic structure on M. A smooth vector field on M which is uniquely defined by the relation a>(v, sgrad / ) = v(f), where v runs through a set of all smooth vector fields on M and « ( / ) is the value of the field v on the function / (that is, a derivative of the function / in the direction of the field v), is called a skew-symmetric gradient sgrad / of the function / (a skew gradient). D e f i n i t i o n . A smooth vector field v o n a symplectic manifold M which has the form v = sgrad F, is called a Hamiltonian field on M if the smooth function F is defined on the entire manifold M. The function F is then called Hamiltonian (or energy function). Each Hamiltonian vector field can be considered as a Hamiltonian system of differential equations on M (and conversely). Such a system can be written (in corresponding symplectic coordinates) in the form: dpi/dt

= dH/dqt,

dqi/dt = -dH/dPi

.

138 T h e function {/, g} = w( sgrad / , sgrad g) is called the Poisson bracket of smooth functions / and g on a symplectic manifold M with a form w. 2.2. The Liouville theorem. Liouville tori We will say t h a t two functions f and g are in involution manifold M if their Poisson bracket is exactly zero.

on a symplectic

T h e o r e m 2 . 1 . Let a set of n smooth functions f\,... ,fn in involution be given on a symplectic manifold M2n. For example let / i , . . . , fn be the independent involutive integrals of some Hamiltonian system v = sgrad H, where H = f\ is a smooth Hamiltonian on M. We will denote by M j the common level surface given by a system of equations

*(*) = 6,...,/«(*) =6.Suppose that on this surface all the functions fi are functionally independent. Then: 1) The level surface M{ is a smooth n-dimensional submanifold invariant with respect to each vector field vt- = sgrad fi, that is, all these fields are tangent to the level surface M j . 2) If the level surface M$ is connected and compact, it is diffeomorphic to an n-dimensional torus T " . In a noncompact case Afj is diffeomorphic to a factor of the Euclidean space Rn by a certain lattice ("cylinder"). S) If the level surface is a torus T", then in a certain open neighbourhood of this surface, one may introduce regular curvilinear coordinates S i , . . . , s„; *Plt'" t'Pn called "action-angle", which have the following properties. Sa. The functions si(x),... ,sn(x) set coordinates in the directions transversal to the torus Tn and are functionally expressed through the integrals fit--- tfn- In these coordinates, the equation of the torus is given as Si = ... = sn=0. Sb. The vector field v has the simplest form on the torus Tn in coordinates
139 slightly deform and are covered by almost periodic integral trajectories. These tori are "typical" in the following sense: the measure of the complement to their union is small when the perturbation is small. If a Hamiltonian system is close to integrable system (and is nondegenerate on isoenergy surfaces), the invariant tori cover the most part of the manifold of constant energy. Thus, all Hamiltonian systems which are close to integrable ones are characterized by the existence of "rich" set of invariant tori on isoenergy surfaces. Small perturbation of an integrable Hamiltonian system induces some small diffeomorphism of its isoenergy surface. As a result we obtain the isoenergy surface of nonintegrable system (in the case of perturbation of general type). Con­ sequently, the information on the topology of isoenergy surfaces of integrable systems gives us the information about the topology of isoenergy surfaces of all nonintegrable systems close to integrable ones. The study of this topology is the aim of our work. 2.3. Isoenergy surface of integrable system, i.e. constant-energy

surface

Let M* be a four-dimensional (compact or noncompact) symplectic man­ ifold, and v = sgrad H be a Hamiltonian system with the Hamiltonian H, having a second supplementary independent smooth integral / , such that {H, / } = 0. We will study the integrability of the system on a single iso­ lated constant-energy surface Q3 = {H = const.}. We should emphasize that, in our opinion, special attention should be given to integrability on a separate fixed constant-energy surface. The reason is that in mechanics and physics one often has to deal with a Hamiltonian system inte­ grable only on one isoenergy surface and nonintegrable on the others. Analytic systems are usually integrable either simultaneously on all Hamiltonian regular level surfaces or on none of them. For this reason we think it is instructive to consider smooth systems and smooth integrals which admit a simultaneous presence of both "integrable" and "nonintegrable" constant-energy surfaces inside M. In this connection, the following general problem arises. Let a Hamiltonian system with a Hamiltonian H be given. One should establish whether among the constant-energy surfaces of this system there exists at least one on which the system is integrable. It is natural to conjecture that many Hamiltonian systems (nonintegrable globally) do have such a surface. Clearly, a smooth (or analytic) integral / , which integrates a system v = sgrad / / only on one Hamiltonian level surface, satisfies an equation weaker than the ordinary equation {H, / } = 0. Namely, /

140 is an integral on the surface itself, whereas outside this surface, it does not yet have to commute with H. In the simplest form, this condition can be written as follows: {H, / } = A(JJ), where the function \{H) is such that A(0) = 0 and A'(0) = 0. We assume that the isolated Hamiltonian level surface of interest is given by the equation H = 0. Thus, the general equation {H, / } = A ( # ) deserves a most thorough exam­ ination. On a Hamiltonian level surface, consider two vector fields sgrad H and sgrad / . For these vector fields to commute on a level surface, it suffices that the gradient of the function {H, / } vanishes on the surface H = 0. Indeed, we know that [sgrad H, sgrad / ] = sgrad {H, / } . Thus, the function X(H) must be quadratic in i f in the neighbourhood of the value H = 0. In particular, it would be possible first to investigate the general properties of the equation {H, / } = sH2, where e is a nonzero constant. From the Liouville theorem it follows that all nonsingular two-dimensional compact level surfaces of a second integral / (on a constant-energy manifold) are unions of tori. It turns out that the structure of singular level surfaces of integral f can also be completely described. Since H is an integral of the system v, it follows that the field v may be restricted to an invariant three-dimensional isoenergy surface Q, that is, Q — {x g M : H(x) = const.}. Being a symplectic manifold, M is orientable, and therefore the isoenergy manifold Q is also orientable. Consider anoncritical level surface Q, that is, on which grad H ^ 0. Then sgrad H ^ 0 in all points i 6 ( J , i.e. « / 0 on (J. Let us consider the second independent integral / (see above). We restrict it to the surface Q and obtain a smooth function (we denote this function again by / ) . As has already been mentioned, we will consider integrability of the system only on one separate isoenergy surface. 2.4. Bott

integrals

D e f i n i t i o n (see [14], [15]). We will call a smooth integral / a Bott integral on an isoenergy surface Q if the critical points of the function / form on Q nondegenerate critical smooth submanifolds. The general properties of such functions were studied in the well-known papers of R. Bott (see for example [5]). Based on this, it is appropriate to call such functions Bott functions. Recall that the critical submanifolds for a function / is called nondegen­ erate if the Hessian d?f of the function / is nondegenerate on normal planes to the submanifold. In other words, the integral / can be considered as a Morse function in all normal directions to the critical submanifold. In our

141 case, nondegenerate critical submanifolds for the integral f on Q may be onedimensional and two-dimensional. We do not have a zero-dimensional critical submanifold, because o / 0 on Q and / is constant on all integral trajectories of the field v. The field v does not have the isolated points as the integral trajectories. An important "experimental fact": the investigation of concrete integrable systems (see, for instance, the works of M. P. Charlamov [9]-[ll], T. I. Pogosyan [50], A. A. Oshemkov [46]-[49], A. T. Fomenko, A. V. Bolsinov, L. S. Poryakova, M. R. Vovchenko [4], V. V. Kozlov [30], etc.) has shown that the overwhelming majority of the discovered integrals in concrete mechanical and physical systems are Bott integrals on almost all regular isoenergy surfaces Q3 and M*. Consequently, the class of Bott integrals introduced in our paper is suffi­ ciently large. C o n j e c t u r e . Bott integrals are the integrals "of general position" (of general type) in integrable Hamiltonian mechanics. This conjecture is not yet proved because we need here a strong notion of "general position". It would be very interesting to describe the set of integrable systems having the Bott integrals on almost all isoenergy surfaces. D e f i n i t i o n . The subdivision of the phase space M* and the isoenergy surface Q3 into the union of the Liouville tori and the connected components of the critical surfaces of the Bott integral / will be called Liouville foliation of M and Q. 2.5. Separatrix diagrams of the critical submanifolds of Bott integrals Consider the critical nondegenerate submanifold L of the integral / on isoenergy surface Q. Let us fix some (arbitrary) Riemannian metric on Q and consider the vector field w = grad / . The reader will recall that a separatrix diagram of the critical submanifold L is a union of all integral trajectories of the field grad / incoming in L and outcoming from L. In view of this, we shall speak of an incoming separatrix diagram and an outcoming separatrix diagram. In a small open neighbourhood of a critical submanifold L, both inand out-diagrams are smooth submanifolds. They may be either orientable or nonorientable. D e f i n i t i o n (see [14], [15]). We will call the Bott integral / orientable on isoenergy surface Q if all separatrix diagrams of its critical submanifolds are

142

orientable. If at least one of its separatrix diagrams is nonorientable, we say that the integral / is nonorientable. 2.6. Classification of critical submanifolds of Bott integrals Lemma 2.1. The critical connected submanifolds of a smooth Bott in­ tegral on a compact nonsingular isoenergy surface Q3 are either circles, twodimensional tori, or Klein bottles. Proof. If / is a Bott integral of the system v and L is its critical submanifold, then L admits the nondegenerate tangent vector field (the restriction of the field v on L). Thus, its Euler characteristic is equal to zero. It turns out that in some strong sense the critical Klein bottles are ''noninteresting submanifolds". Statement 2.1 (see [15]). Let Q3 be a nonsingular compact isoenergy surface in M* and f a Bott integral on Q having the critical Klein bottles among its critical submanifolds. Let U(Q) be a sufficiently small tubular neighbourhood of the surface Q in M. Then there exists a two-sheeted (two-fold) covering w:

(U(Q),B,f)^(U(Q),H,f)

(with a fiber Z%), where U(Q) is a symplectic manifold with a new Hamiltonian system v = sgrad H with the new Hamiltonian H, which is of the form H = if*[H). This new system v is integrable on the manifold Q = ir~1(Q) by means of a Bott integral f = *"*(/)• All critical submanifolds of the integral f are orientable (i.e. circles and tori). In this case, all the critical Klein bottles in Q "unfold" into critical tori t2 in Q. The manifold H(Q) is a tubular neighbourhood of the manifold Q. Consequently, if we consider the integrable systems up to two-fold coverings of their isoenergy surfaces, we can assume that all critical submanifolds of the Bott integral / are orientable (no Klein bottles!). It should be mentioned here that our experience of investigation of concrete mechanical systems shows that practically in all cases we do not see critical Klein bottles in real physical situations. Thus, every integrable system with Bott integral can be covered by some new integrable system, where a new Bott integral does not have critical Klein bottles. From this it follows that if / is a Bott integral (on Q) with critical Klein bottles, then *i(Q) ^ 0 and the group nx (Q) contains a subgroup of index two. Here n\ (Q) denotes the fundamental group of the manifold Q. If, for instance,

143

isoenergy surface Q is homeomorphic to the 3-sphere S3 (an important case in mechanics), then any Bott integral f on the sphere S3 does not have critical Klein bottles. 2.7. The elementary bifurcations of Liouville tori The basic idea of the papers [14]-[17J is as follows. We investigate the motion of Liouville tori inside the manifold Q, induced by the change of the value of the integral / . The tori can transform, disappear and appear. This process is the particular case of the general effect of dependence of the solutions of differential equations on the initial data (see details below). It should be mentioned here that today there exist powerful computer programs which allow us to see on the computer screen the transformations of Liouville tori in phase space (see, for example, the excellent paper of P. J. Channell and C. Scovel [8]). The general theory of transformations of Liouville tori and their classification was constructed by the author in [14]-[19]. Let us describe the transformations of Liouville tori when they pass through the critical value c of the integral / . Let us denote R the connected component of the set / - 1 ( c ) containing the critical points of / and suppose that the critical submanifold L C R is connected. We will call such an integral / simple on R. It turns out that only the following cases are possible. Theorem 2.2. (see [15]). a) L is a maximal circle (the local maximum of the integral f). Then there exists the one-parametric family {T, C / - 1 ( c — e),e > 0} of the Liouville tori T, which decrease when e decreases and which coincide with the circle when e = 0. In the case of minimal circle L (the local minimum of the integral f) the Liouville tori transform in a similar way: the critical circle transforms (blows) into one-parameter family of Liouville tori. b) L is the maximal or minimal two-dimensional torus (the local maximum or minimum of f). Then there exist two one-parameter families of Liouville tori, which draw together (when the value of the integral tends to the critical one) and finally stick together in the torus L. (It is important that from the topological point of view we do not have any transformation of a single Liouville torus. Consequently, in further considerations we include the minimal and maximal tori in the set of "usual" Liouville tori.) c) L is the maximal or minimal Klein bottle (the local maximum or mini­ mum off). Then L is the limit of one-parametric family of Liouville tori. (As we mentioned above, in real physical systems critical Klein bottles practically

144

never appear. Thus usually we will suppose that our integral does not have Klein bottles. Of course, the general theory in [15] was constructed in the case of an arbitrary Bott integral.) d) The connected component R of the set / _ 1 ( c ) contains precisely one hyperbolic critical circle L. In that case R is the compact piece-wise-smooth two-dimensional polyhedron which has singularities of the multiplicity four: the singular curve can be locally represented as the transversal intersection of two planes. Each such singular curve is closed and coincides with one of the hyperbolic (saddle) critical circles of the integral f. Let X be the "cross" — the bouquet of 4 segments with common end. The neighbourhood of this circle S in R is obtained from the direct product X x I of X on the segment I, when we identify two crosses on the boundary using either identical mapping or the symmetry with respect to the vertex of the cross. In the first case we will speak about orientable saddle, in the second case — about nonorientable saddle. In the case d) the separatrix diagram of the critical circle can be orientable and nonorientable. T h e o r e m 2.2*(see [15]). In the case of an arbitrary Bott integral every bifurcation of Liouville torus is the composition of the elementary bifurcations of the types a ) - d ) . 2.8. Nonresonance integrable Hamiltonian systems Let us consider an integrable system of arbitrary dimension and the Liouville foliation of its isoenergy surface, i.e. the set of Liouville tori Tn (the manifold M2n has dimension 2n). D e f i n i t i o n . Let us call the Hamiltonian / / and the system v = sgrad H nonresonance on a given isoenergy surface (H = const.) if on this surface everywhere dense are Liouville tori on which integral trajectories of a system v form a dense irrational winding. In the nonresonance case the closure of the integral trajectory of a gen­ eral position is, consequently, the Liouville torus containing this trajectory (or initial point of the trajectory). The experience of the study of concretely known physical systems on shows that on four-dimensional manifolds Hamiltonians are mainly nonres­ onance ones on almost all surfaces Q3 (Oshemkov, Charlamov, Fomenko, Bolsinov, Kozlov, Tatarinov). In the multidimensional case, i.e. on M 2 n where n > 2, interesting examples are known where a Hamiltonian H is "essentially"

145 resonance on almost all surfaces Q. This happens, in particular, when a system is integrable in a noncommutative sense, i.e. it has the complete set of inte­ grals which form a noncommutative Lie algebra. For an analysis of the main cases of noncommutative integrability (starting from the works of E. Cartan) see the books [18], [19] and also the reviews by Trofimov and Fomenko in [56], [57]. In the case of noncommutative integrability the system is "degenerate" (resonance) in the sense t h a t its trajectories are everywhere dense only on "fewdimensional tori" (dim < n ) . They are organized into "large* Liouville tori (in a compact case). Consequently, an initial Hamiltonian system is irrational only on small tori, but not such on "large tori". In the four-dimensional case, however, the nonresonance nature of the system is a typical situation, because here we do not have real noncommutative integrability: every semi-simple Lie algebra of dimension two is commutative. In the non-resonance case the closure of the integral trajectory, passing from the point x, is the Liouville tori Tn which we denote T " . Thus, we have the correspondence x —► T". Changing x, we change the torus T£. These tori can be transformed (by some bifurcations) for some special values of x. Consequently, the deformations and transformations of the Liouville torus T " as a function of x, show the dependence of the solutions of our Hamilto­ nian system of the initial data x. Let us consider the four-dimensional case. In non-resonance system the Liouville tori and all our theory do not depend on the choice of concrete Bott integral / , that is, only the fact of its existence is important. See the proof and discussion in papers [16], [19]. Really we must speak about integrable Hamiltonian H, because the function H "knows everything" about its integrals / . The choice of the second integral / is nonunique. On the other hand we cannot change the form of the Hamiltonian H, because this energy-function is given by the nature of the physical or me­ chanical problem. The function H describes in some sense the energy of the mechanical system and consequently, the form of this function is given to us by God. Nevertheless, it is convenient to fix some concrete integral / . In this case the formulations of some theorems will be simpler. We will assume in our paper that all Hamiltonians considered (Hamiltonian systems) have Bott integrals and are nonresonance. 2.9. Topological equivalence

of integrable Hamiltonian

systems

Let us consider two integrable Hamiltonian systems: vi on some isoenergy surface Ql"'1 (i.e. Q i = {Hi - const.}) in phase space M\n and u 2 on some other isoenergy surface Q\n_1 (i.e. Q2 = {H2 = const.}) in phase space

146

Affn. Let us fix the orientation on both manifolds Qi and Q2. The following definition was introduced in [16]. Definition. We will call integrable nonresonance Hamiltonian systems vi on Qi and t>2 on Q2 topologically equivalent (or geometrically equivalent) if there exists a diffeomorphism r : Qi —► Q2, which transforms Liouville tori of the system vi into the Liouville tori of the system t>2 and preserves the orientation of the isoenergy manifolds. C o m m e n t . It is convenient to formulate a stronger notion of the topologi­ cally equivalent systems and add the condition of preserving the orientation of all critical submanifolds of the Bott integrals. These submanifolds are closely connected with singular fibers of Liouville foliation on Q. Integrable nonresonance system defines the foliation of the isoenergy mani­ fold, where nonsingular fibers are Liouville tori and singular fibers are the orbits of Poisson action of the commutative group R" on the manifold Q. Because we fixed the orientation of the group Rn, we obtain the induced orientation on all its orbits (regular and singular). The Abelian group Rn is defined by the set of commutative integrals / 1 , . . . , fn. Now let us consider the four-dimensional case Af4. We know here the structure of all critical submanifolds of the Bott integral / on Q3 (see Lemma 2.1). They are: tori, Klein bottles and circles. We will assume that our systems do not have critical Klein bottles (see above). Then we include critical tori in the set of Liouville tori. Each critical circle of the Bott integral is some periodic solution (integral trajectory) of the Hamiltonian system. Thus, this circle has the uniquely defined orientation (defined by the system v). Definition. Two integrable nonresonance systems with Bott integrals will be called topologically equivalent on orientable isoenergy manifolds Qf and Q\ if there exists a diffeomorphism r : Q\ —► Q3, which transforms the Liouville tori of the system t>( into the Liouville tori of the system v2 and preserves the orientation of the isoenergy surfaces and the orientation of critical circles of the integrals. As we know, every nonresonance Liouville torus can be characterized as the closure of any integral trajectory lying in the torus. Consequently, the set of all nonresonance tori does not depend on the choice of the second Bott integral / . Every resonance torus T and every connected component L of arbitrary critical level surface of the integral / can be approximated by the nonresonance tori.

147

Consequently (see [16], [19]), the surfaces T and L do not depend on the choice of the integral / . Thus, Liouville foliation on Q3 (and, .in particular, the critical isolated circles of the integral fj do not depend on the choice of the integral f. The following definition is evidently equivalent to the previous one, but emphasises the fact that the equivalence or nonequivalence of two integrable systems does not depend on the choice of concrete Bott integrals. Definition. We call two integrable nonresonance Hamiltonian systems vi and »2 on the isoenergy surfaces Q\ and Qa topologically equivalent if there exists a diffeomorphism r : Qx —* Q2, which preserves the orientation of the isoenergy manifolds, transforms the Liouville foliation of the system «i into the Liouville foliation of the system v2 and preserves the orientation of all isolated critical circles of the integrals. Important remark. The level surface of the integral f\ on Qi can be non-connected, i.e. can be the union of several Liouville tori. These tori can be mapped in different level surfaces of the integral / 2 . Consequently, the diffeomorphism r does not preserve, generally speaking, the levels of the integral but must map the Liouville tori into some Liouville tori. S. Formulation of the Problems We will consider the Hamiltonian system on a symplectic four-dimensional manifold M* (compact or noncompact) v = sgrad H with a smooth Hamilto­ nian H. Let Q 3 be the compact connected isoenergy surface in M. Suppose that v is nonresonance and integrable on Q and / is the second independent Bott integral. Let us formulate the main problems of the present paper. 1. Enumeration problem. Does there exist an algorithm which enumerates all integrable Hamiltonian systems up to topological equivalence? 2. Recognition problem. Does there exist an algorithm which solves the problem: are two integrable Hamiltonian systems topologically equivalent or not? 3. Problem of algorithmical classification of all integrable Hamiltonian sys­ tems of general position up to topological equivalence. Does there exist such an effective algorithm which can be realized on the computer? To be meaningful in the formulation of all these problems, we must com­ ment — how we can define the Liouville foliation in a finite way. We suggest

148 the following procedure. Let us call the piece-wise-linear three-dimensional manifold Q P L with a fixed subdivision into the union of sub-polyhedrons the piece-wise-linear Liouville foliation, if there exists a piece-wise-smooth homeomorphism of the manifold Q-pi, on the "smooth original", t h a t is, on the manifold Q with a given integrable Hamiltonian system v. This homeo­ morphism must transform the elements of the subdivision of the manifold Q P L into the fibers of the Liouville foliation of the system v. It is evident t h a t we can define the piece-wise-linear Liouville foliations in a finite way (algorithmically). We prove t h a t all these problems have positive answers. Theorem S.l. 1) There exists an algorithm of enumeration of all classes of topologically equivalent integrable Hamiltonian systems (of general position and with two degrees of freedom). 2) There exists an algorithm of recognition of topologically equivalent (and topologically non-equivalent) integrable Hamiltonian systems. S) There exists an algorithmical classification of all integrable Hamiltonian systems up to topological equivalence. 4 . S k e l e t o n s a n d C o m p l e x i t y of I n t e g r a b l e H a m i l t o n i a n S y s t e m s 4.1.

Skeletons The proof of Theorem 3.1 and concrete classification of all integrable Hamil­ tonian systems of low complexity is based on the important notions of the skele­ ton and complexity of integrable Hamiltonian systems and on the classification theorem of isoenergy surfaces of integrable systems. The notion of the skeleton of the system was discovered by A. T. Fomenko in [15], [17], [19] and it coincides with the notion of topological invariant I(H, Q) of integrable system, introduced and investigated in these papers. The notion of marked invariant I(H, Q)* was introduced by A. T. Fomenko and H. Zieschang in [27]. The notion of the complexity of integrable system is sim­ ilar to the important notion of the complexity of a compact three-dimensional manifold, introduced by S. V. Matveev [34], [36]. In both cases the complexity is measured by the number of singularities and in both cases the important role graphs of degree 4 play. The classification theorem for isoenergy surfaces was proved by the author

in [15].

149 Let us consider the Liouville foliation of the three-dimensional manifold Q. Let us cut the manifold Q along some Liouville torus T and then glue the two copies of torus T (which appear after cutting) using some diffeomorphism (preserving the orientation). We obtain some new three-dimensional manifold Qi with some new Liouville foliation. We will say that foliation on Qi is obtained by the twisting from the foliation on Q. D e f i n i t i o n . Let us call two integrable Hamiltonian systems roughly equiva­ lent if their Liouville foliations are obtained one from another by some twisting along Liouville tori. D e f i n i t i o n . We will call the class of all systems roughly equivalent to a given system the (abstract) skeleton of a system. Indeed, the notion introduced has a very simple and visual geometrical sense. Let us introduce the notion of geometric skeleton. D e f i n i t i o n . Geometric skeleton is the pair (P, K), where P is a compact closed orientable two-dimensional surface and K is the graph in P The graph i f is as follows: 1) Each vertex of the graph K is either isolated or has the degree 2 or 4. 2) The surface P — K (the complement to K in P) is homeomorphic to the union of several open rings S1 x (0,1). Two skeletons are considered identical if they are homeomorphic as two topological spaces (with preserved orientation). R e m a r k . Graph K can be non-connected and can contain loops and mul­ tiple edges. As we will see later, the isolated vertices of K represent the minimal and maximal critical circles of Bott integral / on Q. Other connected components of K correspond to the connected components of critical level surfaces of the integral / , which contain saddle critical circles. Thus, the saddle circles cor­ respond to the vertices of the graph. The orientable saddles correspond to the vertices of degree 4, nonorientable saddles — to the vertices of degree 2 (we mark these vertices by stars). The connected components of the manifold P — K (namely, the rings) correspond to the one-parameter families of Liou­ ville tori (including the maximal and minimal critical tori). These tori move from one critical level in the direction of another critical level. The following theorem is one of the central points of the whole theory.

150

Theorem 4.1. There exists the natural one-to-one correspondence between the set of all geometric skeletons and the set of all skeletons ofintegrable Hamiltonian systems. It follows from this theorem that the problems of algorithmical enumera­ tion, algorithmical recognition and algorithmical classification of the skeletons are solved in the positive sense. 4.2. Complexity Let v be an integrable Hamiltonian system on closed orientable threedimensional manifold Q with some Bott integral / . Denote by m the total number of all minimal, maximal and saddle critical circles of the integral. Let us remove from the manifold all isolated critical circles and connected compo­ nents of all critical level surfaces of / containing the saddle circles. In other words, we remove all singular fibers of corresponding Liouville foliation (i.e. the fibers different from Liouville tori). As a result, the manifold Q goes to pieces, transforms into the union of a finite number of open manifolds homeomorphic to the direct product S 1 x S1 x (0,1). Let us denote the total number of such manifolds by n. Definition. The pair of nonnegative integer numbers (m, n) will be called complexity of a given integrable Hamiltonian system v. It follows easily from the definition of the complexity that the complexities of two roughly equivalent integrable Hamiltonian systems coincide. Thus, the complexity is an invariant of the skeleton of the system. The complexity of the geometric skeleton (P, K) is the pair (m, n), where m is the number of the vertices of the graph K and n is the number of the connected components of the manifold P — K. I m p o r t a n t r e m a r k . Thus, the notion of complexity of an integrable system is very natural and reflects the deep properties of the system of Hamil­ tonian differential equations. In some sense this notion naturally appears in a more or less "unique way" from the point of view of symplectic topology of integrable systems.

151

T h e o r e m 4.2. a) The number of different skeletons of a given fixed complexity (m, n) is finite. b) The set of all topologically non-equivalent integrable Hamiltonian sys­ tems with the same skeleton of the complexity (m, n) is parametrized by inde­ pendent parameters »"i, Si, nk,

1 <■ i < n,

1 < k < s ,

where »\ are rational numbers, 0 < r; < 1 or »\ = oo; then e< = ±l;rifc are integer numbers and s < m. The statement a) of this theorem evidently follows from Theorem 4.1 and from the interpretation of the complexity of geometric skeleton (see above). Let us comment on the item b). We introduce the notion of framed geometric skeleton as geometric skeleton with some numerical marks. For example, the rational numbers and the numbers ej = ± 1 (see Theorem 4.2) are located with the open rings which form the manifold P — K. The rule of the construction of the integer marks will be discussed below. Thus, we can say t h a t integrable Hamiltonian system can be uniquely de­ fined (up to topological equivalence) by the framed skeleton. Conversely, each framed geometric skeleton is the skeleton of some integrable Hamiltonian sys­ tem. T h e o r e m 4 . 3 . There exists a natural one-to-one correspondence between the set of all framed geometric skeletons and the set of all integrable Hamilto­ nian systems (considered up to topological equivalence). This theorem gives the answer to the all three questions (see above), be­ cause Theorem 3.1 is the direct corollary of Theorem 4.3. 5. T h e List o f A l l I n t e g r a b l e H a m i l t o n i a n S y s t e m s of Low Complexity 5.1.

Letter-atoms We need some procedure in the coding of the integrable systems. Let us separate the pieces of the geometric skeleton (P, K) into two types: a) the regular neighbourhoods of the connected components of the graph K, b) the remaining rings. Let us call the regular neighbourhoods of the connected components of the graph K letter-atoms. The complexity of the letter-atom is

152 equal to the number of the vertices in its spine. The spine is the connected component of the graph K, which is the deformation retract of the letter-atom. The list of all letter-atoms of complexity no more than 3 can be seen in Table 1 in [4]. Table 2 in [4] shows all orientable spines of the letter-atoms of complexity no more than 5. Here we marked the number of the letter-atoms corresponding to each spine. Orientable spine is the spine without vertices of the multiplicity 2 (i.e. without nonorientable saddles). Nonorientable spines can be easily obtained from orientable ones by introducing one or several sior-vertices on the edges of the orientable spine. Tables 1 and 2 were calculated on computer. 5.2.

Word-molecxilas The rings (which form the manifold P—K) can be interpreted as segmentsedges-connections in some new object which can be called word-molecula. Word-molecula is formed by several letter-atoms connected by edges (which correspond to the rings). The word-molecula defines the geometric skeleton of an integrable Hamiltonian system. I m p o r t a n t r e m a r k . Indeed, the segments-edges-connections, which come out from the letter-atom, are not equivalent in the general case. We need some special enumeration of all edges coming out from the letter-atom to reconstruct (in a unique way) the word-molecula. Here we suppose that all letter-atoms are enumerated in Tables 1 and 2 by some fixed way. It is convenient to enumerate all the edges-connections, coming out from the one letter-atom, in cyclic order. We will start from the edge-connection, which is vertical (upwards), and we will rotate in positive direction. Table 3 in [4] shows the list of all word-moleculas corresponding to the integrable Hamiltonian systems of the complexity (rrt, n), where m < 2. Let us denote by A(m, n) the total number of all skeletons of integrable Hamiltonian systems of a given complexity (m,n). This number is finite, as follows from Theorem 4.2. The list of the values of the function A(m, n), where m < 4, is given in Table 4 in [4]. 5.3. Atomic weight and valency of the word-molecula We will call the total number of the vertices in the spine of a letter-atom atomic weight (of the given letter-atom). Then we will call the total number of edges-connections in the word-molecula valency (of the given word-molecula).

153

The sum of the valency of word-molecula and the atomic weights of all its letter-atoms is always even. Consequently, if m is odd, then all integrable Hamiltonian systems of the complexity (m, n) must have the critical circles of the type of nonorientable saddle. We have calculated the number A(m), where A(m) is an upper bound of all numbers n such that A(m, n) ^ 0 and m is fixed: A(m) = {supn : A(m, n) ^ 0}. It turns out that A(m) = [3m/2]. 6. Real Physical Systems and Their Place in the Table of Complexity of All Integrable Hamiltonian Systems It is extremely interesting and important to calculate the cells (m, n) in our table which contain the real mechanical and physical integrable systems. A. T. Fomenko, A. A. Oshemkov, L. S. Polykova, and A. V. Bolsinov cal­ culated (relying, in particular, on the important papers of M. P. Charlamov and his pupils) the word-moleculas for the following important physical sys­ tems: integrable cases of the equations of the rigid body motion (Kovalevskaya case, Goryachev-Chaplygin case, Sretensky case, Klebsch case, Euler case, Lagrange case); some integrable cases in Toda lattice, integrable cases for the motion equations of the four-dimensional rigid body (the system on the Lie group SO(4)) and so on. In many cases the numerical marks were calculated, i.e. framed skeletons of integrable systems. For some results see Table 5 in [4]. In the previous discussion we assumed that the isoenergy surface Q was fixed. But indeed we need the consideration of all isoenergy surfaces of a given integrable system which correspond to different values of the energy function (Hamiltonian) H. This problem arises naturally when we study a real physical system. Its energy function H is given to us "by God" (we cannot deform this function). Consequently, it is natural to start from the minimum of the Hamiltonian H and then move in the direction of its maximum. The change of the Hamiltonian's value changes the isoenergy surface Q. It is natural to collect all types of surfaces Q which appear in this process. This set of integrable surfaces represents the total topology of a given integrable system. In our language we must collect all corresponding word-moleculas. The word-molecula evidently does not change when we change the value of the Hamiltonian near a regular value. Some transformation of the word-molecula can occur only in the neighbourhood of a singular value of H. The wordmolecula can change only when we cross a singular value.

154 Thus we obtain "one-parameter" family of word-moleculas. Let us repre­ sent these moleculas by the points in the corresponding cells of the plane (each such cell corresponds to the concrete complexity ( m , n ) ) . Then let us connect each two consecutive points by a segment. As a result, we obtain some curve on the plane (m, n). The curve corresponds to the motion of the value of Hamiltonian from the minimum to the maximum. Thus, each real physical system is represented by some curve (sequence of segments) on the plane (m,n). Table 6 contains the curves corresponding to some real physical integrable Hamiltonian systems mentioned above. The cells, corresponding to these systems, are shaded. It is clear that further analysis of other real mechanical systems will fill some new cells on the plane. We obtain, as a result, some remarkable "physical zone". Important problem: describe more or less precise boundary (the form) of this zone. Where are the "real physical integrable systems" lo­ cated? The next problem: how to fill "the empty celb" close to the physical zone or enveloped by this zone? (Fig. 13). There exists the problem of predic­ tion of the properties of integrable systems, close to the "physical zone", the problem of discovering some mechanical systems with prescribed complexity, etc. It seems to us that this table is similar to the Mendeleev table of chemical elements. 7. F o r m a l D e s c r i p t i o n of T o p o l o g i c a l I n v a r i a n t s of I n t e g r a b l e H a m i l t o n i a n S y s t e m We will describe in the present section a slightly different construction of new topological invariant of the integrable system introduced above. S t e p 1. Let us consider the set of all connected graphs Kc with vertices of the multiplicity zero, two, and four. Each such graph is obtained by the following procedure. Let us consider a finite collection of circles and form the graph with the vertices of multiplicity four: we glue these circles in several points, which we call the vertices. Important: only two circles are intersected in each vertex. Then let us put on some edges an arbitrary number of stars, which are the vertices of multiplicity two. The vertices of multiplicity zero are the isolated points. S t e p 2 . Let us construct the Utter-atoms (the alphabet). Consider the graph Kc and its different immersions into standard two-dimensional sphere S2, i.e. x : Kc —* S2. Then we consider the small tubular neighbourhood of such immersion in the sphere. We obtain some two-dimensional surface Pc. This surface is evidently uniquely defined by a given immersion of Kc.

155 The surface Pc has the boundary and contains the graph KC) which is the deformation retract of Pc (the surface Pc can be retracted, deformed on the graph Ke). Now we restrict the class of possible immersions of the graphs Kc in the sphere. We will consider only the immersions with the following property: There exists a Morse function f on the surface Pc such that its critical saddle points are precisely the vertices of the multiplicity four of Kc; then f does not have another critical point and f = const, on the boundary dPc. Figure 16 shows two immersions of the graph Kc into the sphere. The first immersion is admissible, the second one is forbidden. S t e p 3 . We call two admissible immersions *i and %i of the graph Kc into sphere S2 equivalent, if tj can be transformed into t*2 by the following operations: a) smooth isotopy in the sphere, b) operation of removing (and creating) loops from (on) the edges. The equivalence relation is evidently symmetric. The operation b) allows us to remove and include the loops on an arbitrary edge of the immersed graph. S t e p 4 . Let the letter-atom be the class of equivalent immersions of the graph Kc into the sphere. We obtain some set of letter-atoms. They form some "alphabet". Our next step: we "write the words" using this alphabet. We need "grammar rules" to write the words. S t e p 5. Let us consider an arbitrary letter-atom and the corresponding surface Pc. The boundary of Pc is the union of circles. Let us call these circles ends of the letter-atom and let us represent this object by some letter with several segment-edges coming out of the letter. See examples in Fig. 18. Here the letter Ci has two ends and we draw the letter C\ with two segment-edges. The letter Ci has four ends and we draw the letter Ci with four segment-edges. S t e p 6. Let us glue the word-molecula from some finite set of the letteratoms by the following grammar rule: we connect the pairs of free ends of the letters in such a way that the final graph does not have free ends. Two ends belonging to the same letter can also be connected. Let us denote this word-molecula by W. Now we can construct some closed 2-surface P using the iword-molecula W. Because each letter-atom is defined by a 2-surface Pc (with boundary), we can construct the new surface P by gluing the boundary circles of the surfaces Pc in correspondence with the structure of the word W. We will write P = J2 Pc an<^ K — 5Z^c- Here we identify the boundary circles

156

of Pc by some diffeomorphisms. It is evident that we obtain closed compact two-dimensional surface P with some graph K. S t e p 7. Let us construct the marked word-molecula W*. We put on each edge of the word-molecula W a rational number fj, where 0 < r,- < 1 or r,- = oo (here t is the index of the edge in W), and then put on the same edge the number e< = ± 1 . Let us consider the set of all edges of W which have U = 0. We call two letter-atoms (inside W) relative if they can be connected by a sequence of edges with r< = 0 (on each edge). It is clear that different sets of relatives do not intersect. Let us call a connected set of relatives a family. We obtain some set of families inside the word-molecula W. We put a natural number njt on each family in W. Here 1 < t < n, where n is the total number of edges in the word-molecula W; then 1 < k < s, where s is the total number of different families in the word-molecula W. It is evident that s < m, where m is the total number of letter-atoms in W. Finally, the marked word-molecula W* is the following object: W* = (W.{ri},{e,},{n fc }). S t e p 8. The word-molecula W is indeed the pair (P, K), where K is the graph embedded in the closed 2-surface P This embedding is fixed (different embeddings can produce different word-moleculas). Consequently we can put all the numerical parameters r^, e,-, n^ on the surface P. Thus, we can represent the word-molecula in the form: W* =

(P,K,{ri},{ei},M).

We need to define the equivalent marked word-moleculas. Definition. We will call two word-moleculas W and W* equivalent if there exists a diffeomorphism A : P —► P' such that X : K —► K' (i.e. A transforms the graph K in the graph K') and the numerical parameters on the corresponding objects coincide, that is, f\ = r(,ej = ej, n* = n'k. The word-molecula W is the object I(H, Q) discovered by the author in [16], and the marked word-molecula W* coincides with the object I(H, Q)* discovered by A. T. Fomenko and H. Zieschang in [27]. These objects are topological invariants of integrable Hamiltonian systems. Now Theorem 4.3 can be formulated as follows.

157 T h e o r e m 4.3.* There exists a natural one-to-one correspondence between the set of all different marked word-moleculas W* and the set of all integrable Hamiltonian systems (considered up to topological equivalence) with two de­ grees of freedom and in general position. 8. T o p o l o g i c a l S t r u c t u r e of I s o e n e r g y S u r f a c e s of a n I n t e g r a b l e Hamiltonian System 8.1. Class (M) of three-dimensional manifolds. Class (H) of isoenergy surfaces of integrable systems Let us consider the class (Af) of all compact connected closed orientable three-dimensional manifolds Af3. The natural question arises: can arbitrary 3-manifold (from (Af)) be represented as an isoenergy surface of some Hamil­ tonian system (non-integrable in general)? T h e o r e m 8 . 1 . (A. T. Fomenko, S. V. Matveev). Every compact closed orientable S-manifold can be represented as an isoenergy surface of some Hamil­ tonian system v = sgrad H for some smooth Hamiltonian H on some smooth symplectic manifold Af4. The system v is, generally speaking, non-integrable. 4 The manifold Af can be non-compact. This theorem immediately follows from the following result of S. V. Matveev and A. T. Fomenko. S t a t e m e n t 8 . 1 . Let M3 be an arbitrary smooth compact closed orientable S-manifold and I be the interval. Then the direct product M3 xl is a symplectic manifold (i.e., it can be endowed with some symplectic structure). This theorem shows t h a t "non-integrable isoenergy surfaces" cover the total class of all 3-manifolds. The manifold Af4 in Theorem 8.1 is constructed in the simplest way: the direct product Af3 x / , the function H is induced on Af4 by a linear function on the interval / . Let us introduce the new class (H) of all connected closed compact 3manifolds which are the isoenergy surfaces of integrable Hamiltonian systems on Af4 with Bolt integral / . Each such manifold is orientable. We have the inclusion (H) C (Af). Important question: does the class (H) coincide with the class (Af)?

158 T h e o r e m 8.2 (A. T. Fomenko, [16]). The class (H) does not coincide with the class (M). In other words, not every compact closed orientable 3-manifold may play the role of a constant-energy surface of a Hamiltonian system integrated by means of a smooth Bott integral. We obtain an important corollary. S t a t e m e n t 8.2. There are some new topological obstacles to the integrability of a Hamiltonian system of a general type in the class of Bott integrals. Indeed, let us consider some smooth Hamiltonian system. Let us suppose t h a t we can get the information about the topological structure of some isoenergy surface Q3. If we prove that the surface Q does not belong to the class (H), then automatically we find t h a t our system is non-integrable on a given isoenergy surface Q (in the class of Bott integrals). We will demonstrate below a concrete application of this criterion. Consequently we need more topological information about the structure of the isoenergy surfaces from the class (H). 8.2. Five elementary blocks in the isoenergy surface theory We shall describe five types of the simplest three-dimensional manifolds t h a t appear as those ^elementary bricks" of which an arbitrary constantenergy surface Q of an integrable system is glued together. Let Dn be the n-dimensional disk. Type 1. The direct product S 1 X D2 will be called a full torus. Its boundary is one torus T2. Type 2. The direct product T2 X D1 will be called a cylinder. Its boundary consists of two tori. Type S. The direct product N2 X S1 will be called an oriented saddle or, more descriptively, "trousers", where N2 is a two-dimensional sphere with three removed disks (or the disk with two holes). This manifold is homotopy equivalent to the figure eight, i.e. to a bouquet of two circles. Its boundary consists of three tori. Type 4- Consider a full torus which is embedded in R3 in the standard way. Let us drill in this full torus a thin full torus which winds twice around the generator of the large full torus. This manifold will be called non-orientable saddle. Its boundary consists of two tori. Let us denote this manifold by A3. From the topological point of view the manifold A3 is not new, however. It is obtained by gluing a full torus and orientable saddle through a torus diffeomorphism. This may be conditionally written as follows: A3 = I + III =

159 ( S 1 x D2) + (JV2 X S1). R e m a r k . The manifold A3 has another interpretation. It is the tubular neighbourhood of the immersed Klein bottle in i t 3 . It is clear that over the circle there exist only two nonequivalent fiber bundles with a fiber N2. These are the direct product N2 X S1 (see Type 3 above) and a fiber bundle A3. In Type 4 the figure eight moves along a circle in such a way that after a revolution the two circles 1 and 2 exchange places (the figure eight reverses). A small neighbourhood of the circle (base) S 1 is homeomorphic in this case to two Mobius strips which intersect transversally along their common axis. We will denote this fiber bundle by 7 V 2 x S 1 . Type 5. We denote by K2 the Klein bottle and by K3 the space of an ori­ ented skew product of K2 by a segment, i.e. K3 = K2xDx. The boundary of K3 is a torus. The manifold K3 can be realized as the tubular neighbourhood of the Klein bottle K2 immersed in R3. This immersion is different from the immersion of the Klein bottle in Type 4. Thus, two different immersions of the Klein bottle give us two different "elementary bricks* of Types 4 and 5. Prom the topological point of view, the manifold K3 is not new, because it is represented as the following gluing: K3 = I + IV = ( 5 1 x D2) + A3 = 2I+III=2{S1xD2) + (N2xS1). R e m a r k . The manifold of Type 2 can also be represented as the gluing: 11=1+ III. In all cases the sign " + " denotes the gluing of the boundary tori using some diffeomorphisms. Thus, out of the five types of manifolds listed above, only two are really topologically independent: Type 1 and Type 3. The three others are de­ composed into combinations of the manifolds of the Types 1 and 3. But the manifolds of the Types 2, 4 and 5 are of great interest for the analysis of tra­ jectories of the system v, becuase they correspond to peculiar and interesting motions of a mechanical system. Let M* be s smooth symplectic manifold (compact or noncompact) and let v = sgrad if be a Hamiltonian system that is Liouville-integrable on a certain nonsingular compact three-dimensional isoenergy surface Q by means of a Bott integral / . Let m be the number of such periodic solutions of the system v on the surface Q on which the integral / attains a strictly local minimum or maximum. Next, let p be the number of two-dimensional critical tori of the integral / ; q the number of critical circles of the integral / (unstable trajectories of the system) with an orientable separatrix diagram; s the number

~rt3U

of critical circles of the integral / (unstable trajectories of the system) with a nonorientable separatrix diagram; r the number of critical Klein bottles. This is the exhaustive list of all possible submanifolds of the integral / . T h e o r e m 8.S (see [16]). The manifold Q is represented in the form of gluing (by some diffeomorphisms of boundary tori) of the following elementary bricks: Q = mI + pII+

qlll

+ sIV + rV = miS1

+ q{N2 x S1) + si^xS1)

+ r[K2xDx)

x D2) + p{T2 x

D1)

.

Here the numbers m, p, q, r, a tell us how many critical submanifolds of each type a given integral / has on a given manifold Q. If we ignore this interpretation of the numbers and require a simpler topological representation of Q, then we have the following theorem. T h e o r e m 8.4 (see [16]). The manifold Q admits the following representa­ tion: Q = al + pill. Conversely, each manifold of the type aI + f)III belongs to (H) {see [7]). The description of other results of the new theory is seen in the publication which was prepared at the Mathematical Sciences Research Institute, Berkeley, USA (in print). References [l] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed. (Benjamin/Cummings, 1978). [2] V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1978). [3] A. V. Bolsinov, "Criterion of the completeness of involutive sets of functions which are constructed by argument shift method," DoU. Akad. Nauk SSSR 301 (1988) 1037-1040. [4] A. V. Bolsinov, A. T. Fomenko, and S. V. Matveev, "Topological classification and arrangement with respect to complexity of integrable Hamiltonian systems of differential equations with two degrees of freedom. The list of all integrable systems of low complexity," Usp. Matem. Nauk, (in print). [5] R. Bott, "Non-degenerate critical manifolds", Ann. Math. 60 (1954) 248-261. [6] A. V. Brailov, "Complete integrability of some geodesic flows, and integrable systems with noncommuting integrals", DoU. Akad. Nauk SSSR 271 (1983) 273-276.

161 [7] A. V. Brailov and A. T. Fomenko, "The topology of integral submanifolds of completely integrable Hamiltonian systems", Matematicheskii sbornik 133 (1987) 375-385. [English translation: Math. USSR Sbornik 62 (1989) 373-383.] [8] P. J. Channell and C. Scovel, "Symplectic integration of Hamiltonian systems", Preprint, Los Alamos, 1988, LA-UR-88-1828, pp. 1-47. [9] M. P. Charlamov, "Topological analysis of integrable cases of V. A. Steklov", Dokl. Akad. Nauk SSSR 273 (1983) 1322-1325. [10] M. P. Charlamov, Topological Analysis of Integrable Cases in Dynamics of Rigid Body (Leningrad Univ. Press, Leningrad, 1988). [11] M. P. Charlamov, "Topological analysis of classical integrable systems in dynam­ ics of rigid body", Dokl. Akad. Nauk SSSR 273 (1983) 1322-1325. [12] H. Flaschka, "The Toda lattice. P , Phys. Rev. 9 (1974) 1924-1925. [13] H. Flaschka, "On the Toda lattice. I T , Prog. Theor. Phys. 51 (1974) 703-716. [14] A. T. Fomenko, "Morse theory of integrable Hamiltonian systems", Dokl. Akad. Nauk SSSR 287 (1986) 1071-1075 [English translation: Sov. Math. Dokl. 33 (1986) 502-506.] [15] A. T. Fomenko, "The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability", Izv. Akad. Nauk SSSR 50 (1986) 1276-1307. [English translation: Math. USSR Izv. 29 (1987) 629-658.] [16] A. T. Fomenko, "Topological invariants of Hamiltonian systems integrable in Liouville sense", Fund. Anal. Appl. 22 N4 (1988) 38-51. [17] A. T. Fomenko, "Symplectic topology of completely integrable Hamiltonian systems", Usp. Matem. Nauk 44 Nl (1989) 145-173. [18] A. T. Fomenko, Symplectic Geometry, Methods and Applications (Moscow Univ. Press, Moscow, 1988). [English translation: Advanced Studies in Contemporary Mathematics, Vol. 5, Gordon and Breach, 1988.] [19] A. T. Fomenko, Integrability and Nonintegrability in Geometry and Mechanics (Kluwer, 1988). [20] A. T. Fomenko, "Qualitative geometrical theory of integrable systems. Classi­ fication of isoenergetic surfaces and bifurcation of Liouville tori at the critical energy values", in Lecture Notes in Math. Vol. 1334 (Springer-Verlag, 1988), pp. 221-245. [21] A. T. Fomenko, "New topological invariants of integrable Hamiltonians", Baku Int. Topological Conf. Abstracts, Part 2, 1987, p. 316. [22] A. T. Fomenko and V. V. Sharko, "Round exact Morse functions, Morse type in­ equalities and integrals of Hamiltonian systems", in Global Analysis and Nonlinear Equations (Voronez Univ. Press, 1988), pp. 92-107. [23] A. T. Fomenko and V. V. Trofimov, Integrable Systems on Lie Algebras and Sym­ metric Spaces, Advanced Studies in Comtemporary Math. Vol. 2 (Gordon and Breach, 1988). [24] A. T. Fomenko and V. V. Trofimov, "Noninvariant gymplectic group structures and hamiltonian flows on symmetric spaces", Celecia Mathematica Sovietica 7 (1988) 355-414.

162 [25] A. T. Fomenko and H. Zieschang, "On the topology of three-dimensional man­ ifolds arising in Hamiltonian mechanics", Dokl. Akad. Nauk SSSR 294 (1987) 283-287. [English translation: Sov. Math. Dokl. 35 (1987) 529-534. (Preprint published in 1985, Preprint Nr. 55/1985, Ruhr-Universitat Bochum, Germany, pp. 1-10.] [26] A. T. Fomenko and H. Zieschang, "On typical topological properties of inte­ grable Hamiltonian systems'', Izv. Akad. Nauk SSSR 52 (1988) 378-407. [English translation: (Preprint published in 1987. Preprint Nr. 92/1987, Ruhr-Universitat Bochum, Germany, pp. 1-47).] [27] A. T. Fomenko and H. Zieschang, "Criterion of topological equivalence of inte­ grable Hamiltonian systems with two degrees of freedom", Izv. Akad. Nauk SSSR, 1990 (in press). (Preprint published in 1988, Preprint: Topological classification of integrable Hamiltonian systems, Institut des Hautes Etudes Scientifiques 35, Dec. 1988, IHES/M/88/82, pp. 1-45.) [28] M. Gel'fand and I. Ya. Dorfman, "Hamiltonian operators and related algebraic structures", Fund. Anal. Appl. 13 N4 (1979) 13-30. [29] B. Kostant, "The solution to a generalized Toda lattice and representation theory", Adv. Math. 34 (1980) 195-338. [30] V. V. Kozlov, "Integrability and nonintegrability in Hamiltonian mechanics", Usp. Mattm. Nauk 38 Nl (1983) 3-67. [31] B. A. Kupershmidt and T. Ratiu, "Canonical maps between semidirect products with applications to elasticity and superfluids", Commun. Math. Phyt. 90 (1983) 235-250. [32] J. Marsden and A. Weinstein, "Reduction of symplectic manifolds with symme­ try", Rtp. Math. Phys. 5 (1974) 121-130. [33] J. Marsden, T. Ratiu, and A. Weinstein, "Semidirect products and reduction in mechanics", Tram. Am.tr. Math. Soc. 281 (1984) 147-178. [34] S. V. Matveev, "Additive complexity and Haken's method in topology of threedimensional manifolds", Ukr. Mattm. J. 41 N9 (1989) 28-36. [35] S. V. Matveev, "Generalized surgery of three-dimensional manifolds and repre­ sentation of homology spheres", Mattm. Zamttki 42 (1982) 268-277. [36] S. V. Matveev, "One procedure of defining S-manifolds", Vestnik MGU 30 N2 (1975) 11-20. [37] S. V. Matveev, A. T. Fomenko, and V. V. Sharko, "Round Morse functions and isoenergy level surfaces of integrable Hamiltonian systems", Mattmatithtskii tbornik 135 (1988) 325-345. [English translation: (Preprint published in 1986. Preprint 86, 76, Kiev, Institut Matematiki Akad. Nauk Ukr. SSR. 1986, pp. 1-32.)] [38] S. V. Matveev and A. T. Fomenko, "Morse type theory for integrable Hamiltonian systems with tame integrals", Mattmatichtskii zamttki 43 (1988) 663-671. [39] S. V. Matveev and A. T. Fomenko, "Constant energy surfaces of Hamiltonian systems, enumeration of three-dimensional manifolds in increasing order of com­ plexity, and computation of volumes of closed hyperbolic manifolds", Usp. Mattm. Nauk 43 Nl (1988) 5-22. [English translation: Russian Math. Surveys, 43,

163 pp. 3-24.] [40] H. McKean, "Integrable systems and algebraic curves", in Global Analysis, Lecture Notes in Math. Vol. 755, (Springer-Verlag, 1976) pp. 83-200. [41] S. Miyoshi, "Foliated round surgery of codimension-one foliated manifolds", Topology 21 (1982) 245-262. [42] J. Morgan, "Non-singular Morse-Smale flows and 3-dimensional manifolds1', Topology 18 (1979) 41-54. [43] J. Moser, "Various aspects of integrable Hamiltonian systems", in Dynamical Systems, Prog. Math. 8 (Birkhauser, 1980). [44] S. P. Novikov, "Hamiltonian formalism and a multivalued analogue of the Morse theory", Usp. Mattm. Nauk ST N5 (1982) 3-49. [45] P. Orlik, E. Vogt and H. Zieschang, "Zur Topologie Gefaserter Dreidimensional Mannigfaltigkeiten", Topology 6 (1967) 49-65. [46] A. A. Oshemkov, "Topology of isoenergy surfaces and bifurcation diagrams of integrable cases of dynamic of the rigid body on so(4)", Usp. Matem. Nauk 42 N6 (1987) 199-200. [47] A. A. Oshemkov, "Bott integrals of some integrable Hamiltonian systems", in Geometry, Differential Equations and Mechanics, (Moscow Univ. Press, 1986), pp. 115-117. [48] A. A. Oshemkov, "The phase topology of some integrable Hamiltonian systems on so(4)", Baku Int. Topological Conf. Abstracts, Part 2, 1987, p. 230. [49] A. A. Oshemkov, "Description of isoenergy surfaces of some integrable Hamil­ tonian systems with two degrees of freedom", in Proc. Seminar Vector Tensor Analysis S3 (Moscow Univ. Press, 1988), pp. 122-132. [50] T. I. Pogosyan and M. P. Charlamov, "Bifurcation set and integral manifolds in the problem of the motion of a rigid body in linear forces field", J. Appl. Math. Mech. 43 (1979) 419-428. [51] T. Ratiu, "Euler-Poisson equations on Lie algebras and the n-dimensional heavy rigid body", Amer. J. Math. 103 (1982). [52] T. Ratiu and P. Van Moerbeke, "The Lagrange rigid body motion", Annales de Vlnstitut Fourier 32 (1982) 211-234. [53] S. Smale, "Topology and Mechanics: I", Invent. Math. 10 (1970). [54] S. Steinberg, Lectures on Differential Geometry, (Prentice-Hall, 1964). [55] V. V. Trofimov and A. T. Fomenko, "Dynamical systems on orbits of linear representations of Lie groups and complete intergrability of some hydrodynamic systems", Fund. Anal. Appl. 17 Nl (1983) 31-39. [56] V. V. Trofimov and A. T. Fomenko, "Geometrical and algebraic mechanisms of integrability of Hamiltonian systems on homogeneous spaces and Lie algebras", Modern Problems in Math. Fundamental Research, 1987, Vol. 16, pp. 227-299. (Encyclopaedia of Mathematical Sciences, Springer-Verlag). [57] V. V. Trofimov and A. T. Fomenko, "Liouville integrability of Hamiltonian systems on Lie algebras", Usp. Mattm. Nauk 39 N2 (1984) 3-56. [58] F. Waldhausen, "Eine Klasse von 3-dimensional Mannigfaltigkeiten", Invent. Math. 3 (1967) 308-333 (Part 1) and 4 (1968) 88-117 (Part 2).

164 [59] A. Weinstein, "Symplectic manifolds and their Lagrangian submanifolds", Ann. Math. 8 (1971). [60] A. Weinstein, "The local structure of Poisson manifolds", J. Diff. Geometry 18 (1983) 523-557. [61] H. Zieschang, E. Vogt, and H. D. Coldeway, "Surfaces and planar discontinuous groups", in Lecture Notes in Math. Vol. 835, (Springer-Verlag, 1980).

165

FINITE-DIMENSIONAL SOLITON SYSTEMS S.N.M. Ruijsenaars Centra for Mathematics and Computer Science P.O. Box 4079. 1009 AB Amsterdam, The Netherlands

ABSTRACT. We survey recent results concerning Calogero-Moser and Toda systems and integrable gene­ ralizations thereof. We also discuss relations to infinite-dimensional integrable systems.

1. Introduction 2. Classical sdliton systems 2A. Classical integrability and the soliton property 2B. Pure soliton systems 2B1. The Hamiltonian H 2B2. The Hamiltonians S\ Sn 2B3. The action-angle transform and duality 2B4. Soliton scattering 2C. Systems related to pure soliton systems 2C1. Systems with soli tons and antisolitons 2C2. Sutherland type systems and their duals 2C3. Elliptic systems 2C4. Periodic Toda type systems 3. Quantum soliton systems 3A. Quantum integrability and the soliton property 3B. Quantized pure soliton systems 3B1. Preserving commutativity 3B2. The eigenfunction transform and duality 3B3. Soliton scattering 3C. Systems related to quantized pure soliton systems 3C1. Systems with solitons and antisolitons 3C2. Sutherland type systems and their duals 3C3. Elliptic systems 3C4. Periodic Toda type systems 4. Connections with infinite-dimensional integrable systems 4A. Preamble 4B. The classical level 4C. The quantum level Acknowledgements References

* Work supported by the Netherlands Organisation for die Advancement of Research (NWO).

166 1. INTRODUCTION In recent years the well-known nonrelativistic Calogero-Moser and Toda /V-particle systems have been shown to admit integrable relativistic generalizations [1-3]. Results on the former systems and their versions for root systems other than AN-i have been surveyed by Olshanetsky and Perdomov in the early eighties [4,3]. Here, we concentrate on results obtained since that time, especially as concerns the relativistic systems. We presuppose no previous knowledge concerning finite-dimensional integrable systems, but some acquaintance with the surveys [4,5] would probably be helpful. Also, we limit ourselves to the physi­ cally most relevant case of translation-invariant interactions. Thus, root systems other than AN _j are only peripherally mentioned, and no external field couplings preserving integrability are considered. Furthermore, internal degrees of freedom are not discussed and we do not deal with the thermo­ dynamics associated with the systems. Chapters 2 and 3 are concerned with the classical and the quantum versions, resp., of the class of integrable systems just delineated. In both chapters integrability issues and relations between the vari­ ous systems are discussed in some detail. In Chapter 2 we also sketch our results on explicit actionangle transformations [6-8], which lead in particular to duality relations between various parameter regimes. These classical duality properties are of interest not only in their own right, but also because they have obvious quantum translations. This is explained in Chapter 3, in connection with our description of explicit knowledge concerning the joint eigenfunction transform. This transform is the quantum analog of the action-angle transform, and its duality properties agree with those of the action-angle transform in all cases where this has been checked. In fact, our expectation that the classical self-duality of the 'master parameter regime' at the 'one-period' level (the Ilre, regime described below) will survive quantization has been a crucial guide towards finding explicit eigenfunctions, some of which will be reported here for the first time. As it turns out, the quest for a unitary eigenfunction transform for the relativistic systems leads into uncharted territory at the intersection of Hilbert space theory and the theory of analytic difference equations. Here, too, duality properties have been of considerable help. In Chapters 2 and 3 we emphasize explicit knowledge concerning the action-angle and joint eigen­ function transforms, not only because we feel that these maps are the central mathematical objects in the systems at issue, but also because this knowledge is indispensable in making contact with the world of infinite-dimensional integrable systems. Chapter 4 is devoted to a sketch of some of the con­ nections that have emerged thus far. We believe there is a lot more in store here, especially at the quantum level. 2. CLASSICAL SOLITON SYSTEMS 2A. CLASSICAL INTEGRABILITY AND THE SOLITON PROPERTY

To provide some context for our definition of 'soliton system' it is expedient to begin with some remarks on the more general notion of 'Liouville integrable system'. We shall restrict ourselves to the simplest type of phase space < Q , u > , viz., the cotangent bundle over a region C c R * , 0 = {(q, 0)eRw\qeG),

(2.1)

with its obvious symplectic form N

<■> = 2 dqjAdSj. (2.2) V=i Then a Hamiltonian H on 8 defines a Liouville integrable system whenever there exist N independent functions 51, . . . , SN on fi such that {H,Sj}=0,

J = l,...,N

(2.3)

(Sj,Sk)

j,k = \,...,N.

(2.4)

=0,

167 It is important to note that in the very general, mathematical context of this definition the construc­ tion of integrable systems is just as easy (or hard) as the construction of canonical transformations. Indeed, let Q be a region in R2" with coordinates (q,9) and symplectic form N

u=24/A<W/-

(2.5)

Now let S be a canonical map from B onto Q, and define

H(q,k=fm Sk(q,6) =

2

k

K'K

=1

(2.6) (2-7)

N

l « i , < ■ ■•
where fe Cg (R). Then the pullback Hamiltonians H = H-<S>

(2.8)

Sk = Sk°*

(2.9)

$ = S" s ,

(2.10)

where

yield an integrable dynamics fl on Si with commuting integrals S,, . . . ,SN. Conversely, the Liouville-Arnold theorem [9] implies that any integrable system # i , S i , . . . » % may be viewed as such a pullback. (Of course, the structure of > will be far more complicated than just assumed, in general.) The upshot is, that the class of Liouville integrable systems is in essence equivalent to the class of canonical transformations. (The qualifier cannot be omitted due to topological niceties we have no occasion to recall here.) We shall now head for a much stronger notion of integrability, which is how­ ever tied to a quite special, physical context. To this end, we consider a Hamiltonian H(q, 8) that models the dynamics of N interacting particles on the line. We assume that the interaction has a repulsive character. More precisely, we assume that the wave transformations [10,11] S±: < » * , « * > - » < 8 , 4 j > , (q±,0*)>->(q,0)

(2.11)

exist and are canonical maps onto S; moreover, we assume that the incoming and outgoing momenta satisfy

« * > • • • >«,", « £ < • ■ <»t-

(212)

Since the asymptotic Hamiltonians H-=H-S,±

(2.13)

depend only on 6±, it is easily seen that the choices H,Sf" ville integrable systems when one defines Si=

2

< ■ < .

«=+.-

S™' and H,Sf

SJJ yield Liou­ (2-14)

K i , < ■••
St = S f ° S ; ' (2.15) Of course, one can choose other functions of ^ ± that generate the same maximal abelian algebras on 12. For instance, one could replaceffj-in (2.14) by expOSSj) with j8e(0,oo). Such choices should not be viewed as different from the previous ones. However, one can just as well introduce new momenta p(8+) and canonically conjugate positions x(q + ,8+) such that the transform of H equals the function p\\ then the pullbacks to Q of the functionsp\,x2, ■ ■ ■ ,xN yield a Liouville integrable system that is

168 very different from the two previous ones. This goes to show that any repulsive particle dynamics satisfying the above assumptions gives rise to a plethora of Liouville integrable systems. Let us, therefore, introduce a far stronger notion of integrability (which goes back to [12, p. 339]). We shall say that a dynamics H with the above proper­ ties defines a pure soliton system whenever »t = «*-*+..

*=1

N

(2.16)

(conservation of momenta). Notice that this holds if and only if the incoming and outgoing integrals are equal i.e., S? = Sf*.

*=1

N

(2.17)

cf. (2.11H2.15). Pure soliton systems as just defined do exist, and will be studied in Section 2B. However, they are by no means a common occurrence. In fact, within the confines described in the Introduction, the sys­ tems considered in Section 2B are the only ones for which the pure soliton property has been proved. We shall extend the term 'soliton system' to Liouville integrable systems obtained from pure soliton systems H,Sj, . . . ,SN byfinite-parameterdeformations and/or analytic continuation. Admittedly, this sounds like a somewhat loose characterization. However, it does serve to single out the systems studied in Section 2C. Physically speaking, these systems are characterized by attractive and/or confining interactions. In this physical context existence of nontrivial integrals (for the center-of-mass Hamiltonian) is highly exceptional In fact, the IV^ soliton systems defined in Subsection 2C3 are (to date) the most general Liouville integrable iV-particle systems for which the interaction depends only on interparticle distances. 2B. PUKE SOLITON SYSTEMS

2B1. The Hamiltonian H. For the six classes of pure soliton dynamics detailed in this subsection the phase space is given by (2.1), (2.2). The configuration space G can be taken to be G = { 9 eR J V | ? A ,< ■ • • < ? , }

(LJI)

(2.18)

(VI)

(2.19)

for the Calogero-Moser type systems I, II and G = R"

for the nonperiodic Toda type systems VI. For the three nonrelativistic classes the dynamics is of the form

ff=T2^W

(2-20)

Specifically, one has V(q) = g2 V(x) = 1/x

2

n°j-qk\

«eR*

2

K(x) = ^ / 4 s h 4 f « .

/ie(0,oo)

(2.21) (LJ

(2.22)

(IIm).

(2.23)

Obviously, the lm case arises from the II,,, case by taking n to 0. Substituting ? y -» 9 ,+2/>-'lnc

(2.24)

£ -» 1/J«

(2.25)

in the IInr Hamiltonian, and taking the strong coupling limit c->0, one obtains the nonperiodic Toda Hamiltonian, for which

169

!/(,)I S= 2

«PW$--ii-

(VW

i)]

(2.26)

We shall presently discuss the above dynamics and their commuting integrals S j , - . » , S # in more detail. At this point we only note that among the Hamiltonians in the associated maximal abelian algebras the above Hamiltonians H are singled out not only by their obvious physical interpretation, but also by the fact that their flows are most easily studied. The Hamiltonians that follow now are chosen for similar reasons: Their general structure reads

tf=r'2«p03W?),

(2.27)

;=i

and the three classes of potentials are given bv Vj = U k*i

»5=**>n' [

1+

At* Klt-lkY

sh^rtft-ft)

V

J

£eR-

ftd)

(2.28)

(Hrd)

(2.29)

-11/2

sh2z

x

, /}G(0,oo),

ft/ie(0,oo),

±*ei(0,ir)U(-y-+R)

(2.30)

=M9i+t~9j)Mqj-9j-i)

fj{x) = [1 +0 i expO«)]" : ! ) 9w+i =

—

(2.31)

/S,Me(0, oo) (VI,,,)

°o. 9o = °o

(2.32)

Note that the restrictions on the parameters guarantee real-valuedness of the Hamiltonians. The subscripts "nr" vs. 'rel' refer to the fact that the extra parameter fi may be viewed as 1/c, where c is the speed of light. This is explained in detail in [1,2,13]. Here, we only point out that one has HrA = Nfi~l + 2 Oj+PHn + Oifi1),

/8-0

(2.33)

for the cases I and VI; after the substitution 3.34)

z -»1'ft»S

this expansion follows for II, as well. Notice that with this substitution one can obtain //(I„i) from H(II_i) by taking u to 0. Moreover, after the substitutions (2.24) and (2.35)

z -»-riir—Inc+ln/J one obtains //(VI„,) from #(11,^) by sending e to 0 .

2B1 The Hamihoniwu S\,... ,SN. For the nonrelativistic dynamics detailed above the existence of integrals was first Droved bv finding a Lax pair representation fH,L)

= [L,M]

(2.36)

for Hamilton's equations. Specifically, one can take as Lax matrices [4] iflLJ/* =

8jk»k+igQ->jkW2s*T2l<1i-9t)

(2.37)

(from which LCL.) is obvious) and LQfl^jk

= Sjk«k+Sj.k-i

+&j,k+\e**W.1j-
(2.38)

170 Then one has in all three cases H =-jTrL1.

(2.39)

Thus, it follows from (2.36) that the symmetric functions S,. . . . ,SK of the Lax matrix L are con­ served under the H flow. Since the particles repel each other, one also concludes L -»diag(*f

ȣ),

r-*±oo

(2.40)

Sk -*

«? • ' ' * f •

'-±«>

(2.41)

2

where 0* denotes the asymptotic momenta. But the spectrum of Z. is conserved due to (2.36), so that conservation of momenta (2.16) results. Moreover, (S,-,54) is an integral by the Jacobi identity, and it has limit 0 for /-»±oo since the interparticle distances diverge. Consequently, the Hamiltonians S, Ss are in involution. The arguments just presented are physically convincing, but a rigorous proof of the soliton property involves more work. For the above nonrelativistic systems such a proof was first given by Moser [14,15]. Let us now consider the nrel case. Here, it can be shown directly that the functions 11/2

S,=

2

7C(1 N) |7|=l

«P(2/8#y)IT lei

/«/ *«/

sh2r

(2.42)

^1^~9k)

commute with H =f} ' S , . These Hamiltonians are the symmetric functions of the matrix L(llrt)jk = exp(£*,)K,(?)shz/sh(* +-jMty - ? t ) )

(2.43)

(recall (2.29)), as follows from Cauchy's identity. Hence, the spectrum of L is conserved under the flow generated by H. Moreover, Moser's argument can be adapted to this flow. This yields (2.40), (2.41) with $j replaced by txp(fi8j), and entails again involutivity and the soliton property. The same reasoning applies to the Ircl case, obtained by substituting (2.34) in (2.42), (2.43) and taking n to 0. (For more details concerning the assertions in this paragraph we refer to [1].) For the case Vlrd a Lax matrix can be obtained from (2.43) by first substituting (2.24) and (2.3S) in a suitable similarity transform L of /.(Il„.j) and then taking c to 0. Specifically, setting Ljk = fh - J exp^rt$/ - ?*)U-(H™iV

(2-44)

one obtains 0

k<j-\ k=j-\

£(VI*V =
k >j

(2.45) -1

where aj = ^exp[M%-? / -i)K'+/8 2 expW? / -%-i)D" 1

(2.46)

bj = ex?(fi$j)Vj(q)

(2.47)

cf. (2.30H232). The symmetric functions of this matrix read

S,=

IC{\

2 |/|=/

N)

exp(2^)II/r(?/ + .-?/) HMqj-qj-il /./

Jel y + i«;

j,I j-iti

(2.48)

Adapting Moser's argument, the soliton property follows again. (The assertions just made are proved

171 in [3]; the opposite ordering is used there.) With the substitution (2.34) in r.(TT_.) one easilv verifies that

ird = l«+j8inr+0(/3 2 ),

(2.49)

0->O

in all three cases. Using this expansion one can obtain the involutivity of the symmetric functions of Lm as a corollary of the involutivity of the L-i functions. Specifically, from (2.49) one infers

S*.„ = l i m / r * 2 ( - ) * + ' w - f e S w

(2.50)

from which the claim readily follows. Note that this formula for S^ „, is far from explicit, in contrast to the formulas (2.42) and (2.48) for S, „,. 2B3. The action-angle transform and duality. Let us now turn to the construction of an explicit action-angle map for the above systems. We begin by recalling that the Liouville-Amold theorem guarantees the existence of such a diagonalizing map under certain conditions [9]. But these are actu­ ally not met for the above systems: The radii of the invariant tori are infinite. Furthermore, even in situations where the intersections of the relevant level sets are compact and connected (as is the case for the type MI and TV systems discussed in Section 2C), the Liouvule-Arnold theorem is of little help in obtaining explicit information, in much the same way as the spectral theorem is of no use in obtaining the eigenfunctions and spectrum of a concretely given self-adjoint differentia] operator. However, it turns out that for the above pure soliton systems and for the related systems of type II and III (cf. Section 2C) one can construct an action-angle map * whose relevant features can be esta­ blished in much more detail than one might reasonably expect. For all of the pure soliton systems the action-angle phase space can be taken to be 0 = ((q,9)zRw\eN< u

(2.51)

■ ■<*,}

= 24/Arftf,.

(2.52)

Comparing (2.51) with (2.1), (2.18), one sees that Q and Q can be identified in an obvious way for the rational and hyperbolic systems I and II. Doing so, these systems can be defined on Q, too. It now turns out that the inverse & of 9 may be viewed as an action-angle map for one of the latter systems. Specifically, the duality thus obtained reads IQT

lllT»

Mar — Ird>

Irel — ! ! „ ,

Hrel -

"rcl-

(2.53)

(I he notation will be clear trom context.; Action-angle maps for the systems VI„r and VIrc, can also be constructed explicitly. Again, certain duality properties arise, but these are less useful than (2.53). The maps may be viewed as limits of those of the II,,, and Ilrci systems, resp. However, this limit is hard to control and yields less informa­ tion than the direct construction. We refer to [3] for further details on the type VI systems. The key to the construction of * for the systems of type I and II is a certain commutation relation of the Lax matrix with an auxiliary matrix-valued function A on 0. We shall now sketch this con­ struction for the II„i case, picking ±ze;(0,ir). (The following is taken from our paper [6], where also previous work on special cases [16-18] is discussed.) In this case one has (2.54) = SytexpOiflt)

4*

and the commutation relation reads -jcth z[A,L] = e®e- -+(AL + LA). lere, the vector-valued function e on Q is defined by

(2.55)

172 ej = txVC1iu!j+^l}6j)Vj()i,z

;q?'*

(2.56)

cf. (2.29). Combining this with (2.43), one readily verifies (2.SS). By exploiting (2.55) and Cauchy's identity, it can now be proved that L has positive and simple spectrum on Q, and that any unitary U such that UL(fi,t,z;q,0)lT = diag(exp(0»,).... .exptfff,,)), 9N< ■ ■ - <«, (2.57) must satisfy (Ue)j=£0, j = 1,...,N. The gauge freedom in the diagonalizing unitary can then be fixed by requiring (Ue)j>0,j = 1 , . . . ,N. This entails the existence of a unique vector qeR" such that (Ue)j = e x p ( - ^ , +\fqj)Vj{fi,

-z ;ff)u2

(2.58)

and then one obtains the relation UA Qi;q)ir = LQi,fi, -z fi,q) = A(fi,p,z ;q,f).

(2.59)

As a consequence, a well-defined map *:0->Q, (q,9)»(qM (2.60) emerges. From the duality relations (2.58), (2.59) it now readily follows that 9 is a bijection with inverse S(fi,n,z ;q,fi = P°9QL,P, - * M) where f is theflipmap P(x,y) = (y,x),

(2.61)

x,yeH»

(2.62)

Our proof that the bijection 9 is in fact real-analytic and symplectic is rather long and arduous, and we shall not describe it here. (An important ingredient is scattering theory, to which we shall presently turn.) We finish this subsection by noting that the canonicity of 9 together with its con­ struction as sketched above entails that 9 linearizes the flows generated by Hamiltonians of the form #»=TrAtf-'lnL),

AeCff(R>

(2.63)

2A(« y ).

(2.64)

Indeed, from (2.57) one has (Hk°Sfo8) = Hk(q,h= Since S is canonical, this entails txf{lHhy& = Soexp(///4).

(2.65)

Hence, the nonlinear flow (q, 9) H. atffHMq, 9) = maps into the linear flow

fo(0.«(0)

(q,9) » exp(/ff*X?.») = (.h +'*'(«i). ■ ■ - ,qN + th'(eN),i) = (q(t),h as advertised.

(2.66) (2.67)

2B4. Solittm scattering. As we have just seen, one has (q(t)At)) = <%q('),b (2.68) for Hamiltonians of the form (2.63). For the IIrel systems the quantities exp(jujj(t)) are the (ordered) eigenvalues of the matrix

173 A(fi,lk,z;qi +th'(e{), . . . ,qN+th'(BN),ff)

(2.69)

cf. (2.59). Hence, the long-time asymptotics of qJ$} boils down to a problem of spectral asymptotics. Specifically, assuming henceforth h"(x)>0,

VxeR

(2.70)

one needs the spectral asymptotics of matrices of the form At exp(tD), where M is positive and D is of the form D = diag(d,

dN),

d„< ■ ■ ■ <
(2.71)

This problem can be solved under less restrictive assumptions on M and D, and the generalization thus obtained is essential in our proof that $ is canonical. (We mention in passing that the set of matrices M that are allowed (cf. [6, p. 157]) may be viewed as the big cell in the Bruhat decomposi­ tion of GL(N,C). This observation is possibly useful in tackling other root systems. However, the use of upper/lower factorization appears not to simplify the proof of I.e. Theorem A2.) As a consequence of the spectral analysis just referred to, it can be concluded that the assumption (2.70) entails

9,

(<)-?,(<)-> ^ i V * ) -

'->±°°

(2-72)

(/)-5,-»0

/->±oo

(2.73)

S-j + \

Sj Ar-y + l

uniformly on compacts. Here, one has A,(0) = 2 W j - » k ) ~ 2 W j - » * ) k<j

(2-74)

k>j

8(6) = /i~'ln(l -sh 2 z/sh 2 -j0#).

(2.75)

Thus, the scattering map S:Q-

= {(
)eR2JV|^>

>Bi" } - ►a + ^ a

(2.76)

is given by

,#D. (2.77) BN) = (q'n +AA,(#' ) , . . . > and <(2,u> having both the duality (as involution) and the asymptotic con­ stancy properties? 2C. SYSTEMS RELATED TO PURE SOLITON SYSTEMS

2C1. Systems with sotitons and antisolitons. Following Calogero [20], let us substitute qj-^qj+iir/^ (with \
j=n + \,

N

— 1) in the !!„. Hamiltonian. This yields

(2.78)

174

1

*

»

1

2

1

2

1

(2.79) (On,)ch -ft) -?*) "2^%Thus, the resulting systems may be viewed as nonrelativistic interacting particle systems with n parti­ cles of positive charge (solitons) and N —n particles of negative charge (antisolitons); particles with the same/opposite charge repel/attract each other. If we make the substitution (2.78) in the more general IL^ dynamics (2.27), (2.29), then we obtain a real-valued smooth Hamiltonian on ff =

i2«j+^V J=l

l
QW = {(q,B)eRw\qN<

2

sh

irtft-

, 1 «=/
2

• • • < ? „ + 1, « , < • ■ • < ? i }

(2.80)

provided z (2.81) s. = {weC||Imw|
=iw

(Hm)

(2.82)

(2.83) a = (Hd> 1*1 Non-real eigenvalues occur in complex-conjugate pairs, and each such pair corresponds to a solitonantisoliton bound state. All channels one can envisage occur, but they do not couple. That is, there is a region in H*"' where L has real and simple spectrum, corresponding to an asymptotics of freely moving solitons and antisolitons, and there are regions with k<min(n,N —n) bound pairs. There also exist points in $"' for which L is not diagonalizable; such points do not behave as scattering states. Moreover, there are points that behave as multi-body bound states; this phenomenon depends on the dynamics one chooses. The case z =iir/2 is different only in as much as singularities occur when soliton and antisoliton positions coincide. Technically speaking, this entails that the H flow (for instance) is not complete. However, the action-angle map enables one to define the flow for all times in a natural way, but for a discrete set of times where solitons 'overtake' antisolitons. The singular character of z = ittll is most easily seen for N =2. Then H can be written H

= 2/8 ' e x p l - j f l ^ + ^ l c h - ^ ,

-«2)|th-j/i(9i - ? 2 ) | .

(2.84)

Thus, the set q\—qi plays a special role. In particular, by energy conservation 8^ —62 diverges for ?i -?2->0Instead of substituting (2.78), one might also substitute 8j^>0] + iir/p,

;'=n+l, .

. ,N

(2.85)

in the II„i Lax matrix (2.43). Indeed, this also yields real-valued symmetric functions, cf. (2.42). This type of Lax matrix actually arises on the subset of the action-angle phase space corresponding to the 'no bound state' subset nj,"1 CO*"'. (This fact can be understood from the self-duality of the llre| regime described in Subsection 2B3.) As a consequence, one can infer that the H flow arising via (2.85) is not complete. Indeed, the corresponding dual flow on the subset of U" just mentioned pulls

175 back to a linear flow on tf"'; for an open set of initial points this flow leaves 0(T' in finite time. This unstable behaviour can be physically understood by noting that (2.85) amounts to flipping the sign of the particle mass. cf. (2.27). One can also continue both some qj and some Oj, since this still leads to real-valued commuting Hamiltonians, cf. (2.42). However, in this case the Lax matrix appears not to have any properties that can be used to elucidate the relevant features of the dynamics. As a preparation for the next subsection, we now rephrase the mathematical setting of the above results. We have at our disposal a symplectic diffeomorphism * from < B , u > (cf. (2.1), (2.2), (2.18)) onto (cf. (2.51), (2.52)), which is real-analytic in the parameters and in the canonical coordi­ nates, cf. [6, App. B,C]. This map can be analytically continued in q and then restricted to certain real, ^-dimensional submanifolds of C", cf. (2.78). These give rise to new phase spaces (cf. (2.80)), whose images under * can be explicitly determined. Correspondingly, one can obtain detailed infor­ mation concerning associated Hamiltonian flows. More generally, singularity and monodromy properties of * can be determined rather explicitly, but the structure of the image set appears too complicated to answer with ease every question one may care to ask. For instance, the behaviour under the continuation (2.85) can be understood from duality considerations, but we have no information concerning simultaneous continuation in q and 6. The feasibility of answering concrete questions hinges on special features of the Lax matrix, restricted to the real, 2Af-dimensional submanifold of Cw which is chosen as a new phase space. 2C1 Sutherland type systems and their duals. The preceding considerations suggest a systematic approach: Find the physically interesting and mathematically manageable phase spaces embedded in C , as obtained from the pure soliton regimes via analytic continuation in q,6 and/or the parame­ ters. As a first example along these lines, one may start from the IL^ systems and continue the positive parameter ft to the imaginary axis. (Equivalently, one may keep ft positive and take g and qlt . . . ,qN purely imaginary.) Then one obtains again a self-adjoint Lax matrix, with

Hs\tO?^\^-i

/=«

T*V K/
1

(IHnr)-

(/.BO)

sffl*TlM%- -?*)

The obvious physical interpretation of the Him systems thus obtained is now in terms of particles on a ring. On account of energy conservation the angular distances \q — qk\ remain finite for all times. Omitting the center-of-mass motion, an appropriate configuration space is the 'Weyl alcove' G =

{xeRN~

'|*„.

■ ■ , * J V -

, >0, 2 */<2»/|/«|}, Xj=qj-qJ

+ l

(2.87)

and each state in the corresponding (2N — 2)-dimensional phase space is a bound state. The action-angle map 9 can be determined explicitly for the III,,, systems [8]. In particular, one can nr/wA ttiat th*» /VirHmwI'V pi 0PT1 VfllllftS Of J.(ITI__1 Satisfv

•r -«, + , >

(2.88)

Wr

The exceptional set 0, where one or more equality signs are realized is nowhere dense and has. meas­ ure varrf it fnntnins thp Miuilihrium configuration

* = -k\(N+x-1J)' *> = 0> '=1-

(2.89)

!„,-» I l k : A?, «-»/•,«,?■ jiei'(0,oo).

(2.9U)

,.,JV

for which all equality signs in (2.88) apply. The map $ is defined on the complement of iie, and sets up a duality with IIL, systems that can be obtained from the above I^ systems, as follows:

176 At first sight the III,,, systems thus obtained appear unphysical. Indeed, due to (2.88) the 8dependent potentials in 5] SN are positive on S(IIInr), but the exponentials in (the I„, speciali­ zation of) (2.42) turn into phase factors after the substitution (2.90). However, this is easily remedied. What matters is whether there exists a real form for the maximal abelian Poisson algebra. To explain that this is indeed the case, we need a piece of information from [1,2] we had no occasion to mention so far. This is the fact that the functions Si(fi) given by (2.42) commute with Sk{—p). Therefore, we may replace exp by ch at the rhs of (2.42), and then we do get real-valued ^-dependence under the suhstitiirinn 10 <MW In particular, the dynamics

ff=/r'2>s
', **

IKP (S/-9*) 2

i

(2.91)

(III-,)

is real-valued. The flow it generates has a quite different character compared to the flow generated by (2.86). In fact, the I11M systems may be viewed as pure soliton systems, in a sense detailed in [8]. As a second example of the program described in the first paragraph of this subsection, one may continue the parameter g in the 11^ systems to the imaginary axis. (In fact, this can already be done for the special case Inr.) However, this leads to 'physically interesting' systems that seem not to be 'mathematically manageable'. In the same spirit one can obtain from the ll„i pure soliton systems seven other regimes of physical interest, hv first narametriTine z as

z -~iWm

pyli,ge(0,x,)

(2.92)

(Urd)

and then taking /8,/i and/or g purely imaginary. However, we can only handle the three cases that arise when one keeps g real, for reasons just mentioned. The systems obtained by taking fie/(0,oo)//i,fiei(0,oo) will be denoted I\\,a/\l\,^b. When one only takes /}e/(0,oo), one obtains sys­ tems that may be viewed as the IIIrd systems (i.e., the systems dual to the III„i systems, cf. also i2.90^. A The salient features of the IIIrel and IIIr(.i systems are essentially the same as those of the IIIm and III,,, cases already described. In particular, with (2.34) in force, one can prove the inequalities (2.88) for the eigenvalues expOSfy) of L(IIIrel), and for the equilibrium (2.89) one again obtains equality signs. The self-dual regime IlIreLt is quite different from all previous ones. As before, we should replace exp by ch in (2.42) to obtain real-valued dependence on 6. However, to also ensure real-valued qdependence, one now needs a center-of-mass configuration space {XGR"- ' ! » . . .

Gb =

■ ■

, * A F -

i>lftl

;= I

*7 = *~ft+l-

(2.93)

As a consequence, Gj, is non-empty if and only if N
=^ifrgei(0,-1v).

(2.94)

Therefore, not only the configuration space, but also the allowed particle number is bounded. In fact, the regime IIIrci b has yet a third boundedness property: The natural center-of-mass phase space is not r c ^ C X H " - ' , but rather GXU<\f~l [81. 2C3. Elliptic systems. The nonrelativistic Hamiltonians considered thus far may all be regarded as snerial rjispfi fif

H=

i

"

/=!

+g2

2 %;-?*;«,«'), geR*. «. -i'u'e(0,oo] !
(IVn,).

(2.95)

177 (Here, 9 h the Weierstrass "^-function; the restriction on the primitive periods 2u,2u' guarantees positrvity on R.) More precisely, this is true for the 1^,11^, UL^ and VI^ Hamiltonians; one needs to shift ft + ,qn over u' (obtaining a real-valued IV^ Hamiltonian) to ensure that the II„r Hamiltonian (2.79) results (up to a constant) when one sets w'=i'ir//i and takes u t o o o . The commuting Hamiltonians that generalize those of previous sections can be taken to be the sym­ metric functions of the matrix

L(Wnr)jk : =

8jk»k+igV-8jk

.
(2.96)

Here, o is the Weierstrass a-function and XeC an auxiliary parameter. The Lax matrix (2.96) was introduced by Krichever [21]. He shows that the H flow linearizes on the Jacobian of the genus N curve defined by

\L(\)+alN\ --= 0

(2.97)

Krichever's results pertain to complex qj and 6p and it seems not an easy matter to obtain results for real 2JV-dimensional phase spaces of physical interest As the relativistic generalization of (2.95) one can take [1,2] the dynamics (2.27), with V

i

(2.98)

== IIA(*-- ft) (IV*,).

( Y X ^ Y ) --*WJ, ±/ e(0,- -2jw') rw= : ^'TrL, Then one hastf = where L is the Lax matrix [22, 2] T

=r

Ml v rel)/A: — expt/Wy) rfig)

'

'V

-j^v

avt) a(Oi-Qk+y)

(2.99)

(2.100)

A Lax pair formulation for the H flow has been found by Bruschi and Calogero [22], whereas in [2] it is proved that the symmetric functions of L are given by

ca-yy-'aa+a-ih) ofxy

2,

2

/ C ( l , . . . ,N)

1/1=/ and are in involution. Note that after the substitution

exp(2/ty)IT/<%- "ft) Jzl

(2.101)

jel IctI

(2.102)

Y -"ft?

the expansion (2.49) again holds true. Therefore, involutivity of the symmetric functions Sk of L(IV n l ) follows as a corollary, cf. (2.50). Moreover, since 2 , depends on X only via a multiplicative factor, the functions S*(X|) and S/(X2) also commute pairwise when Xi^=X2. For the TV^ case the curve (2.97) appears to have (generically) genus 1 +N(N —1)/2 [2]. There is yet another parameter regime of physical interest, which generalizes the I I I ^ j regime. It is defined by taking

fien;

ye(0,u)

(IVrd,,,).

(2.103)

The generalizations of (2.93) and (2.94) read G„

=

N-l

{xeRN- ' ( * , , . • • ,xN- 1>Y. "2 xj<2u-y),

Xj,

= qj-qJ

+t

(2.104) (2.105)

N<2a/y

and commuting real-valued Hamiltonians can then be defined via (2.101), with the prefactors omitted and exp replaced by ch. 2C4. Periodic Toda type systems.

The periodic Toda systems can be defined by

178 H = T207+*2

2

«pWty- ~ii- ,)]+** exp[K?i -?*)]

I!
;=i

(V„,)

reR , ue(0,oo)

(2.106)

(The systems denoted type V by Olshanetsky and Perelomov [4,5] may be viewed as type I systems perturbed by an external field.) As Lax matrix one can take [23,4] = «/*«*+*/.*- \+r8j,k

L(Vm)jk

+ i exp[K?j-?y-i)]

- r V r " * - 1 * , ! * ^ expW?! -?*)], XeC"

-(irfUpSki

(2.107)

and then (2.39) again holds true (up to a constant when N =2). Substituting mT

?r»ft-2/M

.

I =

1, ■ .

(2.108)

.,N

one obtains LfVI^) in the weak coupling limit T-»0, cf. (2.38). The commuting V,,, flows can be linearized on the Jacobian of the algebraic curve (2.97), which is hypereUiptic and has (generically) genus N -1 [24,25]. Just as for the rv^ case, little seems to be known for real q and 8. The Vjd generalization of (2.106) can be taken to be (2.27) with Vj given by (2.30), but now (2.31), (2.32) should be replaced by

M*) =E l l + ^ e x p O i x ) ] " ?* + i =1\

.

?0

2

(2.109) (V,d)

=?AT

(2.110)

This again entails the expansion (2.33). The commuting integrals are still given by (2.48), but now with (2.109), (2.110) in force [3]. Bruschi and Ragnisco [26] found various Lax pair formulations for the H flow, and used one of these to obtain a linearization on a spectral curve defined in analogy with (2.97). inozemtsev recently ODserved mat tne vM tiamutonian (i. iut>) may be obtained as a strong cou­ pling limit of the IV^ Hamiltonian (2.95) [27]. A priori, his argument sheds no light on the behaviour of the conserved quantities in his limit. However, we shall now show how one can obtain the Vm Lax matrix (2.107) from the 1VM Lax matrix (2.96). Moreover, we shall obtain a V„, Lax matrix from the IVrd Lax matrix (2.100). On account of the mode of convergence and the argument embodied in (2.50), the Liouville integrability of the systems of type V may then be viewed as a corol­ lary of the involutivity of the functions 2 t given by (2.101). The limits to be detailed now can be easily controlled and verified bv using the reoresentation

o(q;ui,iv/n) —exp (l'W2 /2i") sh 2 )Uj r j | 1 i

- e x p ( / K 7 - -Tkmim-

-exp(--n-

-Ikfia)]

[1--exp(--2kiw>)f

(2.111)

for the Weierstrass o-runction. (Here, the notation of [2s| is used.) ror both limits one needs position shifts given bv qj^>qj-2ju/N, / = 1, . ..,N (2.112) Thefirst-mentionedconnection can be made by starting from the similarity transform Ljk

= i<W«)> expl •^iv*r -?*)+(/■- * > * ]

(2.113)

Substituting (2.112) in L, together with

g

—> T/X

X-> u+ one obtains

exp(/i«/7V) 117

P

lnX

(2.114) (2.115)

179 limZ.,, = L(ym)jk(iTy-k

(2.116)

(as anticipated in the choice of spectral parameter in (2.107)). For the second transition we make use of a renormalized Lax matrix L'jk = LQV^

e x p ^ V t ^ J V - 1)+(X-TX* - f t ) D

(2.117)

Substituting (2.112) in V', together with *-»T7 + — + 1 " 0 8 T )

(2.118)

X ^ - ^ + ^UIOST)- — l n ( / » - - l n X JV

/i

(2.119)

/t

M

one obtains lim i r = B(A + A) = /.(V^)

(2.120)

i»=diag(6,,...,^) 4 » = fiM Ay* =

(2.121)

+ 1[l+(1»"X]a>-fi/15iAr[l+(i»-''X-

1

1

]ai

(2.122) (2.123)

*>/-l

a, = 0 V expW$,-<&-,)Kl+/S 2 ' 2 e x p ^ - ? ; - , ) ] ) - '

(2.124)

6, = expG80,Xl+/?V exp[M(?; + , - * ) D W ( I + 0 V expWft-?,-!)])" 2

(2.125)

Taking a—>oo in the symmetric functions of / / (which follow from (2.101)) one obtains the symmetric functions of L(Vrel). Explicitly, one finds

tt\+(ifrfXl-xSi, 2 , = U+(iprf*f~lllHifrf*~l]S , K

1=1, l=N

N-\ (2126)

where S\, . . . ,SN are given by (2.48) (with (2.109), (2.110) in force, of course). Thus, involutivity of Si, . . . ,SN may be viewed as a corollary of the involutivity of the elliptic Hamiltonians (2.101), as announced above. We add four more remarks. First, up to a constant and a similarity transformation, the matrix L(V^)T equals the Lax matrix of a Lax pair for the H flow that was recently obtained by Oevel and Ragnisco [29]. Secondly, setting l
(2.127)

and substituting (2.108), one obtains lim I(V„,) = L(VIrel)

(2.128)

cf. (2.45)-(2.47). Thirdly, one has LWrt) = lAr+j8L(Vor)+0(J82),

0->O

(2.129)

cf. (2.107). Therefore, the Liouville integrabihty of the nonrelativistic periodic Toda systems may be viewed as a corollary of the Liouville integrabihty of our relativistic generalizations, cf. (2.49), (2.50). Fourthly, from (2.129) and (2.126) one infers (by using (2.50) with S / r d replaced by 2 ; ) that the func­ tions Sk(Vm) are X-independent for k
180 3. QUANTUM SOLITON SYSTEMS 3A. QUANTUM iNTEGRABiLrrY AND THE SOLTTON PROPERTY

A Hamiltonian on a symplectic manifold does not have any (nontrivial) integrals, in general. Therefore, the notion of Liouville integrability is a restrictive and hence useful one in the context of general classical Hamiltonians. In contrast, a self-adjoint Hamiltonian on a Hilbert space % always has nontrivial integrals (assum­ ing its spectral multiplicity is greater than one). Indeed, this is an obvious consequence of the spectral theorem. Therefore, no abstract analog of Liouville integrability exists for general quantum Hamil­ tonians. However, starting from a concrete Liouville integrable system H(q,$), St(q,B),... ,SK(q,8), the question whether a quantization of these Hamiltonians exists such that they still commute is a welldefined one. (Here and below, 'quantization' will mean the substitution »-»» = -»ftV,,

(3.1)

where h denotes Planck's constant.) Whenever the question has an affirmative answer, we shall refer to the operators thus obtained as defining a quantum integrable system. In this sense, all of the sys­ tems discussed in Chapter 2 turn out to admit integrable quantum versions. Next, recall that the notion of Liouville integrability is useless for classical Hamiltonians describing systems of repelling particles on the line, it being always satisfied under mild conditions on the forces. Therefore, we singled out the systems studied in Chapter 2 by first introducing the notion of pure soliton system, cf. Section 2A. In the same physical context, this notion has a quantum analog. Specifically, let us assume that the dynamics is sufficiently repulsive for the wave operators [10] S ± : %* = L2(G-,dN0)-+X=L2(G,df'q),

G-.GCR*

(3.2)

to exist and be isometric from 30* onto X Then we shall say that the dynamics gives rise to a pure soliton system when the S-operator S = S;'£_

(3.3)

conserves momenta. (This notion is worked out in more detail in [12].) Observe that the requirement is only useful when more than two particles are involved (just as in the classical case). Note also that we are not requiring that a quantum pure soliton system be a quantum integrable system arising from a classical pure soliton system (this would exclude examples that will be discussed in Section 4C). However, the term 'quantum integrable system' will be used for any quantum pure soliton system, whether it has a classical version or not. As already mentioned, the classical pure soliton systems H,SU ... ,SN of Section 2B can be quan­ tized in such a fashion that the commutativity property is preserved. Except for the VIrd case, the commuting operators are (formally) self-adjoint on L2(G,d"q), where G denotes the classical configuration space. Under such circumstances one expects to obtain quantum pure soliton systems as just defined. To explain why it is plausible that the soliton property survives quantization whenever the classical Hamiltonians admit commuting, self-adjoint quantizations H, S\, . . . ,SN, let us first note that the spectral theorem guarantees the existence of an isometry S that simultaneously diagonalizes the quan­ tum Hamiltonians on an L2-space %. On physical grounds one expects to be able to choose X = L2(G,dN8),

(3.4)

where G denotes the definition domain of the classical action variables. Also, the kernel of S should be a joint eigenfunction E(q,S), qeG, OeG. Taking interparticle distances to oo, the operators Si, . .j ,SN reduce to the symmetric functions of the operators tt,. . . ,6N and exp(y80|), . . . , exp (fi$n) for the nonrelativistic and relativistic systems, resp. Thus, one expects E(q,8) to have plane-

181 wave asymptotics. A consideration of the long-time asymptotics then leads to the expectation that the momenta are conserved. In a similar vein, one can argue that the spatial asymptotics of E(q,8) should factorize into two-particle quantities, and hence the S-operator should factorize too, as a quantum analog of the factorized scattering described in Subsection 2B4. (For more details on these heuristics, see [19].) Let us now sketch the contents of Section 3B in relation to the above notions. We first detail how the pure soliton systems of Section 2B can be quantized in such a fashion that commutativity is preserved. Then we shall try and describe to what extent the scenario of the previous paragraph has been realized for the resulting quantum integrable systems. Let us mention here that even for the sim­ plest case I,,, the scenario has not been completely filled in yet. Thus, it is not yet certain that the quantized pure soliton systems of Section 3B are quantum pure soliton systems as denned above. (To be sure, finite-dimensional quantum dynamics do exist for which the soliton property has been rigorously proved. We shall return to this in Section 4C.) In Section 3C we shall be concerned with quantum integrable versions of the systems discussed in Section 2C. 3B. QUANTIZED PURE SOLITON SYSTEMS

3B1. Preserving commutativity. The quantization substitution (3.1) leads to unambiguous, (formally) self-adjoint operators H on the Hilbert space L2(G,d"q) for the cases Inr.IIn, and VI,,,., cf. (2.18)(2.26). This also holds true for the symmetric functions of the Lax matrices (2.37) and (2.38). Indeed, one obtains sums of terms each of which is a product of commuting operators; then self-adjointness is clear from the fact that the classical functions are real-valued. In [5] it is proved that the operators thus obtained commute. The idea of the proof is to exploit classical results: One need only show that the extra terms (compared to the classical case) arising when partials are pushed through sum to zero, since the remaining expression is already known to vanish. Let us now turn to the relativistic cases. Here, ordering problems already occur for the dynamics H, cf. (2.27). We shall first consider the ll rel case, cf. (2.29). As can be seen from explicit calculations for N = 3, various 'obvious' choices for the ordering in H and S\, . . . ,SN (cf. (2.42)) spoil commuta­ tivity. In this connection it should be pointed out that no a priori argument is known from which the existence of an ordering choice entailing commutativity would follow. However, for the systems at hand such a choice does exist: For any h the operators

■s/=

<35>

2 n/-(%-fc)«p(02«/)n/+
jel

jel ktl

commute when one sets yl(f) = sh^^+zyslrjw,

z=\ifyg

This claim can be shown to follow from the functional equations V rrsh(x,-^-T)sh(JC,-^+y-p) tell ...Jf) hi sb(xt-Xj)sh(Xi-Xj-p)

(3.6)

= 0

(3.7)

which hold for any W > 1 , * e { l , . . . ,N), xeCN, y.peC [2]. Classical commutativity can now be obtained as a corollary of quantum commutativity. Indeed, the functional equations (3.7) reduce to the functional equations expressing involutivity when one divides by p and takes p to 0 [1,2]. Moreover, quantum commutativity for the Il nr case may also be viewed as a corollary. To explain the latter statement, we introduce the commuting operators

Ak(P)=^(~f+' 1=0

N-k

S,(fi),

k = \,...,N

(3.8)

182 We have made the ^-dependence explicit, since we are going to consider the nonrelativistic limit /3-»0. (The formulas (2.49), (2.50) should be recalled at this point.) Thus, we expand Ak(fi) as a formal power series Ak(fi =--

2AP/r,

/8-»0

(3.9)

m=0

and calculate

4f == 0,

A\»- -Ifij

4 0 ) ==4'> ==o, 4

2)

(3.10)

=j

"2 4sbf^K?y- - * )

(3.11)

Here, the replacement of g2 by g(g —ft)as compared to the classical case (recall (2.21)) is due to con­ tributions arising when the partials in (3.S) act on the potentials. Next, we set 1

Hm = - j / i 2 A + - ^ ( g - / > ) M 2 2 ^fiqj-

(3.12) -qk)

and note that

^ M-

-A2(fi)-- --l?Hm +

0(f),

P-»0

(3.13)

Therefore, setting »t = min {/|i4jt'»7t0},

we may conclude that Hm commutes with Ak.

AV =

2

(3.14)

Jfc = l,. .,N Noting that

»,,■ •*...

g=o,

(3.15)

i, < ■ • ■ < i .

we infer nk*Zk. Now assume nk
* = !,.

14*W1==o,

k,l=l,.

.,N .,N

(3.16) (3.17)

However, the above arguments do not yield an important piece of information on Akk): it must be the k'h symmetric function of the quantized Lax matrix (2.37), with g replaced by \e(g — ft)]1'1 Indeed, as already mentioned, the latter operator commutes with Hm [5]. Subtracting A$ yields an operator commuting with Hm that must be zero by the above reasoning: The point is again that the difference consists of (2m//i)-periodic terms that vanish for \qj—qk\-^aa. Thus far we have restricted ourselves to the systems of type II. The relevant information for the type I systems now easily follows from the above by taking y. to 0. We proceed by considering the

183 quantization of the VI„, functions (2.48). Here, the ordering choice that 'works' is given by

s,= 7C(1 2

(3.18) I I / r t y + i - ^ e x p C S S * , ) I I / r ( * - ? / - .) If) lei jel Jml v\=i ;+<« j-Ul This can be proved either directly [3] or by reduction to the 11,^ case via the substitutions (2.24), (Z3S) and the limit «->0. In the same way as for the IL^ case, VI^ quantum commutativity now leads to commutativity for the classical VI^ and quantum VIm systems. The Iljd operators (3.5) are (formally) self-adjoint, provided zsi'R. In contrast, the VI^ operators (3.18) are not self-adjoint (except when hPy.=2irk, keZ). When one takes z=y+iir/2 with ye=R" in (3.5), self-adjointness is also violated. (Recall the VL^j case arises for y-»oo.) Thus, if a joint eigenfunction transform exists for the latter cases, it is not going to be isometric. From now on we use the convention h=\. Since g has the dimension of action, it should cause no surprise that the above operators turn out to have special properties for gelM. We note in particular that (3.5), (3.6) entail Sk =

2

«PtM, +

•' +*(.)].

£=1

(3.19)

Hence, the quantum IL^ systems ate free for z =-pfti. 3B1 the eigenfimction transform and duality. For special values of g (in particular Vi, 1 and 2) the commuting 1^/11^ operators can be related to the radial parts of the Laplace operators on zero/negative curvature Riemannian symmetric spaces with restricted root system AN-\. (More pre­ cisely, this is the case when the center-of-mass coordinate is omitted.) The VI„ systems can also be tied in with harmonic analysis on symmetric spaces. For the cases just mentioned the eigenfunctions are known explicitly and have been studied in great detail. (A lucid and nontechnical survey of results obtained in the symmetric space context can be found in [5]; systematic and detailed treatments are presented in [31-33].) In particular, for each per­ mutation a there exists a unique function £„ satisfying SkE.(q, 6) = E.(q, 6)

2

'h ' ' * h>

k

=l

N

(3.20)

/ , < • ■ « ,

£.(?. *) ~ exp(ty-«„),

qN«qn-i

(3.21)

■ •■ « ? i

where

». = IKm

(322>

W

Moreover, a suitable linear combination E(q, 9) of these AM joint eigenfunctions solves the Plancherel problem. That is, the operator S: X = Ll(G,d"8)-+X = L\G,dNq),

f~ fdN6E (•,»)/(»)

(3.23)

is an isometry from % onto % where G, C = (yel»'V<-■■<}'■}.

G =

(lor, Um) (VW

(3.24)

(cf. (2.18), (2.19), (2.51)). Of course, the case g = \ is trivial: Then (3.12) corresponds to the radial part of the Laplace-Beltrami operator on GL(N,C)/SU(N), and one obtains E.(q,0) = txpdqt.),

E = W ^ - r E . ,

g = \.

(3.25)

184 Recently, joint eigenfunctions for the IInr case with g taking arbitrary values have been constructed by Heckman and Opdam; moreover, their results apply to arbitrary root systems [34-37]. The above picture for the symmetric space values of g is not substantially modified, but the Plancherel formula ('orthogonality and completeness') has not been proven yet Let us now turn to the relativistic cases Irel, 11^ and VTre|. Here, the quest for explicit joint eigen­ functions leads to problems of a novel nature, most of which have not been solved to date. To explain some of the difficulties involved, and to sketch some of our results obtained thus far, we specialize to the 1^ and H^j cases with N =2. (Even for N = 2 we have no information on eigenfunc­ tions for the VI,,) case.) Then the problem is to find eigenfunctions for the operator H =

shy ( q - i ft?) shv<7

T,$

shvfa+ifa) shvg

+(fi^-p).

(3.26)

Here, we have introduced the formal translation operator (7VX?) = / ( ? - © .

feC

(3.27)

and the new parameter v = fi/Z

(3.28)

(When we would stick to p., various factors 2 would arise due to our use of asymmetric center-of-mass variables q = q,-q2,

9 = («,-« 2 )/2,

(3.29)

cf. also (3.5), (3.6) with N =2, l = \, k—l.) More in detail, for N =2 the key question is, whether a function E exists such that ■7jHE(q,0) = E(q,B)chfie

(3.30)

and such that the operator 00

(S/Xf) = jdBE(q, ff)f(6), /eC3°((0,oo)) o gives rise to an isometry between the Hilbert spaces %= L2Q0,aa),dff),

%= L2(i0,ao),dq).

(3.31)

(3.32)

There appear to be no results in the vast literature on eigenfunction expansions that have a direct bearing on this question. More generally, the (once very active) study of analytic difference equations (A &Es) and analytic difference operators (A AOs) (such as (3.30) and (3.26), resp.) seems to have been abandoned, by and large, before functional analysis and quantum mechanics really got off the ground. Indeed, the last full-fledged monograph devoted to Ad.Es appears to date from 1924 [38]. (More recent studies do exist, cf. e.g. [39-42]; however, just as in [38], die questions dealt with are of a quite different character.) To explain the key difference between a second-order ODE or discrete difference equation and the second-order A A£ (3.30), we recall that die solution space is two-dimensional in die former context, and that the Plancherel problem is solved by the Weyl-Kodaira-Titchmarsh dieory. In contrast, for the A AE (3.30) even the existence of any solution is a priori unclear. However, this problem can be solved by invoking the extensive lore gathered mostly in the 19th century. Specifically, solutions can be constructed by recursive procedures [38]. Unfortunately, such solutions are often badly singular; even natural boundaries can easily occur. But the existence of even one non-trivial solution E(q, 0) entails the existence of an infinite-dimensional solution space. Indeed, for any F(q, 8) with period i'/5 in q the function F(q, 8)E(q, 9) is another solution. As a more restrictive fact, we may mention mat

185 the solution space is a two-dimensional vector space over the field of i/3-periodic functions [38], but all of these results fall far short of settling the key question mentioned above. A closely related question is: How can one turn the A AO (3.26) into a self-adjoint operator on X? Again, we have no general answers to offer. Recall in this connection that the operator —id/dq on the dense domain Co°((0,oo))C3C is the standard example of a symmetric operator without self-adjoint extensions. Thus, even for the simple A AO Tlfi = exp(-iPd/dq)

/3e(0,oo)

(3.33)

an interpretation as a self-adjoint operator on % is problematic. Having provided some context, we are now prepared to present some explicit answers to the above question for the I„i and 11^ cases. First of all, it so happens that the answer for the N = 2 Ird case (obtained by taking v—>0 in (3.26)) can already be found in the literature (once one realizes where to look). To substantiate this assertion, we invoke the (known) eigenfunction transform for the II,,, Hamiltonian

-^r-p-gts-u-1 Ae-^+ftfr-D^.

lA

(3-34)

whose kernel reads

E{fi,g;q,8) = 2 * + ! rte+-j)-' iF^ig

T(-iq/P)T(iq/ft)

+ iq/fl),"J(g -iq/P), g + ^ ; ~sh2/W),

q,6>0.

(3.35)

The crux is, that the contiguous relations for the hypergeometric function and the A A£ T(l +z)=zT(z) entail that the function (3.35) satisfies the_/4 A£ (3.30). Moreover, for g>Vi the opera­ tor 6 defined by (3.31) extends to an isometry from % onto % by virtue of the Weyl-KodairaTitchmarsh theory. Therefore, the above key question has an affirmative and explicit answer for the N=2 I rf case. We conjecture that the N > 2 existence question for the Irel regime has already been answered in the literature too. Specifically, we expect that the 11^ eigenfuncuons of Heckman and Opdam [34,35] are joint eigenfunctions for the commuting Ird operators, acting on the spectral variables. More in detail, this should hold after an obvious similarity transformation turning the Lebesgue measure used here into the Plancherel measure used in harmonic analysis, and after a suitable normalization of the dependence on the spectral variables. (Also, we should repeat at this point that the Plancherel prob­ lem has only been solved for the group values of g.) Of course, what is being said here, is that we expect the duality properties of die classical level to survive quantization, cf. (2.53). For the arbitrary N IM case this conjecture can easily be verified for the group values of g: It amounts to the fact that E(q, 8)=E(0,q). Furthermore, for N =2 this selfduality property is evident without restriction on g. Indeed, then the desired isometry is in essence the Hankel transform, whose kernel depends only on the product qB. Similarly, we expect the Ilrcl transform to satisfy E(fl,v,g ;q, 9) = £(v, P,g ;6,q\

(3.36)

This holds true for all cases where we have found explicit solutions. In particular, it holds when g

= A/ + l = l,2,3, •••

(3.37)

For these g-values the following functions solve the A A£ (3.30) and have the self-duality property (3.36): F„(q, 0) = \A(vq)A {fi0)\tXf{iqe-Mvq -M0$) 2 ( - ) * +,Guexp(2fcvg +2108)

(3.38)

186 Here, the function A (x) and the matrix Qkl depend on v and ft only via their product a = 0v

(3.39)

Hence, self-duality amounts to Qu being symmetric. Explicitly, one has M

(3-40)

A(x)= II[2sh(x + ij <*T' /-I

Qu = exp(i aM(M +1)/2)

2

exp( - 2i o[i, + • • • +i*])

K i , < ■■•
2

expC-2,a[/-, + ■ ■ ■ +y,D

(3-41)

-M<jt< ■ • • <j,<M j.t{-M+k,...,-l+k,k)

(Here, empty sums equal 1 by definition, so that F0(q, ff)=exp(iqtf), e.g.) We have not found a mani­ festly symmetric formula for Qw. However, it is easy to check symmetry for small M, and we shall prove symmetry for arbitrary M elsewhere, as well as the claim that FM(q, 8) solves the A A£ (3.30). _ We shall also omit the proofs of the assertions that now follow [43]. First, not only FM, but also FM solves (3.30). Second, the real-valued kernel EM(q, 8) = O r * ( - , r + l [ F M ( q ,

ff)-(-fFM(q,

ff)},

MelM

(3.42)

defines an isometry &M from % onto % for any ae(0,ir/M], thus solving the Plancherel problem posed above, cf. (3.30)-(3.32). Third, the operator +i

(<5Mf)(.q) = Q.'n)-\-if

jdeFM(q,ff)f(ff),

/eL 2 (R),

MelM"

(3.43)

— oo

preserves parity and amounts to SM on the subspace L2(R)a of odd functions. However, 9j, is not isometric on L2(R)S, even though the even kernel has properties similar to the odd one: It has in essence the same plane-wave asymptotics for \q\—>oo, is real-analytic on R, and solves the self-adjoint A A£ (3.30). (In fact, its deviation from being an isometry can be explicitly described.) 3B3. Sotiton scattering. As mentioned in the previous subsection, a linear combination E(q, ff) of the functions (3.21) solves the Plancherel problem for the group values of g. Specifically, one may take

£(,,*) = o r ™ 2 (-r oeSH

n

i
«&*-•>»* n

i<j »"'(0>»"'0)

utiw-»,))-»£„(
0.44)

where u(6) is the two-particle S-malrix,

u(8) =

expMl-j)] TO+2iff/ti)T(g-2iff/ii) r(l-2i»//i)r(g+2/'»//i) J & ^

exp[4,(lnM)#/M]

(I-r) flU

(3.45)

(VU

More generally, for any g>Vi the Heckman/Opdam function with the same structure is the obvious candidate for an isometry from % onto X Reading off the S-matrix from (3.44) according to the stan­ dard recipe of formal scattering theory, one finds that 5 is the unitary multiplication operator S(f)=

II

^%-ffj))

(3.46)

To show that the !„,, \\m and VI,,, systems are quantum pure sob ton systems as defined in Section

187 3A, one would have to prove that the wave operators S± exist, that they have range % and that S =£+'(£_ is given by (3.46). We expect that this is true for any g>Vi, the kernels of i± being given by the incoming and outgoing eigenfunctions EUa.ff) = f ( o . M ( r "

(3.47)

W11C11 U11G UULG3 d » WJUipOllMJU UL»C1 AIU1 LUC gCUCJOUZCU S111G 11IU1MU1II1 I

J.^J).

Let us now turn to die relativistic cases L^ and IL^, taking N=2 from now on. Using the known asymptotics of the T-function and the hypergeometric function, one obtains from (3.35) u(tf) = exp[ii7(l-s)]

(Ird)

(3.48)

(!!„,)

(3.49)

Continuing with the II„j case, we obtain

sh(fi0+ika)

u(9) =

k = \ sh(yW-1* a) '

g=M + \

from the explicit transforms (3.38H3-42). We can also determine u for arbitrary g, provided some assumptions are made. Suppose E(q, 0) is a solution to (3.30) with asymptotics E(q,0) ~ (2 W )- 14 [u(») ,4 exp(i ? ff)- U (fl)- l4 exp(-i g #)],

q->ao

(3.50)

If we assume in addition that E is self-dual (i.e., that (3.36) holds true), then it follows that E satisfies an A A£ inff,too. Specifically, one must have

o.5n

\HE(q, 9) = chvqE(q, ff) where H is the A AC dual to (3.26), H

=/_(tf)f jv / + («)+/ + (ff)7'_ /v /_(ff)

/.•-hs*

(3.52) (3.53)

Let us now consider (3.51) for q-*oo, using the asymptotics (3.50). Comparing leading terms, it follows that u (ff) satisfies the A &£ u(0+-pv) = F(ff)

(i.S4)

u(0—Viv) where F(8) =

A(6-\tv)f-(ff+\iv)

(3.55)

The point of all this is, that first-order A A£ of the form (3.54) can be solved explicitly for large classes of right-hand sides. Moreover, the solution is unique when certain analyticity requirements are imposed. In particular, for the special case just encountered, we find [44] that the solution satisfying u(Q)=l isravenbv

■c-tfOff *£,* " g ^ n

(3.56)

provided the parameters a = 0v=-jfti,

T = ae = — K

(3.57)

188 belong to R = {(«,T)eR2|oe(0,2ir], T—joe[0,ir]}

(3.58;

The S-matrix (3.S6) has various remarkable properties that do not meet the eye. For instance, it is an elementary function for a dense set fyCR. In particular, one has u(B)-

1

'

M <MfiO-ika) f-J

shM/r+i/irVa) chOrA/v — 111? /ml

(3.59)

for points in the set <5, = Ua.T)eRW = (M + Vm-Lii. M>\.

£>0)

(3.60)

which is already dense. For (a,T)e3)Dl3)1 there exist elementary self-dual solutions to the A A£ (3.30) with the asymptotics (3.50), (3.56). However, the corresponding transforms are not isometric, in gen­ eral. To date, we have not been able to prove or disprove the existence of self-dual periodic multi­ pliers that correct this. On the other hand, a natural generalization of the Harish-Chandra c-function does exist for any (a,r)eR [44]. 3C. SYSTEMS RELATED TO QUANTIZED PURE SOLITON SYSTEMS

3C1. Systems with sotitons and antisolitons. The operators associated with the II,,, and llrel cases are obtained from the commuting operators of the 11^ and 11^ cases via the substitution (2.78). Since this amounts to an analytic continuation, it is obvious that the former^ operators also commute (as for­ mal PDOs and A A0s, resp.). For special values of g the commuting II„r operators can be tied in with harmonic analysis on pseudo-Riemannian symmetric spaces. However, even in these special cases much less is known concerning the joint eigenfunctions than for the Ilg, case. In particular, one is still far removed from a verification of the picture of factorized scattering pertaining to the 11^ systems. This picture is taken for granted in the physics literature (cf. e.g. [45]), and will now be summarized. First, recall we have already discussed the corresponding classical situation in Subsection 2C1. The multi-channel scattering described there is believed to persist at the quantum level (for g > l ) . How­ ever, in any channel with asymptotically free solitons and antisolitons (from now on s and 7, resp.) there occurs a key difference with the classical case: an si collison leads to a non-zero reflection for gelM. Because it is still assumed that a multi-body scattering amplitude can be written as a product of 2-body amplitudes corresponding to an arbitrarily chosen temporal order for the 2-body collisons involved, the product must be independent of this choice. Consequently, one obtains constraint equa­ tions for the ss and 77 reflection coefficient u and the si transmission and reflection coefficients t and r. These cubic equations are the well-known Yang-Baxter equations, and they are indeed satisfied for the functions uj and r of the N =2 case. This is a conseauence of the relations r(#) =

sh(ir«/v) sh(i?re— itS/v)

"(*)■

e>o

(3.61)

r(9) =

miirg) sb(ivg—v8/v)

u(B),

e>o

(3.62)

which can be verified from the 55 eigenfunction transform. (The latter is obtained by taking a suitable linear combination of E(v,g;0,q±iv/2y), cf. (3.35).) Consider now, more generally, the 11^ case. We expect that the picture just sketched for the n ^ case applies here, too, with t,r and u still related via (3.61) and (3.62). In particular, for the solutions detailed in Subsection 3B2 we have g eN, so that (3.62) says r =0. This is indeed the case for the si eigenfunction transforms following from (3.38H341), The physical fact that the interaction between s and 7 is attractive for g > 1 finds its mathematical expression in the occurrence of bound states. For the II„. operator

189

H*=

-v2g(g-\)/ch2vq

~ -JY dq

(3.63)

these have energies E„ = ~v2\g - ( » + l)f,

n =0

Jf - 1 ,

ge(M,M +1]

(3.64)

n =0,. . . , M - 1 ,

g e(ilf,M +1]

(3.65)

and the corresponding wave functions read *„(?) = P„(/shv9>to(9),

Here, P„ are Gegenbauer polynomials, and the ground state may be taken to be Uq)

= [2chv9]'-s

(3.66)

Since one has ^(?)~exp[v|9|(l-^)],

tat-

(3.67)

the functions >l>„(q) are indeed in L2(R). For the IIrel Hamiltonian chvfa-ife) chvf

T„

chvfa+ife) chv^

+<0-*-0>

(3.68)

the odd and even eigenfunction transforms can be found explicitly for g = M + 1 eN via analytic con­ tinuation of the solutions (3.38), and their isometry properties are known [43]. There are M bound states and the ground state wave function can be taken to be
(3.69)

/=]

More generally, explicit square-integrable and pairwise orthogonal eigenfunctions of the formal A A0 (3.68) can be found for g > 1. The ground state, normalized to satisfy (3.67), reads [44] sh(T—a)xchTxcos2sv<7 «(T—a) <M?) = exp[(a-T)/ir+ fdx (3.70) 2 xshirxshax irsh ax o and then the excited states are given by (3.65), where the P„(x) ate again polynomials of degree n and parity (—f. (These polynomials are in essence ^-Gegenbauer polynomials, cf. Subsection 3C2 below.) As the generalization of (3.64), we obtain the eigenvalues 2cos(T-(n+l)a),

n =0,. . . ,M-1,

gt(M,M + 1],

r=ag
(3.71)

The function (3.70) is an elementary function for a dense set in a region obtained by shifting (3.58). In particular, when T=W/2 (3.70) can be written ip0(q) = (sh2vq/shirq/P)

T=IT/2

(3.72)

and when T=(M + l)a (3.70) reduces to (3.69) [44]. 3C2. Sutherland type systems and their duals. Just as for the llnr and IIrd cases, commutativity of the quantized Hamiltonians associated with the regimes III„, IIIrd and IIlre|> is clear from the fact that these operators can be obtained by analytic continuation from the II,,, and IIrel regimes, resp. We continue by sketching the state of the art as concerns their joint eigenfunctions. To this end it is expedient to discuss the N = 2 case first. Omitting the center-of-mass motion, the relevant operators and their duals read H =

-

dq1

+ v2g(g-l)/sm2vq

(Ulm)

(3.73)

190

it =■ H =

I

■if-- If-

''_, 11 - ^ ' ++((vv-_>»--v-v,)) T-,

H

(III-,)

W p T . fsiny( M«th±!M +!MP

.si sinv^ savq

ail™,)

(3.75)

-(v-»—v)

(rii-i)

(3.76)

' +CJ--0

(HI,,,,*)

(3.77)

(riw)

(3.78)

+ (fi^~P)

! anvq

p

I

. ^ rhfie+ «J ~ - > *- *B*P-P H =

ff =

H =

H =

shfie

shiB

UJ*7' - . r y f f '

s Jtvfa+fl nv(q+fe)Fr_ sinvo smvq

sinv^ s

up F

« 1 S I sin/W an5»

sinv( »y(q-fl?)

T

'

5

^

\i

vnfi$ sin sin«

(3.74)

+»-.-»

+

<—>

+(v->-v)

Here, we are taking flve(0,oo) in all cases, in contrast to Subsection 2C2. The change g1-*gig — \) in (3.73) (as compared to Subsection 2C2) is a natural consequence of the IHrei—►IHnr transition at the quantum level, cf. Subsection 3B1. Recall also that Te and T( denote translation of q and $, resp., cf. (327). For f 3»3/2 the onerator |\17T> is essentially self-adtnint on OPttOir/vWrV. where

X=

L2Q0,v/v],dq)

(3.79) 2

2

Its closure has purely discrete spectrum {v (g + n ) | n e \ j and a corresponding basis for % can be taken to be (3.80)

*■(«) ~ ^.(cos»?)*ii(?) where the rn are Uegenbauer polynomials with weight function

Mqf

= (smvqf*

(3.81)

Correspondingly, the dual space may be taken to be

% = l\G),

G = {vg +vn\n e N }

(3.82)

and the duality of the classical level is preserved. Indeed, when one discretizes the 0-variable in (3.74) by taking $eG, then the function E(q,0m) ==*„(?),

«„ = V£+vn

(3.83)

is an eigenfunction of H with eigenvalue 2cosv^ by virtue of the three-term recurrence relation of the (suitably normalized) Gegenbauer polynomials. For the III,,) regime this state of affairs persists, in essence. The dgenfunctions are again of the form (3.80); the weight function (3.81) and its associated orthogonal polynomials should be replaced by certain ^-analogs. Specifically, the ^-Gegenbauer polynomials studied in [46,47] arise in this way. (This was pointed out to the author by T. Koornwinder [48].) The parameters employed by Askey and Ismail [46] are related to ours by qju = exp(-2j8v),

flu

= exp(-2flvy),

A,, = g

(3.84)

Hence, the limit qxi-*l may be viewed as the nonrelabvistic limit c-»oo. Again, the duality of the ;lassical level survives quantization: The three-term recurrence relation for the (appropriately normalzed) ^-Gegenbauer polynomials implies that the generalization of (3.80) is an eigenfunction with agenvalue 2cosv^ for the operator (3.76), viewed as a discrete difference operator on the Hilbert space

(3.82).

191 An interpretation of the operators (3.77) and (3.78) as Hilbert space operators is less straightfor­ ward. We see only one sensible way to do this without loosing formal self-adjointness: Both % and % should be finite-dimensional (This can also be understood from the fact that the natural classical Mreu phase space is a bounded subset of R 2 " - 2 [8].) More in detail, it appears inevitable to impose a quantization constraint on the parameters /},v,ge(0,oo): They should satisfy 2r+la = it,

a = /3v,

r = ag,

/elM*

(3.85)

and then the Hilbert spaces are given by % = l\G),

X = l\G)

(3.86)

where vG = /8G = {T,T+a, . . . , T + / O } = M(1,T)

(3.87)

Indeed, restricting q and 8 to the points in G and G, resp., given by qm=Pg+Pm,

6n=vg+m,

m,n=0

/,

(3.88)

the operators (3.77) and (3.78) have a well-defined self-adjoint action on % and % resp. It so happens that the eigenfunctions of the operators just defined are again already known, in essence. Indeed, now one has

_!_ E(qmA)

= lw(yqm)w(fien)]' P„(cosv9m),

m,n =0, . . . ,/

(3.89)

*=0,...,/

(3.90)

where w is the weight function w SUIT

n

^'"ft"-

JJQ sm(/ + l)o

on M(I, T) and the P„ are corresponding orthogonal polynomials. These may be viewed as special cases of discrete ^-polynomials and corresponding weight functions obtained in [49]. Specifically, the relation of the Askey-Wilson parameters [49] to ours can be taken to be N-*l,

q-*exp(ia),

a =fc = - c = -d-»exp(/T-io/2)

(3.91)

and then one obtains />„0i(m))-»/7„(2exp(iT)cos(T-t-ma)) = CP„(COS(T+ma))

(3.92)

where c is a positive normalization constant chosen such that (3.89) defines an orthogonal matrix. The self-duality of the classical 111,4,6 regime is again preserved under quantization, since one has [49] P„(cos(T+mo.)) = Pm(cos(T+na))

(3.93)

For geM the desired eigenfunctions for the N =2 I1I„| and III,^ cases can also be obtained via analytic continuation of our explicit IIre| solutions (3.38)-(3.42). The representations of the abovementioned orthogonal polynomials that arise in this way are new. In this connection it is to be noted that the various representations for the f-Gegenbauer polynomials in terms of basic hypergeometric functions (in which the index n can be taken to be complex) do not admit a continuation to the II,d regime. The difficulty at issue here has already been discussed in Subsection 3B2, in another guise: One would have to factor off (unknown) periodic functions that obstruct analytic continuation. Next, let us briefly sketch what is known concerning arbitrary N eigenfunctions. First, the work of Heckman and Opdam [34-37] mentioned before yields (on specialization to -4/v-i) a complete solu­ tion for the III,,, case. The joint eigenfunctions can be written (in the center-of-mass system) as pro­ ducts of factorized weight functions and polynomials in N — 1 variables. The Plancherel problem is now much simpler than for the II„r case, since one is dealing with orthogonal polynomials. It has been solved (for arbitrary root systems) by Heckman [35].

192 The answers to (most of) the analogous questions for the Ul^ regime can be found in recent work by McDonald [50]. He also considers arbitrary root systems, and for AN-i his fundamental shift operator may be viewed as a transform of our operator S{ (with flj + • ■ ■ + qN=0). The structure of the joint eigenfunctions following from his work is the same as for the III,,, case. (His work and the connection to our commuting III^ operators were pointed out to us by T. Koornwinder [48].) As concerns eigenfunctions for the Ill^t case with 2<JV <W/T (recall (2.94)), there are only specu­ lations: We expect that they can be obtained via analytic continuation of McDonald's polynomials, and that they will turn out to be self-dual. Finally, we would like to describe some results by T. Koornwinder that have a bearing on the IIL^ eigenfunctions. First, he has proven [48] that they have the duality properties expected from the classi­ cal level, by means of an induction argument which applies to all root systems considered in [SO]. Second, he has tied in the g-Legendre polynomials (i.e., the ^-Gegenbauer polynomials with g = Vi) with Woronowicz' impressive work on compact quantum groups [51-53]. Specifically, he has shown [54] that they may be viewed as zonal spherical polynomials associated with Woronowicz' version of SqU(2) [51,52], as a natural ^-analog of the relation between Legendre polynomials and ■SI/(2)=:S| J/(2). However, the definition of 'spherical' is no longer unambiguous for qe(Q,\) [54]. It can be expected that McDonald's g = 'A AN-\ polynomials may be similarly tied in with harmonic analysis on SqU(N) [52,53]. 3C3. Elliptic systems. As already mentioned, there is no guarantee that one can find a quantization of a classical integrable system for which commutativity is preserved. However, no ordering ambiguities occur when one quantizes the symmetric functions of L(TVm\ cf. (2.96), and the resulting operators do commute. This is proven by Olshanetsky and Perelomov [5] in the same way as sketched above for the liar case. Next, let us consider the IV^ case. It so happens that there again exists a factorization of the potential yielding commuting operators Si Sf/. This factorization involves the Weierstrass ofunction. Specifically, if one replaces (3.6) by

f± (q) = a(q±y)/a(q),

y = ife

(3.94)

then the operators (3.5) commute. This follows from the fact that the functional equations (3.7) still hold true when sh is replaced by a [2]. The latter functional equations are the most general expression of complete integrability for all of the systems considered in this survey (except possibly for the quantum IV^ systems, cf. below). Indeed, involutivity for the classical IV,el case (and hence for all classical cases I-VI) finds its expres­ sion in a sequence of functional equations for the 9-function that arise when one divides the o-analog of (3.7) by p and sends p to 0. (In fact, this is the only known proof of Liouville integrability for the rVfd case, thus far.) Moreover, quantum commutativity for the relativistic cases follows by taking lim­ its (the Vrcl case will be detailed in the next subsection). This entails commutativity for the cases Um and Vlaj by virtue of the reasoning in Subsection 3B1. Then the cases I„ and III^ follow from II„. The Vm case will be dealt with in the next subsection. Unfortunately, we have no complete proof that commutativity for the quantum IV^. case may be obtained as a corollary. When one replaces S,(/J) in (3.8) by 2,(j8) (given by (2.101), (2.102)), then the difficulty is that the coefficients of the Tkl (cf. the reasoning below (3.15)) might be non-zero con­ stants, due to functional relations between the various functions involved. (Note that any such con­ stant must vanish when u or uf is taken to oo.) However, for k = 1,2,3 one does obtain /i* =k, cf. the explicit formulas (3.33)-(3-35) in [2], and we believe that, more generally, nk =fc for k*£N. If one could prove this conjecture, then one would obtain (3.16), (3.17). Note, however, that the k =2,3 formulas just cited entail that A^ and A^1 are not equal to the second and third symmetric function, resp., of the quantized Lax matrix (2.96) with g replaced by \g(g — l)]" 2 At this point we would like to repeat that the fundamental role of the functional equations (3.7) for the o-function pertains to the root system AN-y considered throughout this survey. However, the

193 obvious conjecture is now, that it should be possible to generalize the functional equations to other root systems, thus yielding quantum and classical integrable systems generalizing those considered in [4,5]. Finally, let us turn to explicit eigenfunctions. Since (3.19) still holds at the elliptic level (recall (3.94)), we shall take g^=Q,l. Then the eigenfunctions are only known for N = 2 and g =2,3,4, • ■ ■ Specifically, the relevant TV,,, operators read H„ = —jj+gfg-l^q), «=2,3, ••• (TVm) (3.95) dq Their eigenfunctions are the well-known Lame functions [28]. The relativistic A hOs generalizing Hm read H =

<*q)

T„

*«+'*) <*?)

+C8--0),

£ = 2,3, •••

(IVrf)

(3.96)

For these A AOs we have found explicit eigenfunctions reducing to the Lame functions in the nonrelativistic limit /)-»0. Details concerning these functions and their Hilbert space properties will appear elsewhere [55]. 3C4. Periodic Toda type systems. When one quantizes the Hamiltonian (2.106) and the symmetric functions of the Lax matrix (2.107) associated with the periodic Toda systems V„, then the resulting operators commute. This can be proven in the same way as for the case IIM [5]. For the relativistic case V^ one encounters the same ordering ambiguities as for the case VIre]. However, the same ord­ ering choice as made in (3.18) (but now with (2.109), (2.110) in force) yields commuting operators H,S\ SN. This can be proven directly [3], but it is also possible to obtain this as a corollary of the quantum IV^ commutativity. Indeed, we may replace the a -functions in the quantum IVrd opera­ tor S/ (given by (3.5), (3.94)) by the rhs of (2.111) with the first exponential omitted, since these exponentials only give rise to an overall multiplicative constant in S,. If we then substitute (2.112) and (2.118), and take u—>oo, we obtain the V„, operator (3.18); The 'VIrel part' comes from the sh-factor at the rhs of (2.111), whereas the infinite product supplies the extra terms needed to turn the VIrel into the V^ operator. In a similar fashion, the o-analogs of the functional equations (3.7) turn into the functional equations expressing V^ quantum commutativity [3]. When one substitutes (2.108) in the V^ operators and takes T to 0, then one obtains the Vlrel operators, as is easily verified. The Vm transition involves more work. When we start again from (3.8), with S/{fi) replaced by the Vrd operators derived from 2(> cf. (2.126), then the reasoning below (3.15) can only be followed til the point where interparticle distances are taken to oo. There is no analog of this for the V„, case, since the functions e, = exp[/i(9y— qj-\], j — l N, cannot simultaneously go to 0. Indeed, this is clear from the relation e, • • • en = 1. Even so, we can again obtain a contradiction, as follows. For nk where P is a polynomial without constant term, we must have P(X{,... ,Xi/)=^M_ Cj(x\ • ■ ' xN)l. But the minimal order in fl at which such contributions to Ak(fi) can arise equals IN. Since nk equals the fc-th symmetric function of the quantized Lax matrix (2.107). Indeed, the difference commutes with Hm [5] and consists of terms that arise when partials act on potentials. Therefore, the above arguments can be used once more. Finally, let us describe the state of the art as regards joint eigenfunctions. This is quite similar to the nonperiodic case: For Vre] one is dealing with non-self-adjoint operators and nothing is known,

194 whereas for Vm the dgenfunctions have been found and studied in considerable detail (by Goodman and Wallach [56]). 4. CONNECTIONS WITH INFINITE-DIMENSIONAL INTEGRABLE SYSTEMS 4A. PREAMBLE

In recent years it has been widely advertised that there are intimate relations between infinitedimensional integrable system theory and various subdisdplines of mathematics and physics that would appear to be rather far removed from this area at first sight. The latter include the representa­ tion theory of Kac-Moody and Virasoro algebras and generalizations thereof, soluble models in twodimensional classical and one-dimensional quantum statistical mechanics, quantum group theory, knot theory, conformally invariant field theories, string theories A recent 'flow chart' of the intercon­ nections between the various fields can be found in [57]. This flow chart is quite extensive and the fields it covers are currently investigated by large numbers of researchers. Our purpose in this chapter may be phrased as adding yet another box to the flow chart of [57], for which we propose the label 'Finite-dimensional Soliton Systems'. Some of the contents of this box have been sketched in the previous two chapters, but it should be repeated that we have restricted ourselves to the root system AN ,, absence of integrable external field couplings and internal degrees of freedom, and zero temperature. We shall continue to do so in this chapter. 4B. THE CLASSICAL LEVEL

The llrel systems are intimately connected to the pure soliton solutions of various soliton PDEs and infinite soliton lattices. The latter include the sine-Gordon equation, the /^''-reductions of the KP equation (yielding KdV, Boussinesq, . . . for n = 1,2, • • • ), the modified KdV equation, the infinite Toda lattice Similarly, solutions describing solitons, antisolitons and their bound states can be obtained from the Ilre] systems. The latter situation is discussed in [7]; here, we shall restrict ourselves to the pure soliton case [1,58]. We begin by recalling that we have already considered quite general one-parameter flows associated with the lire] systems, cf. (2.63H2.69). Here, we have occasion to study two-parameter flows. Thus, we now work with i(l,U;$,+fV<S,)-*ft 1 '(8t), ■ ■ ■ ,?* +
(4.1)

(where A0,A! are real-valued) instead of (2.69). If we then set # , = TrA,(lnL)

;=0,1

(4.2)

and q(t,x) = exp(tffo -xHftq,

SW

(4.3)

(where conf. denotes the projection on configuration space), it follows as before that the dgenvalues of (4.1) are given by expfai (<,*)], . . . , explqN(t,x)]. Therefore, one may conclude ln(det(l+i)) = 2ln(l+exp[ ?/ (<,x)D /=1

(4.4)

H

Tr(Arctg^) = SArctgtexp^r.aOD.

(4.5)

The crux is now, that the functions at the rhs or certain derivatives thereof are pure soliton solu­ tions to the above-mentioned soliton PDEs for an appropriate choice of A0 and A] and of the 'cou­ pling constant' z. (For infinite lattices one should take xsZ.) The details are spelled out in [1,58]. As a consequence of the formulas (4.4), (4.5) one may view soliton solutions to the abovementioned infinite-dimensional integrable systems as linear superpositions of single soliton terms.

195 Each of these terms gives rise to a uniquely determined space-time trajectory Xj(t), obtained via the requirement ty(f,x,<(r))=0. The asymptotics of these trajectories is the same as for the soliton solutions [58,7]. In this way an intuitive picture suggests itself of solitons being deformations of an elastic medium that conceals an underlying point particle dynamics. We proceed by discussing an issue that is relevant to the problem of quantizing those infinitedimensional dynamical systems whose particle-like solutions can be obtained from the TV-particle sys­ tems discussed in Chapter 2. This concerns Hamiltonian formulations of the Inverse Scattering Transform (1ST) for the former systems. Such formulations lead to soliton action-angle coordinates that can be compared to the variables q,8, cf. [1, Subsection 6D]. The result is, that the angle vari­ ables coincide (in essence) with qit . . . ,qN for the (m)KdV, sine-Gordon and Toda cases, whereas f j , . . . ,fg coincide with the soliton action variables only for the mKdV and sG cases. For the KdV and Toda cases the action variables of [59,60] and [61,62], resp., are given by Pj

= exp(20,)

pj = -2cth(fl,)

(KdV)

(4.6)

(Toda).

(4.7)

To specify and discuss the soliton 5-map for these two cases, it is expedient to first identify $2" with fi + ~ Q in the 11^ equations (2.74)-(2.77) (via reversal of ordering). Moreover, we put /?=/»= 1 in the latter and denote the 5-map o n f l " ~ f l thus obtained by S(z). Then the soliton 5-map for the KdV and Toda cases (with the parametrizations (4.6) and (4.7), resp., in force) is given by S(iv/2), cf. [1]. Now one readily verifies that S(z) is canonical w.r.t. a symplectic form

£=24 A *(«/)

<4-8)

if and only if p(tf) is linear in 0. In particular, the KdV and Toda JV-soliton 5-map S(iir/2) is not canonical when one employs cjj and pj (given by (4.6) and (4.7), resp.) as canonical coordinates on the scattering data. (In the Toda case the y'-th angle variable used in [61,62] is not equal to qj, but the difference depends only on #,, so our arguments do apply.) The violation of canonicity just established may not look startling at first sight. However, it does appear bizarre when viewed from another perspective. Indeed, one would be inclined to expect that the asymptotics of a Hamiltonian (and hence canonical) flow is coded in an 5-map that is also canon­ ical. (On a personal note we might add that we were very puzzled when we noticed the non-canonicity of the KdV soliton 5-map some ten years ago. When we asked H. Segur for advice, he was not puz­ zled: He simply did not believe us!) We would like to clarify this issue here, since it has a bearing on the soliton sa particle correspon­ dences sketched above. (We should mention at this point that the non-canonicity of soliton 5-maps for the KdV, Toda and finite-density nonlinear Schrodinger (NLS) cases has been observed and dis­ cussed in [62,63]. However, we feel that both the diagnosis and the remedy presented in [63] are far from compelling.) First and foremost, it should be recalled that neither the 1ST nor the soliton 5-map involve any Poisson or symplectic structures. Such a structure is needed only when one wishes to write the non­ linear evolution in Hamiltonian form. More specifically, the equation of motion gives rise to a vector field X on the manifold Q of Cauchy initial values, and the structure then ensures that X may be viewed as the vector field associated with a Hamiltonian H on fi. Now the direct transformation <S> from Q onto the manifold Q of scattering data transforms X into a vector field X on fi, whose flow is linear on one half of the variables coordinatizing Q and trivial on the other half In such cir­ cumstances there are uncountably many different symplectic forms a on £2 such that X is the vector field arising via u from a Hamiltonian HonQ.A finite-dimensional example (which encodes the KdV JV-soliton situation) may be in order: With it equal to (2.51) one may take X=

2e*P<3»./)9i> y=i

«=2/v«/)4AAy=i

" = 2

/<*
y = i-oo

(4.9)

196 for any nonvanishing/such that the integral makes sense. Of course, one wants u and H to be smooth in a suitable sense, but in the absence of a clear picture of the topology of the scattering data manifold Q it is impossible to pin down just what restrictions this entails. In fact, to proceed we shall also assume that 9 is 'smooth'. (Again, this has not been sorted out with the standards of rigor that are taken for granted in modern global analysis, as far as we know.) Accepting this, it is obvious that the pullbacks under * of the above-mentioned forms a and Hamiltonians H give rise to symplectic forms u on Q (since 'd commutes with pullback') and Hamiltonians H that all lead to the same vector field X on U one started with. The upshot is, that there exist uncountable many different Hamiltonian formulations of the 1ST. Let us now explain why the soliton S-map need not be canonical for any of these formulations, a priori. This becomes evident when soliton scattering theory is formulated in a mathematically precise way. Such a formulation should involve a comparison map J: Q-»Q that identifies a point in Q with N bound states and vanishing reflection with a function in Q that equals a linear superposition of N one-soliton functions depending on the N energies and norming constants. (Cf. [12, Subsection 2F] for more details on this picture.) Since one is comparing two very different symplectic manifolds (intertwined by the 1ST), it is in fact extremely optimistic to expect that J will be asymptotically canonical, in the sense that the wave transformations in the two-space scattering theory picture [10,12] not only exist, but are also canonical. However, if the wave maps are not canonical, then S is not likely to be canonical. On the other hand, whenever one succeeds in parametrizing soliton solutions with angle parameters q],. . . ,qN and action parameters 0,,. . . ,6N in such a way that the soliton S-map is given by S(z), —ize(0,>r/2], then the S-map is canonical w.r.t. the symplectic form (4.8), provided one takes p(0) linear in 6. Then the soliton part of the Hamiltonian is (in essence) uniquely determined (cf. (4.9) for the KdV case). Of course, the radiation should be taken into account as well, but it so happens that this can be done in a very natural way for the cases at hand. Indeed, let us show thisfirstfor the KdV case, where we need

» = i 2/>j /2= T 2 > P ( 3 » / )


( 41 °)

when we use the form (2.S2), cf. (4.9). This corresponds to the soliton part of the Hamiltonian H =\fdxuHx)

(KdV)

(4.11)

on Q, i.e., the Hamiltonian —I\ in the well-known hierarchy. Therefore, one gets a smooth extension of H to all of Q by picking it equal to —/|°S, where S = * ~ ' denotes the 1ST. The requirement that the Hamiltonian H thus obtained generate the KdV evolution of the reflection coefficient (as found via the direct transformation) now fixes the symplectic form on the radiation, in the following sense. When we keep the customary angle variables
{
*=<J.,p, x=q,8.

(4.12)

Proceeding in this way, we find p\k) = (4k2)~'p(k)

(KdV)

(4.13)

when p(k) and ip(k) are given by [63, Eq. (8)]. With the symplectic form thus fixed, one readily verifies that the complete S-map (soli tons + radiation), as specified in [63, Eqs. (21), (22)], is canonical. (Note that the qk employed in I.e. corresponds to —qk/2.) Furthermore, the higher flows in the hierarchy are now generated by —/i,-i instead of J'a. + i (f° r n > 0 ) , whereas the Zakharov-Faddeev momentum Hamiltonian I) should be replaced by

197 P = - \ fdxu(x)

(4.14)

(Here, the sign conforms to the sign in the KdV equation u — (>uux + u,*, = 0 employed in most of the literature.) The latter facts make it very plausible that the new symplectic structure on $2 just defined coincides with the 'second Hamiltonian structure' introduced by Magri [64,65] (up to a sign that corresponds to the different sign of our angle variables qt compared to the angle variables used in the literature). The reason that we are not sure that (minus) Magri's structure results is that we are requiring that soli ton and radiation variables commute, cf. (4.12). To our knowledge, the relations (4.12) have not been pro­ ven for Magri's structure; possibly, terms propotional to S(k) are present. Such terms must occur for the 'first Hamiltonian structure' (Gardner's structure [66]), since otherwise blatant contradictions arise [67]. (We feel that the presence of such terms is a highly undesirable feature of the Gardner choice.) We can see only two drawbacks of the new symplectic structure: It is far less 'obvious' than Gardner's structure (even when it does equal Magri's structure up to sign), and it is not likely to admit an r-matrix formulation, in contrast to Gardner's structure (cf. [62, p.467]). However, in our opinion these liabilities of the new structure are negligible compared to its assets: (i) By definition, it decouples soliton and radiation modes; (ii) It gives rise to a canonical scattering map; (iii) It obviates certain unphysical characteristics of the energy and momentum Hamiltonians associated with the Gardner choice. To substantiate the latter claim, we point out that the Gardner energy (momentum) has opposite (the same) sign for radiation and soliton solutions. In contrast, the energy (4.11) is always positive (as expected for any disturbance), and the momentum (4.14) is negative/positive for radiation/ soli tons (as should be the case, since radiation/soliton solutions move from right to left/left toright[68]). Next, let us consider the Toda case. When one combines [62, p. 504, Eq. (4.42)] with the parametrization Zj = th($j/2) corresponding to [1, Eq. (6.17)], then one obtains qj = 2/sbSj, cf. also [1, Eqs. (6.6), (6.18)]. Thus, the Hamiltonian corresponding to the choice (2.52) reads H = 21n(zj)=-2Mcth2-j#/) j=\

(Toda).

(4.15)

j =\

Again, there is an obvious way to extend to the radiation: If one compares with [62, p. 504, Eq. (4.36)], one sees that H may be viewed as the transform of the 'total elongation' functional H = - lim q„

(Toda).

(4.16)

n-*ao

Therefore, we can proceed just as for the KdV case: We require (4.12) and then take ~p(8) = (2sin2fl)-'p(fl)

(Toda)

(4.17)

where p(ff) is given by [62, p. 503, Eq. (4.28)]. This ensures that <J<0) satisfies <(>=2sinS, as desired (cf. [62, p. 504, Eq. (4.41)]). There appears to be no obvious candidate for the new symplectic structure on Q that corresponds to the symplectic structure on Q just defined. Admittedly, the new structure and the corresponding Hamiltonian (4.16) are not exactly the obvi­ ous ones for the Toda lattice. Possibly, the situation can be better understood by answering the analo­ gous questions for the relathistic Toda lattice, and then taking the speed of light to infinity. Again, there are an 'obvious' Hamiltonian and symplectic structure that arise from viewing die lattice as the JV-»oo limit of the V^ (or VL^) systems, cf. Chapter 2. Moreover, an 1ST formulation and JV-soliton solutions are known [29]. However, the soliton 5-map has not been determined yet. As we shall argue in the next section, the quantized nonlinear Schrodinger equation (NLS) may be viewed (in more than one way) as a degenerate case of the quantized particle systems of Chapter 3. We suspect that the classical NLS breathers (i.e., the sohtons in the attractive and rapidly decreasing case [62]) may be tied in with degenerate particle systems, too. However, their explicit space-time dependence (as specified e.g. in [62, p. 132, Eq. (5.38)]) appears incompatible with (4.4) or (4.5). This

198 also holds true for the repulsivefinite-densityNLS soliton solutions detailed in [62, p. 170, Eq. (8.33)]. On the other hand, we have found a clearcut connection of the latter solitons to the II„i systems at the level of the S-map. Just as for the KdV and Toda cases, the soliton S-map is not canonical wj-.t. the symplectic structure arising via the r-matrix formulation (cf. [62, p. 266]). However, this can again be remedied by changing the symplectic form on Q. Indeed, as the analog of the reparametrizations (4.6), (4.7) one now needs Pj = ^Hiij

(f.d.NLS>

(4.18)

Then the soliton S-map is again given by S(iir/2) (as is readily verified from [62, p. 266]), and hence is canonical w.r.t. the symplectic form (2.52). In the same way as for the KdV and Toda cases we now infer that we should take H=

2<^" 2 -« 2 /'))" 2 = 2 ^ / c h « ,

(f.d.NLS)

(4.19)

as new soliton Hamiltpnian, cf. [62, p. 264, Eqs. (9.100), (9.101)]. Comparing to [62, p. 2S8, Eq. (9.64)], it follows that H can be smoothly extended to radiation by taking H = -«*/,,„

(f.d NLS)

(4.20)

as the analog of (4.11) and (4.16). When we now insist on (4.12), then we need p(X) = «o(« 2 -X 2 r , p(X)

(f.dNLS)

(4.21)

with p(X) and
199 general When appropriately parametrized, pure soliton solutions to completely different-looking evo­ lution equations (including sG, mKdV, KdV, Toda, finite-density NLS, . . . ) all yield the S-map S(i'v/2) of the hyperbolic 11^, systems, whereas the A<„''-reductions of the KP hierarchy lead to the S-tnaps S(in/(n +1)) [58]. Moreover, the full panoply of particle-like solutions to the sG and mKdV equations can be modeled when one includes the fi,^ (z =nr/2) particle systems [7]. In point of fact, there is also strong evidence that the particle-like solutions to the fully anisotropic Landau-Lifshitz equation lead to an S-map that occurs in our systems. The systems just referred to are the systems dual to the elliptic systems of Chapter 2, with y equal to <■>'. (Note this value is the elliptic analog of the value z =i>/2); In the absence of an explicit action-angle map for the IV^ case, we do not know what these systems look like. This is a matter of considerable interest, and even more so when the quantum situation is taken into account. The evidence we have in mind is the fact that it is possible to parametrize the particle-like LandauLifshitz solutions (as detailed in [69]) in such a way that the function e n of [69] can be written C12 = ^k2[9(2iK'; K, HK')-^

-02; K, 2iJf)]-'.

(4.22)

Specifically, defining the elliptic modulus by

k = (l-b2/a2f,

a>b>0

(4.23)

and substituting kj = -adnffij.k)

(XYZ-»XXZ)

(4.24)

kj = -bnd(0j,k)

(XYZ->sG)

(4.25)

or

in C)2, one can verify that (4.22) holds true. Of course, these two substitutions must then be linearly related, and indeed one can get from (4.24) to (4.25) by shifting 6t over K. However, both are useful: One has lim c,j = t h ^ i ~h),

(426)

but the limit b-*0 can only be taken in the complete solution [69, Eq. (3.12)] when one employs (4.24). This yields one of the two partially anisotropic cases, viz., the case where the two equal cou­ plings are greater than the third one. To handle the 6-»0 limit when (4.25) is used, one must rescale x and / by a factor b, and then one can reach the particle-like sG solutions by proceeding as indicated in [62, pp. 459-460, Eqs. (8.15H8.19)]. It remains to explain the connection of (4.22) to soliton and particle S-maps. On the soliton side, the S-map seems not to be available in the literature. (We have not found any specification of soliton action-angle variables, either.) However, from [69, Eq. (3.12)] it is very plausible that the shifts of the soliton position parameters 2, = R e { ? + ~ j l n ( n < »

(4-27)

**/ are factorized, with two-soliton shift given by ln(l/ci 2 ) (up to functions of the form fj(8j)). On the particle side, asymptotics as just described result from the eigenvalue asymptotics of LQVriMqi

+'"i

qN+ta„),

aN< ■ ■ ■
|<|->oo

(4.28)

in the same way as sketched in Subsection 2B4. The point is, that if one takes the spectral parameter X equal to B + U ' , then one can show that a diagonal similarity transformation turns L(IVrd) into a positive matrix. Therefore, the asymptotics follows from [6, Th. A2] by using the generalized Cauchy identities of [2], yielding factorized shifts as just described.

200 4C. THE QUANTUM LEVEL

The particle picture of solitons that emerges from the considerations in the previous section is surpris­ ing and intriguing at the level of classical field theory, but not more than that from the viewpoint of physical applications. For instance, a solitary water wave is vastly different from a point particle. However, at the level of quantum field theory the physical status of a soliton-particle correspondence is quite different. To see this, recall that the physical content of a (temperature and density zero) quantum field theory is completely determined by the S-operator and the bound-state spectrum. Quantum fields are auxiliary entities serving as a convenient vehicle to arrive at a theoretical understanding of experimen­ tal data concerning particle characteristics. We are not aware of alternative descriptions for quantum field theories in which particle annihilation and creation occurs. However, soliton quantum field theories such as the quantized sine-Gordon theory are characterized by an S-operator that conserves particle number. Therefore, any quantum dynamics that leads to the same 5-operator (and boundstate spectrum) is physically indistinguishable from the quantum field dynamics. We have worked out these considerations in more detail some ten years ago (cf. the Introduction of [13]). They served as motivation for the second part of [12], where we constructed relativistic particle dynamics leading to the S-operators of the Federbush and continuum Ising field theories. (To date, these are the only positive-energy relativistic quantum field theories for which not only nonperturbative existence, but also the soliton 5-operator ascribed to diem have been rigorously confirmed [70-74].) The ^-particle dynamics involved in ttiis alternative description (cf. [12, Section 3]) give rise to the simplest examples of what we have dubbed 'quantum pure soliton systems' in Sec­ tion 3A, the ^-particle S-matrix being given by the multiplication operator (SfW>=

II

exp[i,)*(0,-«*)]/■(»),

4>6[0,2,r)

(4.29)

\
on L2(RN,dN8). (Here and from now on, e denotes the sign function.) As Af-particle dynamics on L2(RN,dNq) yielding this soliton S-operator one can choose H = M( • X 2 exp(/3«,))M( • )*,

(4.30)

;=> where exp(fi$j) denotes the Fourier transform of multiplication by expGSfl,) (recall (3.1) and our standing convention h = 1), and where M(q)=

n

exptiX^-^)/^

(4.31)

Kj
(This is possibly not clear from [12, I.e.], but can be gleaned from [75, Section 7B].) It is to be noted that the classical version of H is free, and that one may rewrite H as " = 2 j=\

nexP['X%-?*)/2]exP03»;)nexP[-'X?/-?*)/2] k^j

(4-32)

k^j

Therefore, one may regard H as a degenerate case of the operator S ^ I ^ ) , obtained from (3.5), (3.6) by fixing z and taking /i to oo. The limit just indicated amounts to taking g to 0 with fig fixed. Thus, even for TV = 2 we do not know whether this limit can be given a rigorous sense (in terms of strong convergence of eigenfunction transforms, say). At any rate, one would have to deal with eigenfunctions that are not invariant under parity (since (4.32) is not). When one takes the known N =2 U^ case as a lead, this is certainly not as preposterous as it may appear from (3.26) (which is formally invariant under parity). The point 1 3

is, that the II,,, Hamiltonian (3.34) is not essentially self-adjoint on 'C<S°(R*) for gefx.-r), and 'almost all' self-adjoint extensions will violate the formal parity invariance of (3.34). The issue of self-adjoint extensions just mentioned is the key to the connection between the IInr sys­ tems and the quantized nonlinear Schrodinger (NLS) theory. We proceed to describe this relation. To

201 this end, we first recall that the latter quantum field theory is characterized by an explicitly particle number preserving Hamiltonian. Specifically, the restriction to the TV-particle sector reads H=—jA+X

2

%-ft).

(4.33)

\<J
where S denotes the Dirac delta function. As is well known, H can be given a rigorous meaning via quadratic form techniques, and then amounts to an unusual self-adjoint extension of the Laplacean restricted to C™ -functions whose support does not meet the hyperplanes qj=qkWe shall first discuss the features of H as an operator on L2(RN), (die symmetric L2-functions). For X>0 the interaction is repulsive and one obtains yet another quantum pure soliton system without a classical version. The analysis involved in proving this can be found in the Ph.D. diesis of Oxford [76]. In fact, he handles die far more complicated attractive case X<0. The complications are due to die presence of Af-body bound states for any M<JV, whose center-of-mass wave function reads

#?„...,?«) =

II

«p
(4.34)

We are now prepared to detail the relation to die II,,, particle systems. To this end we begin by considering the TV = 2 center-of-mass situation. The key observation is, that die kernel (3.35) can be continued to ge(—Vt,Vi); It tiien corresponds to a self-adjoint extension of die restriction of (3.34) to Co°((0,oo)) tiiat differs from die (Friedrichs) extension associated with the choice ge('/4,3/2). For ge(-*4,0) the operator S defined by (3.31) witii E(q,8)->E(v,g;8,q) (cf. (3.35)) does not extend to an isometry onto X This is due to the presence of a bound state orthogonal to its range, viz., \Kq) = (shvqy,

?>0

(4.35)

Setting g = X/ ft

/i=2v

(4.36)

and taking p->oo, die kernel E(ji/2,\/n;6,q) converges to

E(q,e) = (2w)-

ie-i\ ie+t\

M

exp(iq0)+c.c.

q, 0>O

(4.37)

and die bound state (4.35) to 4iq) = exp(\q/2),

q>0

(4.38)

These are precisely die NLS transform and bound state, transformed from i 2 (R), to L2([0,oo)) in the obvious way. Notice tiiat die (formally) attractive/repulsive II„, potentials lead to repulsive/attractive NLS 'potentials'! More generally, we expect (strong) convergence of the ILj, arbitrary N transforms to die NLS boson transform in die same scaling limit. Apart from the rule of tiiumb that in soliton theory the two-body situation extends to arbitrary particle number, there are two more solid hints that this should be true. First, die parameter ji sets the Iengtii scale: When it goes to oo, die II U potentials converge to 0 inside die Weyl chamber G, cf. (2.18)-(2-23). Hence, one expects mat the asymptotics following from (3.21) and (3.44), viz.,

E(q,S)~(2«)-Nn?.(-y

II

«Clto-»;»*

«"'(i)
■ exp(iq• «„),

q„«

• • ' «qi

II

■&*-•/»"""

- <■)>» (/)

(439)

extends to all of G in the limit. Accepting this and noting - l i m U ( I W =\/li;9) = - ^ 7

=«(NLS,X;0)

(4.40)

202 (cf. (3-45)), one obtains the NLS transform corresponding to the channel without bound states (up to an overall phase). Indeed, the latter is given by the rhs of (4.39) with u-m (NLS) and the factor (—f omitted. (The natural comparison operator for the NLS boson case considered here is the generalized cosine transform.) The second indication of convergence for general TV is the bound state behavior for >i-»oo: For g < 0 and \g\ sufficiently small, the II,,,. dynamics has an Af-body bound state
II

sbitiqi-qjY,

qu<--
(4-41)

which converges to the NLS bound state (4.34) in the above scaling limit. On account of arguments similar to the ones just presented, we expect that the NLS boson transforms can also be obtained as scaling limits of the g = 2 IIre| transforms. Indeed, from (3.49) we have -limu(IL d ,e=2,«=»7-0\/2;fl) = u(NLS,A;0)

(4.42)

fi-A

Recalling a = j3v = fin/2, we see that fi-»oo when /?->0, so that this limit is analogous to (4.40). In fact, it is straightforward to verily that for q, 8>0 the N -2 kernel Ej(q, 0) (given by (3.38H3.42)) converges to the NLS kernel (4.37) in the limit just detailed. Provided one restricts attention to the TV =2 case, one can also reach the NLS transform (4.37) via a j = 2 Ilrej transform. Moreover, this can be done in two essentially different ways. The first scaling limit is the one specified in (4.42), whereas the second one consists in taking a to n/2, ft to 0 and (hence) /? to oo; Specifically, one should set a = (ir-vX)/2. Also, in the second scaling limit one gets convergence to (4.37) with q and 6 interchanged. It is not clear to us whether similar relations are going to persist in these two limits when N is greater than 2. At any rate, since solitons and antisolitons are distinguishable, one would need some very special linear combinations of the eigenfunctions (which, it should be recalled, have not yet been found for N >2). At this point we should mention that the eigenfunctions of the NLS Hamiltonian (4.33) for distin­ guishable particles are also known [77-79]. (We are not aware of a proof of the Plancherel formula for this case.) However, for g = 2 the si reflection coefficient r vanishes, in contrast to r (NLS). Therefore, it is not likely that the g = 2 II„i eigenfunctions can lead to the non-symmetrized eigenfunctions of (4.33). On the other hand, it follows from (3.61), (3.62) that one does obtain the NLS transmission and reflection in the limit (4.40). Thus, one may expect to get the complete L2(R) transform via the N =2 IInr eigenfunctions. (We have not checked this.) Let us add one more remark concerning the situation for distinguishable particles. As already men­ tioned, we have trouble seeing how the L2(RN) transform might be reached via type II systems. How­ ever, it may well be that the desired transform can be obtained via a linear combination of IIm eigen­ functions, as a generalization of the picture for the NLS boson transforms already sketched. (The latter picture, to be sure, is very plausible but not proven.) Again, for ff=2 this can be achieved. Indeed, the kernel E(q, 8) = £(ji/2,\/M;|»|,|?|)+f(?M«)£0'/2,l -X//i;|»|,| 9 |),

q, 0<=R*

(4.43)

(recall (3.35)) converges to the even extension of the kernel (4.37) on Z.2(R), and to the kernel (2/»)wsin
203 papers [81-84] not only proved the Plancherel formula, but also obtained mathematically rigorous results concerning soliton scattering and conserved quantities. Presently, work by Babbitt and Gutkin is under way [85] which promises to elevate the XXZ model to a comparable mathematical level. It so happens that the XXZ two-magnon S-matrix can be transformed to a difference kernel (cf. e.g. Ch. 1 in Gaudin's monograph [86]). It then amounts to the g = 2 IL^, two-soliton S-matrix (3.49) for anisotropy parameter A E ( 0 , 1 ) and to the g =2 IIL^ two-soliton S-matrix [44] for Ae(l,oo). This may be compared to the equality of the NLS two-soliton S-matrix (4.40) and the g = 2 II^ S-matrix (3.45). This is only one reason why we believe that there might be a way to tie in scaling limits of 'relativistic" eigenfunctions with the Bethe transforms of the XXZ model, as a generalization of the links between degenerate II,,,. particle systems and the NLS Bethe transforms sketched above. Since one is dealing with lattice models, one would probably need limits of systems of type IILa or TV^. How­ ever, to date we have not been able to find a clearcut connection even for N = 2. By now, those readers still with us may well be tired of our hunches. Instead of indulging in further speculations, let us finish by pointing out some connections that do have solid proofs (though one may question the assumptions on which these are based). First, the S-matrix/(3.56) with T=IT/2 is just the soliton-soliton S-matrix of the quantum sine-Gordon model [45]. As pointed out by Zamolodchikov [87], this S-matrix is very closely related to the 6-vertex model free energy (cf. e.g. p. 148 in Baxter's monograph [88]). Second, when one proceeds as sketched in the paragraph containing (3.54), but now for the systems dual to the IVrcl systems, one obtains [44] an S-matrix that has a specializa­ tion related in a similar way to the 8-vertex model free energy [87,88]. Third, for T=W/2 the boundstate spectrum (3.71) amounts to the sine-Gordon soliton-antisoliton bound state spectrum [45]. We do not know how to compare the T=W/2 soliton-soliton wave functions (3.38)-(3.42) or the lowest-energy soliton-antisoliton bound state (3.72) to any previous results in the physics literature. Indeed, physicists appear to be convinced that a relativistic quantum mechanics description of the sine-Gordon/massive Thirring model is impossible. ACKNOWLEDGEMENTS We would like to thank R. Goodman, G. Heckman, E. Thomas, L. Woronowicz and, especially, T. Koomwinder for illuminating discussions on harmonic analysis, special functions, and quantum groups. We also take this opportunity to thank S. Albeverio and A. Hurst, who responded to a Query in Notices of the AMS, making us aware of NSrlund's monograph [38] and the references [39-41]. REFERENCES 1. S.N.M. RUUSENAARS, H. SCHNEIDER, A new class of integrable systems and its relation to solitons, Ann. Phys. (NY) 170, 370-405 (1986). 2. S.N.M. RUUSENAARS, Complete integrability of relativistic Calogero-Moser systems and elliptic function identities, Commun. Math. Phys. 110, 191-213 (1987). 3. S.N.M. RUUSENAARS, Relativistic Toda systems, to appear. 4. M.A. OLSHANETSKY, A.M. PERELOMOV, Classical integrable finite-dimensional systems related to Lie algebras, Phys. Reps. 71, 313-400 (1981). 5. M.A. OLSHANETSKY, A.M. PERELOMOV, Quantum integrable systems related to Lie algebras, Phys. Reps. 94,313-400(1983). 6. S.N.M. RUUSENAARS, Action-angle maps and scattering theory for some finite-dimensional integrable systems, The pure soliton case, Commun. Math. Phys. 115, 127-165 (1988). 7. S.N.M. RUUSENAARS, Action-angle maps and scattering theory for some finite-dimensional integrable systems. II Solitons, antisolitons, and their bound states, to appear. 8. S.N.M. RUUSENAARS, Action-angle maps and scattering theory for some finite-dimensional integrable systems. Ill Sutherland type systems and their duals, to appear. 9. V.I. ARNOLD, Mathematical methods of classical mechanics, Springer, Berlin (1978). 10. M. REED, B. SIMON, Methods of modem mathematical physics. Ill Scattering theory, Academic Press, New York, (1979).

204 11. W. THTJUUNG, Lehrbuch der mathematiscben Physik. I. Klassische dynamische Systeme, Springer, Berlin (1977). 12. S.N.M. RUUSBNAARS, On one-dimensional integrable quantum systems with infinitely many degrees of freedom, Ann. Phys. (NY) 128, 335-362 (1980). 13. S.N.M. RUUSBNAARS, A positive energy dynamics and scattering theory for directly interacting relativistic particles, Ann. Phys. (NY) 126, 399-449 (1980). 14. J. MOSER, Three integrable Hamiltonian systems connected with isospectral deformations, Adv. in Math. 16, 197-220 (1975). 15. J. MOSER, Finitely many mass points on the line under the influence of an exponential potentialan integrable system, pp. 467-497 in: Dynamical systems, theory and applications (J. Moser ed.), Lect. Notes in Physics Vol. 38, Springer, Berlin (1975). 16. H. ATRAULT, H.P. MCKEAN, J. MOSER, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Commun. Pure Appl. Math. 30, 95-148 (1977). 17. D. KAZHDAN, B. KOSTANT, S. STERNBERG, Hamiltonian group actions and dynamical systems of Calogero type, Commun. Pure Appl. Math. 31, 481-507 (1978). 18. M. ADLER, Completely integrable systems and symplectic actions, J. Math. Phys. 20, 60-67 (1979). 19. P.P. KULISH, Factorization of the classical and the quantum S matrix and conservation laws, Theor. Math. Phys. 26, 132-137 (1976). 20. F. CALOGERO, Exactly solvable one-dimensional many-body problems, Lett. Nuovo Cim. 13, 411-416(1975). 21. I.M. KRICHEVER, Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles, Func. Anal. Appl. 14, 282-290 (1980). 22. M. BRUSCHI, F. CALOGERO, The Lax representation for an integrable class of relativistic dynami­ cal systems, Commun. Math. Phys. 109, 481-492 (1987). 23. H. FLASCHKA, The Toda lattice. II Existence of integrals, Phys. Rev. B9, 1924-1925 (1974). 24. M. ADLER, P. VAN MOERBEKE, Linearization of Hamiltonian systems, Jacobi varieties, and representation theory, Adv. in Math. 38, 318-379 (1980). 25. M. TODA, Theory of nonlinear lattices, Springer, Berlin (1981). 26. M. BRUSCHI, O. RAGNISCO, Lax representation and complete integrability for the periodic rela­ tivistic Toda lattice, Phys. Lett. A134, 365-370 (1989). 27. V.I. INOZEMTSEV, The finite Toda lattices, Commun. Math. Phys. 121, 629-638 (1989). 28. E.T. WHITTAKER, G.N. WATSON, A course of modern analysis, Cambridge University Press, Cambridge, (1973). 29. O. RAGNISCO, private communication. 30. F.A. BEREZTN, Laplace operators on semisimple Lie groups, Amer. Math. Soc. Transl. 21, 239-339 (1962). 31. S. HELGASON, Groups and geometric analysis, Academic Press, New York, (1984). 32. H. SCHLICHTKRULL, Hyperfunctions and harmonic analysis on symmetric spaces, Birkhauser, Boston (1984). 33. A. TERRAS, Harmonic analysis on symmetric spaces and applications II, Springer, Berlin (1988). 34. G.J. HECKMAN, E.M. OPDAM, Root systems and hypergeometric functions I, Comp. Math. 64, 329-352(1987). 35. G.J. HECKMAN, Root systems and hypergeometric functions II, Comp. Math. 64, 353-373 (1987). 36. E.M. OPDAM, Root systems and hypergeometric functions III, Comp. Math. 67, 21-49 (1988). 37. E.M. OPDAM, Root systems and hypergeometric functions IV, Comp. Math. 67, 191-209 (1988). 38. N.E. NORLUND, Vorlesungen iiber Differenzenrechnung, Springer, Berlin (1924). 39. R. JOST, Lineare Differenzengleichungen mit periodischen Koeffizienten, Comm. Math. Helv. 28, 173-185 (1954). 40. S. ALBEVERIO, Analytische Ldsung eines idealisierten Stripping- oder Beugungsproblems, Helv. Phys. Ada 40, 135-184 (1967).

205 41. J.B. Mc GumE, C_A. HURST, The scattering of three impenetrable particles in one dimension, J. Math. Phys. 13, 1595-1607 (1972). 42. G.K. IMMIKK, Asymptotics of analytic difference equations, Lect. Notes in Math. Vol. 1085, Springer, Berlin (1984). 43. S.N.M. RUUSENAARS, to appear. 44. S.N.M. RUUSENAARS, First-order analytic difference equations and integrable quantum systems, to appear. 45. A.B. ZAMOLODCHKOV, A.B. ZAMOLODCHIKOV, Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models, Ann. Phys. (NY) 720, 253-291 (1979). 46. R ASKEY, M.E.-H. ISMAIL, A generalization of ultraspherical polynomials, pp. 55-79 in: Studies in Pure Math. (P. Erdos, ed.), Birkhauser, Boston (1983). 47. R ASKEY, J. WILSON, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs Amer. Math. Soc., Vol. 54, Providence (1985). 48. T. KOORNWTNDER, private communication. 49. R ASKEY, J. WILSON, A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols, SIAM J. Math. Anal. 10, 1008-1016 (1979). 50. I.G. MCDONALD, Orthogonal polynomials associated to root systems, Oxford preprint (1988). 51. S.L. WORONOWICZ, Twisted SU(2) group. An example of a non-commutative differential cal­ culus, Publ. RIMS 23, 117-181 (1987). 52. S.L. WORONOWICZ, Compact matrix pseudogroups, Commun. Math. Phys. HI, 613-665 (1987). 53. S.L. WORONOWICZ, Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93, 35-76 (1988). 54. T. KOORNWTNDER, Continuous ^-Legendre polynomials are spherical matrix elements of irreduci­ ble representations of the quantum SU(2) group, CWI Quarterly 2, 171-173 (1989). 55. S.N.M. RUUSENAARS, Generalized Lame functions, to appear. 56. R GOODMAN, N.R WALLACH, Classical and quantum mechanical systems of Toda-lattice type III. Joint eigenfunctions of the quantized systems, Commun. Math. Phys. 105, 473-509 (1986). 57. R.K. BULLOUGH, S. OLAFSSON, Complete integrability of the integrable models: quick review, pp. 329-334 in: Proceedings of the 1988 Swansea LAMP Conference (B. Simon, A. Truman, I.M. Davies, eds.), IOP Publishing, Bristol (1989). 58. S.N.M. RUUSENAARS, Relativistic Calogero-Moser systems and solitons, pp. 182-190 in: Topics in soliton theory and exactly solvable nonlinear equations (M. Ablowitz, B. Fuchssteiner, M. KruskaL eds.), World Scientific, Singapore (1987). 59. V.E. ZAKHAROV, L.D. FADDEEV, Korteweg-de Vries equation, a completely integrable Hamil­ tonian system, Func. Anal. Appl. 5, 280-287 (1972). 60. H J . FLASCHKA, A.C. NEWELL, Integrable systems of nonlinear evolution equations, pp. 355-440 in: Dynamical systems, theory and applications (J. Moser, ed.), Lect. Notes in Physics Vol. 38, Springer, Berlin (1975). 61. H.F. FLASCHKA, D.W. MCLAUGHLIN, Some comments on Backlund transformations, canonical transformations, and the inverse scattering method, pp. 253-295 in: Backlund transformations (R Miura, ed.), Lect. Notes in Math. Vol. 515, Springer, Berlin (1976). 62. L.D. FADDEEV, L_A. TAKHTAJAN, Hamiltonian methods in the theory of solitons, Springer, Berlin (1987). 63. V.S. BUSLAEV, L.D. FADDEEV, L.A. TAKHTAJAN, Scattering theory for the Korteweg-de Vries (KdV) equation and its Hamiltonian interpretation, Physica 18D, 255-266 (1986). 64. F. MAGRI, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19, 1156-1162(1978). 65. B.A. KUPERSHMTDT, G. WILSON, Modifying Lax equations and the second Hamiltonian structure, Invent. Math. (52,403-436(1981). 66. C.S. GARDNER, Korteweg-de Vries equation and generalizations. IV. The Korteweg-de Vries

206 equation as a Hamiltonian system, J. Math. Phys. 12, 1548-1551 (1971). 67. L.D. FADDEEV, LA. TAKHTAJAN, Poisson structure for the KdV equation, Lett Math. Phys. 10, 183-188 (1985). 68. W. ECKHAUS, P. SCHUUR, The emergence of solitons of the Korteweg-de Vries equation from arbitrary initial conditions, Math. Meth. AppL Sci. 5, 97-116 (1983). 69. E. DATE, M. JIMBO, M. KASHTWARA, T. MIWA, Landau-Lifshitz equation: solitons, quasi-periodic

solutions and infinite-dimensional Lie algebras, J. Phys. A16, 221-236 (1983). 70. S.N.M. RUIJSENAARS, Scattering theory for the Federbush, massless Thirring and continuum Ising models, J. Func. Anal. 48, 135-171 (1982). 71. S.N.M. RUIJSENAARS, The Wightman axioms for the fermionic Federbush model, Commun. Math. Phys. 87, 181-228 (1982). 72. J. PALMER, C. TRACY, Two-dimensional Ising correlations: convergence of the scaling limit, Adv. Appl. Math. 2,329-388(1981). 73. R. SCHOR, M.O' CARROLL, The scaling limit and Osterwalder-Schrader axioms for the twodimensional Ising model, Commun. Math. Phys. 84, 153-170 (1982). 74. J. PALMER, C. TRACY, Two-dimensional Ising correlations: the SMJ analysis, Adv. Appl. Math. 4, 46-102 (1983). 75. S.N.M. RUUSENAARS, Integrable quantum field theories and Bogoliubov transformations, Ann. Phys. (NY) 132, 328-382 (1981). 76. S. OXFORD, The Hamiltonian of the quantized nonlinear Schrodinger equation, Ph.D. thesis, Los Angeles (1979). 77. J.B. MCGUIRE, Study of exactly soluble one-dimensional //-body problems, J. Math. Phys. 5, 622-636(1964). 78. C.N. YANG, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19, 1312-1315 (1967). 79. C.N. YANG, S matrix for the one-dimensional iV-body problem with repulsive of attractive 8function interaction, Phys. Rev. 168, 1920-1923 (1968). 80. HA. BETHE, Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Zeitschr. f. Physik, 71, 205-226 (1931). 81. L.E. THOMAS, Ground state representation of the infinite one-dimensional Heisenberg ferromagnet I, J. Math. Anal. Appl. 59, 392-414 (1977). 82. D. BABBITT, L. THOMAS, Ground state representation of the infinite one-dimensional Heisenberg ferromagnet II. An explicit Plancherel formula, Commun. Math. Phys. 54, 255-278 (1977). 83. D. BABBITT, L. THOMAS, Ground state representation of the infinite one-dimensional Heisenberg ferromagnet III. Scattering theory, J. Math. Phys. 19, 1699-1704 (1978). 84. D. BABBITT, L. THOMAS, Ground state representation of the infinite one-dimensional Heisenberg ferromagnet IV. A completely integrable system, J. Math. Anal. Appl. 72, 305-328 (1979). 85. D. BABBITT, E. GUTKIN, The Plancherel formula for the infinite XXZ Heisenberg spin chain, Los Angeles preprint (1989). 86. M. GAUDIN, La fonction d'onde de Bethe, Masson, Paris (1983). 87. A.B. ZAMOLODCHIKOV, ZJ-symmetric factorized 5-matrix in two space-time dimensions, Com­ mun. Math. Phys. 69, 165-178 (1979). 88. R.J. BAXTER, Exactly solved models in statistical mechanics, Academic Press, New York (1982).

207

R E L A T I V I S T I C A N A L O G S OF B A S I C I N T E G R A B L E S Y S T E M S

John Gibbons* and Boris A. Kupershmidt**

* Department of Mathematics, Imperial College, 180 Queen's Gate, London S W 7 2BZ, U K

** T h e University of Tennessee Space Institute Tullahoma, T N

Abstract.

37388, U S A

Relativistic extensions are defined of continuous and discrete

integrable systems associated with scalar Lax equations. These extensions are proved to be integrable. The relativistic Toda lattice is treated in detail.

208

1. Introduction There are a few operations known to be fruitful when applied to vari­ ous theories of dynamical systems. These operations include linearization, complexification, deformation, quantization, relativization. It is the latter operation which is the most dehcate: only exceptional dynamical systems possess relativistic analogs, and those that do form a very special class of "natural" systems. Since integrable systems are rare in spaces of dynamical systems, one should not expect special relativistic analogs of integrable systems to exist. Surprisingly, two integrable relativistic analogs of nonrelativistic integrable systems have been found (by Ruijsenaars and others [1-8]): the relativistic Toda lattice and the relativistic Calogero - Moser system. Since the classical Toda lattice is the simplest basic (i.e., scalar Lax) discrete integrable system, one may hope to extract from an analysis of its relativistic deformation a general ansatz apphcable to other Lax equations, and not only discrete ones. In this paper we present such an ansatz for arbitrary scalar Lax equations, both continuous and discrete. We start with the continuous case first: we show that the relativistic flows are well-defined (Sections 2,3) and that these flows commute and have a common infinite set of conserved densities (Section 4). Then we work out the discrete case (Section 5). An example of the relativistic Toda lattice (Section 6) concludes the paper.

209 2. Relativistic Version of Differential Lax Equations Let K be a differential Q-algebra with a derivation d, and let C — u

K[ i

]» l € I,N G Z + , be a differential algebra built out of K and the

independent variables u,-'s. (Standard facts about differential algebras and Lax equations can be found, e.g., in [9] or [10].) Let O — 0{C) be the ring of pseudodifferential operators with coefficients in C:

0:=\

Z, f.V I / . G C},

'2.1)

s
with £ being the operator version of d. For any r £ Z, we denote by 0>r the following subspace of O:

-= { £ / * n / * e C};

(2.2)

5>r

8s the similarly, we have subspaces 0>r,
0+:=0>o,

O-O-:=O<0.

(2.3) (2-3)

Fix n G N and set n-1 Ui

ue ' 9 = 0 or -oo. L = C + Yl ^

(2.4)

The usual (nonrelativistic) Lax equations have the form L,t=[P+,L] ,L] =

[L,P.},

(2.5)

210

where P G Z(L) C 0(C),Z(L)

being the centralizer of L in 0{C).

In the

scalar case we are concerned with here, P runs over {(L^n)m

=£m + ... | m € N } .

(2.6)

Now let e be a formal parameter (which can be thought of as 1/c2, c being the speed of light). Set

£ = (l-eO"%

(2.7)

(i-eo-^fye*

(2.8)

where

fc=0

Thus,

£ G 0[[e]] =: (5.

(2.9)

Let V G Z(£) C O; we shall show in the next Section that Z{C) is generated by { ( £ 1 / " ) m | m G Z}. Our relativistic equations are d

= [V+ + S(V-),C]

= [C,V- - S(V-)],

(2.10)

where the Symbol map 5 : O<0 := O<0[[e]] —* C[[e]] = : C

(2.11)

is denned by the rule

5

( E ^ J - £>«"*• ^G(5-

(212)

211

Obviously, when e —> 0 relativistic equations (2.10) collapse onto the usual Lax equations (2.5). The main result of this Section is Theorem 2.13. The relativistic flows exist, i.e., equations (2.10) are welldefined. Proof. We have L,t[

by (2.7)] = ( l - < ) £ , 4 =

= (1 -€t)[V+

+ S(V-),C]=

(2.14a)

= {l-e£)[C,V--S(V-)].

(2.146)

Denote V1:=V+ + S(V-), V2:=V--S(V-).

(2.15)

Then, from (2.14a) we get L,t = (1 - ef)7>x(l - e O " 1 ^ " LVU

(2.16)

and since V\ £ O+, we see from (2.16) that {£_=0}^{(£,t)-=0}.

(2.17)

Next, from (2.14b) we obtain L,i = LV2 - (1 - ct)P2(l

- ei)~lL.

Lemma 2.19. Let d € O-, and denote i?2 := $ - S{d).

Mi-eC,-1 ed-

(2.18) Then

(2.20)

212

Proof. If

d = ^2v.c"1, "»ec,

(2.21)

s>0

then 02 = 0-S(0) = Y,MC°-l-e3+1)-

( 2 - 22 )

rs-i_es+i=Qs{ri_e)i

(2.23)

n, = ^ ( r 1 ) " ^ - " e a<0.

(2-24)

Now

where

Hence,

tf2(i

- eo-1 = « r l - e)]-1 = Mr1 - ey'c1 [ by (2.22), (2.23)] =

1

-fen.n.V .

2 25

■ <- )

Denote V:=V2(l-eO-\

( 2 - 26 )

= V(l-eS).

( 2 - 27 )

so that V2

213

Then equation (2.18) becomes L,t = LV{\ - e£) - (1 - etfPL.

(2.28)

By Lemma 2.19, V=PC1+PC2 with some p, p £ C.

+ 0(C3),

(2.29)

Hence, from (2.28) we see that

L,t G 6
(2.30)

so that equations (2.10) do make sense.

|

Moreover, from the right-hand-side of equation (2.28) we can extract the £ n _ 1 -term via the following calculation: LV{\ - e£) - (1 - etfPL [ by (2.4), (2.29)] = = ( C + u^C-1

+ ■ ■ -KPC1 +PC2 + ■ ■ 0(1 -

e

0-

- ( 1 - eOipC1 +PC2 + ■ ■ 0 ( r + Un-iC" 1 + . . . ) = = [ p C - 1 + K - i P + " P ( 1 ) +P)C~2 + •-•](! - ( 1 - eOlpC-1 + (p + pun-X)C-2

-*)-

+ ■■■] =

= [~epC + PC'1 ~ <«n-iP + np^ + p)Zn-x + . . . ] -[-epC

- ep^C'1

+PC-1 ~ e{p + pun^)C-1

= -£(n-l)pWr1+.-,

+•••] = (2-31)

214

so that u B _ M = -e(n - l ) p M .

(2.32)

Hence, for n = 1 we can take uQ = 0. E.g., for the relativistic KP equation we can take

£ = (l-60"1 U + E^r-M

•

(2-33)

In a sense, the KP case is the simplest one, since in this case we do not need to introduce an additional, with respect to the usual nonrelativistic case, variable u „ _ i . 3. The Centralizer Z(C) If V G Z(C) and

P = EV-

(3-1)

i>o then VQ G Z(L): this follows upon taking the e° -terms of the equality [^(l-ee)-1I] = 0

(3.2)

We wish to show that, given P G Z(L), 1 n r

there exists a V G Z(C) with

P 0 = P- It is enough to consider P = (L / ) ,r

G Z, and then it is enough

to show that there exists ( £ 1 / " ) r , which certainly exists if X := Clln does. So, we have to solve the equation

Xn=C,

(3-3)

215

i.e.

(1 - eOXn = L.

(3.4)

Set X = Y,Xjtj,

XjZO,

(3.5)

j">0

so that Xn = £ ( J T V ,

(3.6)

j>0

where

(Xn)j=

J2

X

h--Xjn.

(3.7)

Then the defining equation (3.4) becomes (*")o = L,

(3.8)

(Xn)j+1=t(Xn)j,

J€Z+.

(3.9)

Now, (Xn)0 = (X0)n, so that equation (3.8) yields Xo = L1'n€t

+ 0<0.

(3.

216

We construct the remaining Xj's using induction on j . Suppose

Xo,...,Xj

have been found. Then, for Xj+i, equation (3.9) gives n-l

J^X^Xj+iX^-1-01

= something involving X0,...,Xj,

(3.11)

a=0

and the following well known Lemma [11] furnishes the desired Xj+im. Lemma 3.12. Suppose Y is an invertible element in O, and let m G N . Then the map

p:0-+0,

m—1

p(Q) := Y, YOQY™-1-",

(3.13)

a=0

is an isomorphism. Remark 3.14. Introducing the following grading in O: rk(u\N))

= N + n-i,

rk(K) = 0, rfc(f) = r, r*(e) = - 1 ,

(3.15)

we get rk(C) = rfc(L) = n, rfc(X) = 1, rk(Xj)

= 1+ j,

(3.16)

so that Xj G Cj?+1

+ O^,

(3.17)

where the constants Cj G Q are found from the equality

5 > e i + v = (i - <*)-1/n£ = E c - i y t f v ^ 1 j>0

j>0

(3-18)

217

As a Corallary of Lemma 3.12 we get the following useful Lemma 3.19. Suppose Y^Y'1^ G O, m G N, and (Ym),t=[9,Ym).

(3.20)

Y,t=[8,Y).

(3.21)

Then

Proof. Denote /3:=Y,t-[9,Y).

(3.22)

Then, by (3.20), m-l

[0,Ym] = (Ym),t = £ YaY,t y » - i - « [ by (3.22) ] = a=0

m—1

m—1

a=0

a=0

= Yl Ya(\0,Y) + /3)Ym-1~a = [Ym,0] + J2 y^y" 1 " 1 " 0 , so that m-l

0= ^

Ya$Ym-x-a.

(3.23)

a=0

Since Y = ]C,>o ^' e J

1S

invertible iff YQ is, Lemma 3.12 remains true when

extended from O into O. Hence, equation (3.23) implies that j3 = 0, so that equality (3.21) holds.

|

218 4. Integrabilitv of Continuos Relativistic Systems Let V € Z(£), and denote by d-p : C —> C the evolutionary derivation defined by the formula (2.10). Let TZ G Z(C) be another element commuting with C, and let d-R : C —► C be the corresponding evolutionary derivation denned by the equation dn(C)

= [H+ + S(TZ.),£]

= [C,U- - S(U-)].

(4.1)

We are going to show that the flows (2.10) and (4.1) commute: Theorem 4.2. If V,TZ G Z(C) then [dp,&R]=0.

(4.3)

Proof. We shall show that [dv,dn](C)

= 0.

(4.4)

From Lemma 3.19 we find that dn(£1/n)

= [1Z+ + S(Tl.),C^n]

= [C^n,1Z--S(TZ.)],

(4.5)

and hence dn{V)

= [TZ++ S(TZ-),V]

= [V,TZ. - S(K.)}.

(4.6)

Therefore,

dn(v+) = [v,n. - S(n.)]+ = [v+,n. - s(n.)}+.

(4.7)

219

Now, dndv(C) = dn ({V+ + S(V-), £]) [ by (4.1), (4.7)] =

- [[v+,n. - s(n-)]+,c] + [sdn(V-),c]+ +[P+ + S(V-),[n+

+ S(R-),£)},

(4.8)

where we used the obvious rule Sdn = dTiS on 6<0.

(4.9)

Interchanging R and V in (4.8) we get dvdn(C) = [[ll+,V- - S(V.)]+,£} +

[Sdp(K_),£]+

+[1l+ + S(1l-),[V+ + S(V-),£}}.

(4.10)

Hence, substracting (4.8) from (4.10) we obtain that [d-p,djz](C) equals the commutator with £ of [1Z+,V. - S(V.)]+ + [11- -

S(R.),V+}++

+[Jl+ + S(Tl-),V+ + S(V-)] + S(dv(TZ-) - dn(V-))

= [n+,V-}+ + [ii-,v+}+ + [n+,v+] +

=

(4.ii)

+[n+,-s(V-)]+-[S(Tz.),v+]++ +[n+, s(v-)] + [S(n-),v+] + s (dP(n.) - an{v.)).

(4.12)

Now, the expression (4.11) vanishes [9] being the + - part of 0 = [11, V] = [11+ + 1 1 - , V+ + V-] = = {[K+,V.) + [1l-,V+] + [1l+,V+]) + ([R-,V-]).

(4.13)

220

In the expression (4.12), only the last term S{dv(n.)-dn{V-))

(4.14)

survives: since S(V-) € C, we have [R.+,S(V.)) = [n+,S(V.)]+,

(4.14)

and all the remaining terms cancel each other out. Thus, we have to show that S(dv(TZ.)-dn(V-))

= 0.

(4.15)

By (4.6), dn(V.) = [&n{V)\- = [P,K- - S(1Z.)).,

(4.16)

so that S{dv{Tl.)-dn{V-))

=

= S([R,V- - SOP-)]- - [V,K- - 5(ft_)]_) =

= s([n,-v+]- - \n,s{V-)]. - [V,TIJ\- + [v,s(n.)).) = = s([v+,n.}. - [v,n.]. + [S(v.),n.] + [v.,s(n.)]) = = S(-[V.,TZ.))

+ S([S(V-),K.)

+ [V.,S(Tl.)}).

(4.17)

Hence, the equality (4.15) follows from the following identity:

s([u,v)) = s([S(u),v] + [u,s(v)]), u,ved,

(4.18)

221

provided all expressions m formula (4.18) make sense, e.g., when U, V € C_ To prove formula (4.18), it is enough to consider the case

u = /c, V

= gts,

f,gec,

(4.19)

r, s € Z .

Then [U,V]

= [fC,9ta] = £[/(Ps

(j)

-0(;)/O)r+*-j,

so tha S([U,V))

= E [/O*01~ 3 ( j ) / ( j )

I eJ-r-

5

(4.20)

i>0

Further, [S(U),V]

= [A- r ,^1 = -be 5 ,A- r ] =

= -E^)/(j)^~ie~r> i>o

so that 5 ([S(U), V] + [U, S(V))) = S ([S(U), V] - [S(V), U}) =

U) s j r = s (- E 9(j)f z - t- +E fc^e-^-*) V

j>0

j>0

} i3} r = E[-s(i)/ 0 ) + f(i)9 ¥-*- ,

which is the same as (4.20).

= I

(4.21)

222

Remark 4.22. The proof above remains valid when / and g are matrices. Remark 4.23.

As in the classical (nonrelativistic) theory, formula (4.6)

implies that the set {Res(V)

| V € Z(C)} is a common infinite set of

conserved densities for each of the commuting flows {dn \ 1Z € Z(£)}

given

by formula (4.1). 5. Discrete Relativistic Systems Let A : K —► K be an automorphism of a commutative Q -algebra K and let C = K[q\r\

r £ Z , 0 < i < M , M € NU{oo}, be the corresponding

polynomial algebra built out of K and the variables u,'s. (See [10] for th< basic properties of discrete Lax equations.) Since we shall not return to th< continuos case, we borrow differential notations from the preceding Sections Let

O: ~il*f.C s
(5.1)

\fs ec}

be the associative ring of difference operators, with the multiplication rule

(fO(gC) = f&s(g)C+r, f,gec,

s,rez.

(5.2)

The notations C := C[[e}], O := 0[[e]], 0+, O-, etc. have the same meaning as before, with C, entering instead of £. Let M

L--

=c + £c '?■■ i=o

5.3)

223 The usual (nonrelativistic) discrete Lax equations have the form L,t=[P+,L]

= [L,P_],

P = im,m€N.

(5.4)

Set C =

(l-eO~1L.

(5.5)

Our relativistic version of equations (5.4) is C,t = [V+ + S(V-),C]

= [C,V. - S(7>_)], V = Cm,

(5.6)

where the Symbol map S:6-^C is defined by the formula (5.7)

Theorem 5.8. The motion equations (5.6) are well-defined. Proof. Denote Vi := V+ + S(V-),

V2 := V- - S(7>_).

(5.9)

Then L,t

[by (5.5)] = (1 - eQC,t [ by (5.6), (5.9)] = = (1-60^1(1-60-^-^1 = = LV2

_ (i _ eQV2(l

- eg

^

(5.10a) (5.106)

224 Since Vi E 0+, formula (5.10a) shows that L,t G 6>-M.

(5.11)

Lemma. R 19 For any i? G 0 - , set # 2 := *? - S(i9). Then i?2(l-€C)_16 0 - .

(5.13)

Granted the Lemma, we have P2 = V(l - eC)

(5.14)

with some 7? G 0~: P=PC-1 + 0(C2).

(5.15)

Substituting (5.14) into (5.10b) we get L,t = LV(l-€0-(l-eOVL,

(5.16)

and therefore L,t G O<0. Inclusions (5.11) and (5.17) prove the Theorem. Proof of Lemma, 5.12. Let

0 = Z>.C' _1 , «»eC. s>0

(5.17) |

225 Then, by (5.7), tf2 = i? - 5(i9) = ^ ( C

5

-

1

- e* +1 ) = Y.v.Q.iC1

~ «),

(5-18)

where s

n, = £cv- aa e(5
Since

r1-6 = r 1 (i-o, we get tf2(l-

- O"1 -(£■ .n.)C _1 . a>0

Let now 1Z = Cn be another element of the centralizer Z(C). Denote by d-ji : C —* C the evolutionary derivation defined by the equation

dn{£) = [n+ + S(\R~),£] = [c,n. - s(n.)].

(5.19)

Theorem 5.20. The flows (5.6) and (5.19) commute. Proof. To show that [&p,dn] — 0, we show that [&p,dn](C) = 0.

(5.21)

To prove (5.21) we repeat step-by-step the Proof of Theorem 4.2, until we arrive at formula (4.18). To prove it, let U = fC,V = gC, f,geC,

r,s€Z.

(5.22)

226 Then

S([u,v)) = s([fC,gC}) = = S ([fAr(g) - gA°(f)]C+s)

= [fAr(g) - gA°(f)]e-r-°.

(5.23)

Further, [S(U),V] = [fe-r,gC] = 9*-r(l - A ' ) ( / ) C , so that S([S(U),V]-[S(V),U])

=

= S[ge~r(f - /<<>)<• - fe~a(g - g « ) C ] = = » * - ( / " f(s))e~s

~ M~s)(g ~ 9(r))t-r

= [f9{r) ~ 9fM)t-r-s,

which is the same as (5.23) As in the continuous case, the set {Res(£m)

(5-24) |

\ m £ N} is an infinite

set of common conserved densities of the infinite set of commuting flows

{dR | n G z(c)}. 6. Relativistic Toda Lattice This is a particular

{M = 1, m = l}-case of formulae (5.3), (5.5),

and (5.6): L = ( + q0 + C19.i,

(6.1)

£ = (l-eC)- 1 (C + 9o + r 1 9 i ) ,

(6.2)

227

V. = (/*)_ = [£«*<*« + q0 + C'qi))- = C V = q[-1]C\

(6.3)

fc>0

S(C-)

= q[~1)e,

P2 = £_ - S(£_)

= ^(r

(6.4) 1

- e) = C V a

- <),

(6.5)

i,i = (C + go + C- 1 gi)^ _1> (C _1 - c) - (i - eC)C_1gi(C + go + C%) = = ... = (1 + «fo)(l - A" 1 )(g 1 ) + C _ 1 gi(A - l)[go + e(l + A " 1 ) ^ ) ] , (6.6) and finally f 9o,/ = \qi,t=

(l + e g 0 ) ( l - A - 1 ) ( 9 l ) ,

(6.7a) 1

g i ( A - l ) [ g o + e(l + A - ) ( 9 l ) ] .

(6.7b)

These are relativistic Toda equations; in the variables d = qo+e~l

, c = qu

(6.8)

they become f d,t = ed(l - A - 1 ) ^ ) , U,,=

(6.9a) 1

c(A-l)(cf) + ec(A-A- )(c),

(6.9b)

which, for e = 1, are equations (2.3) in [7]. The conserved densities Ho = lnqi

, Hi = Res(L) = q0, H2 = Res(L2/2)

~

qi

+ ql/2

(6.10)

of the nonrelativistic Toda lattice [10] are deformed into the conserved den­ sities ~Ho = lnquH~i = Res(C) = qo+equH2 +(qo+eqi)2/2

= Res(C2/2)

~

q[~1\l+eq0+e2qi)+ (6.11)

228

of the relativistic Toda lattice (6.7). Interestingly, from equation (6.7a) we see that Hi is also deformed into a conserved density H'1=e-1ln(l

+ eq0) = q0 + 0(e),

(6.12)

which is different from H1=q0

+ eqi =q0 + 0(e).

The problem of the Hamiltonian formalism is the most mysterious. We could derive a Hamiltonian form for none of the commuting hierarchies con­ structed in Sections 2 and 5. Only for the first flow, (6.7), of the relativistic Toda hierarchy 1

2

B,

3

B, B

{(5.6)|A/=I}

did we find the following partial results. Let

be the three Hamiltonian 2 x 2 -matrices [10] of the classical

Toda lattice hierarchy which generate the Toda lattice equations via the rule /
\qiJ

/6H2/6q0\

/6Ht/6q0\

/6H0/6q0\

= B1[ U W )=B3[ . (6.13) \6H2/6qtJ \SHj6qJ \SHg/SqJ t

Recall that the matrix elements of the matrices B1,2,3

are [10]:

*oo = * n = 0> *oi = (1 " A _1 )9i>

(6-14)

B20 = giA - A - V - B n = 9i(A - A " 1 ) ^ , ^ = q0(l - A " 1 ) ^ , (6.15)

B30 = ( 9 l A - A~V)?o + Qo(qiA - A " V ) , Bfi =

9l(A

- l)<7o(l + A " 1 ) ? ! + ft(A + l ) 9 o ( l - A~l)qi,

B301 = (ft A - A " 1 f t ) ( l + A - J ) f t +q2(l

- A-J)ft.

(6.16)

229 The relativistic Toda equations (6.7) can be written in a form similar to (6.13) provided we exchange Hj (6.10) into Hi B*+1(e)J

(6.11) and Bj+1

into

= 0,1,2, where

B ' + 1 ( e ) = B*1

+ eBJ+\

(6.17)

and Bl0 = A~1q1 -

qiA,

B u = B 0 1 = 0,

B 2 = 0,

(6.18) (6.19)

-Boo = 9o(?iA - A _ 1 gi)9o, B3n = qi(A + l ) ( g i A - A" 1 g 1 )(l + A - 1 ) ^ ,

(6.20)

Boi = qo(qiA - A " 1 9 l ) ( l + A " 1 ) ? ! . The matrix B2{e) is Hamiltonian since B2(e) = B2; the matrix S x (e) is linear in the g's and is easily seen to be Hamiltonian; affine transformations of go show that the matrices B1(e) and B2(e) form a Hamiltonian pair; the matrix B3(e) is certainly Hamiltonian but we haven't checked this.

Acknowledgement We are grateful to M. Bruschi and O. Ragnisco for a comment on their paper [6] and to the Science and Engineering Research Council for financial assistance. B.A.K. was partially supported by the National Science Foundation.

230 References [1] S.N.M. Ruijsenaars and H. Schneider, "A New Class of Integrable Sys­ tems and Its Relation to Solitons", Ann.Phys.(NY) 170 (1986) 370-405. [2] S.N.M. Ruijsenaars, "Relativistic Calogero-Moser Systems and Solitons", CWI Report PM-R8702, (Jan. 1987). [3] S.N.M. Ruijsenaars, "Complete Integrability of Relativistic CalogeroMoser Systems and Elliptic Function Identities", Comm. Math. Phys. 110 (1987) 191-213. [4] M. Bruschi and F. Calogero, "The Lax Representation for an Integrable Class of Relativistic Dynamical Systems", Comm. Math. Phys. 109 (1987) 481-492. [5] M. Bruschi and O. Ragnisco, "On New Solvable Many-Body Dynam­ ical Systems with Velocity-Dependent Forces", Inverse Problems 4 (1988) L15-L20. [6] M. Brusschi and O. Ragnisco, "Recursion Operator and Backlund Transformations for the Ruijsenaars - Toda Lattice", Phys. Lett. 129A (1988) 21-25. [7] M. Bruschi and O. Ragnisco, "The Periodic Relativistic Toda - Lat­ tice: Direct and Inverse Problem", Preprint JV615 (1 Agosto 1988) Universita di Roma. [8] S.J. Alber, "On Finite-Zone Solutions of Relativistic Toda Lattice", Lett. Math. Phys. 17 (1989) 149-155.

231

[9] G. Wilson, "Commuting Flows and Conservation Laws for Lax Equa­ tions", Math. Proc. Cambr. Phil. Soc. 86 (1979) 131-143. [10] B.A. Kupershmidt, "Discrete Lax Equations and Differential-Difference Calculus", Asterisque (Paris, 1985). [11] Yu. I. Manin, "Algebraic Aspects of Nonlinear Differential Equations", J. Sov. Math. 11 (1979) 1-122.

232

Liouville Generating Functions for Isospectral Flow in Loop Algebras f M.R.

ADAMS 1 , J. HARNAD 2 and J.

HURTUBISE 3

1

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

2

Department of Mathematics, Coucordia University, Montreal, Que., and Centre de Recherches Mathematiques, University de Montreal, C.P. 6128-A, Montreal, Que., Canada, H3C 3J7

3

Department of Mathematics, McGill University, Montreal, Que., Canada, H3A 2K6 A b s t r a c t , it is shown how the hyperelliptic coordinates leading to complete integrator of certain finite dimensional Hamiltonian systems via a Liouville generating function ma> be generalized to systems admitting a Lax pair formulation in loop algebras. The divisoi coordinates associated to the corresponding line bundles over the spectral curves provide a Darboux system for the polynomial or rational coadjoint orbits in the dual gl(r)+* tc the positive loop algebra, reduced by the natural Gl(r) action. A natural algebro-geometric definition of the symplectic structure on the space of degree g + r — 1 line bundles over s suitable g - parameter family of admissable spectral curves follows from Serre duality. The spectral transform giving an identification between the reduced rational coadjoint orbits ir j//(r) + * and this space preserves the symplectic structure. On certain nonreduced symplectic submanifolds of such orbits, an extended Darboux coordinate system is also introduced. The resulting linearization map involves abelian integrals of the first and second kinds. As ar illustration of the general method, these results are used to obtain the "finite gap" solutions o: the two-component coupled nonlinear Schrodinger equation through the Liouville integration procedure.

1.

Introduction Many integrable Hamiltonian systems can be expressed in terms of Lax equations

£-"■"■

en)

Here L and A m a y b e differential or difference operators, or t h e y m a y b e m a t r i c e s de­ p e n d i n g o n a p a r a m e t e r A, t h o u g h t of (formally or a n a l y t i c a l l y ) as a l o o p variable. T h e nonlinearity i n e q . ( l . l ) results from the functional d e p e n d e n c e of A u p o n L, w h i l e the c o m m u t a t o r structure implies that t h e resulting flow leaves t h e s p e c t r u m of L invariant. ■ Research supported in part by the Natural Sciences and Research Council of Canada and by U S army grant DAA-87-K-0110.

233 In what follows £(A) and A(X) will be elements of a loop algebra fl (either gt(r) = ,A 1]] ]] or <7/(r)®C[[A,A-

some reduction thereof) and, for the most part, they will either

be polynomial or rational functions of the loop parameter A. This framework includes most known finite dimensional integrable systems as well as finite dimensional families of solutions to systems of PDE's in two independent variables, ( x , i ) . The PDE's arise as integrability conditions for commutative pairs of isospectral flows governed by two Lax equations: fx=[A,L]

(1.2a)

~

(1.2b)

= [B,L\.

A large class of such systems, viewed as Hamiltonian flows on the dual (fl + )* of the positive part of the loop algebra (i.e. non-negative powers of A), may be obtained from the Adler-Kostant-Symes (AKS) theorem ([Ad], [K], [S], [RS], [FNR]). For this, one generally makes an identification

fl~fl* via an Ad-invariant metric (X,Y) = tr(X(\),Y(\))0

(1.3)

(where the subscript 0 signifies the A - independent term) and considers Hamiltonian flow in the phase space

(§ + r ~ a-, where fl+ is the subalgebra consisting of sums over nonnegative powers of A only, and fl_ denotes sums over nonpositive powers. The AKS class of Hamiltonians is obtained by taking the restriction to (fl + )* of elements $ 6 ■?(§*) in the ring of Ad*-invariant functions on §*:

^ = *l(S+)-8- •

0-4)

+

Viewing L £ (fl )* ~ fl_ as a Laurent series consisting of nonpositive powers of A , the corresponding Hamiltonian equations become ^

= [W(I(A)))+,L],

(1.5)

234 where the subscript + denotes projection onto j j + . If we think of L(\)

as having a matrix

representation (i.e. embedded in gl(r)), the ring I(fl*) will be generated by functions of the type: #(A,L) = tr(R(A, A - \ L ( A ) ) „ ,

(1.6)

where R(X, A _ 1 , z ) is a polynomial in (A, A - 1 , 2 ) . For purposes of generating non-trivial flow, it is sufficient to consider iZ(A,A - 1 ,z) independent of A - 1 (i.e. polynomials in A), since the projection in (d)+ will eliminate the contributions from negative powers of A. (Note that often the opposite notational conventions are used, with A replaced by A so that £(A) appears as a positive power series or polynomial in A and R = R(X~1,z)

, is

polynomial in A - 1 and z.) The AKS theorem states that Hamiltonian flows on (fl + )* corresponding to any two # 1 , $2 6 I(0") commute. It is then possible to prove for the case where L(X) belongs to a polynomial or rational (and hence finite dimensional) orbit in (fl + )* that, under rather weak genericity assumptions, the AKS ring I+ = I(j}*) |(s+)* provides a sufficiently large number of Poisson commuting elements to ensure that the resulting finite-dimensional systems are completely integrable. To illustrate this approach, let us consider two typical examples of such systems; the first being the Rosochatius system [AHP], a slight generalization of the Neumann oscillator; the second, the two mode coupled nonlinear Schrodinger (CNLS) equation ([AbS], [AHP], [AHH1]), a system of coupled P D E ' s of importance, e.g., in applications t o nonlinear optics.

a) T H E R O S O C H A T I U S S Y S T E M

T h e phase space, as in the Neumann oscillator, is the cotangent bundle TmSn~1 of a sphere, thought of as embedded in R " x R " = {x, y] by the relations | I | 2 = X - I = 1,

xy

= 0.

(1.7)

The Hamiltonian is

ff<*,0 = 5 £ r f + £ > * + £ $

(1.8)

and thus is the sum of the Neumann oscillator Hamiltonian (kinetic energy plus harmonic oscillator potential) plus a nl/x]

potential for each coordinate axis. T h e resulting Hamil­

tonian system may be put in the Lax form (1.1) using the constructions of [AHP] by

235 defining:

^ l1 A AA 2-f-fA-a, -2&X=Z[

JV(A) aWm =

--«?-*£ v ? 2- 4^ 1 1 1 -y)-Vi-^f-

sXjyj + i\/2fij

.

X2-

(1.9)

-XjVj + iVZfij. -xjyj-ttVZHjl

which is viewed as an element m e n t of the F Poisson manifold
\-na(\)N(\)

(1.10)

a(\) = Y[(X-ai).

(1.11)

Z(A) = where

Then on t h e coadjoint orbit of N(\), the Rosochatius flow is obtained via the AKS theorem in Lax form (1.12)

L = [(dH)+,L], where t h e Hamiltonian H is given by:

*w«-*($

G I)]").

N{\) + A

(1.13)

and the flow is subject to the symplectic constraints (1.7). T h e specific form (1.9) for JV(A) may be obtained from a moment m a p embedding (cf. [AHP]) in a coadjoint orbit of (8(1,1) + )*- T h e other commuting invariants are obtained by expanding the characteristic polynomial of N{\).

b) T H E C N L S S Y S T E M

Here, we have two complex functions u(x,t),

v(x,t)

satisfying the coupled system of

equations: iu, + uxx = 2u(\ u | 2 + | v | 2 ) 2

ivt + vxx = 2v(\ u \ +

(1.14a)

2

(1.14b)

\v\ )

These follow as the compatibility conditions for a pair of Lax equations (1.2a,b) with £(A) G s/(3)+* of the form L(\) = L0 + LiX-1 + L2\~'+

...

(1.15)

236 where . /2 Lo = i 0 6 \0 /O Li = I u \v

0 -1 0

0 \ 0 -1 /

(1.16a)

vs 0 0,

u 0 0

(1.16b)

/ | U | 2 + |t,|2 L2 = i I uz V \ vi;zx

-5, - | u |2 -uw -uv

-v_t \ —vu J - | Iv v| 2p /y —

(1.16c)

and A, B are given by A = (
(1.17a)

B = (<*#«)+ = A 2 Io + AXi + L2 .

(1.17b)

The Hamiltonians generating the x and < flows, respectively, are Hx(L)

=

(1.18a)

\[\lTL(\y]g

ift(L) = |[A 2 tri(A) 2 ] o .

(1.18b)

The reductions involved in the specific form (1.16a-c) for L(X) may be shown consistent with the Hamiltonian flow for (1.18a,b) with the loop algebra reduced to 0 = s u ( l , 2 ) . In the following, we shall only be concerned with "finite gap" solutions, where L(X) is assumed to be a Laurent polynomial £(A) = L0 + I i A - 1 + L2\~2

+ ■■■ + £ n _ , A - " + 1 .

(1.19)

To make contact with the construction of [AHP], we assume a factorization of the form: i(A) = where

^V(A)

■\m\2 + \d? A

w(A) = ; £ A- — a j

(1.20)

-niPi

Wi

-\m?

Opi

-n,
-C,P

-viti

1=1

-1 0 I2

(1.21)

237

Pi = VI m I2 + 1 0 I2-

(1.22)

Here 7, = (m,...,,,„)

e C " , C = (Ci, - , C») 6 C "

(1.23)

are two complex vectors providing the complex Darboux coordinates for the coadjoint orbit through N(X) € ( 5 u ( l , 2 ) + ) * , with symplectic structure u = i[driT A drj + dC T A d£] .

(1.24)

If the AKS theorem is applied to determine the flow directly on the orbit through N, rather than L, the same equations arise, provided the Hamiltonians Hx,Ht

of (1.18a,b)

are replaced by:

2 fa(A) H*(N) = l ±lg±tT(N(\) )]o

^ W

(1.25a)

= |[|^tr(JV(A)2)]o,

(1.25b)

where a(\) is again defined as in eq. (1.11). T h e particular structure (1.21) is not the most general that could arise on a poly­ nomial or rational coadjoint orbit. A more complete analysis and discussion of such con­ structions is given in [AHP], [AHH1]. 2.

I n t e g r a t i o n a n d Linearization o f Flows: H y p e r e l l i p t i c C a s e Returning t o the Rosochatius system (1.9)-(1.13), let us examine briefly how the

Lax form of the equation leads to a complete integration of the system in the manner of canonical transformation theory; that is, by providing a complete set of integrals that lead to a Liouville generating function or, equivalently, a complete solution of the HamiltonJacobi equation in separated form. The trick, as always, is to use a convenient coordinate system in which t h e Liouville generating function may be determined as the sum of terms depending only on individual pairs of conjugate variables. For this case (and in fact for most classical mechanical integrable systems) this is accomplished in ellipsoidal coordinates. These are defined as the zeroes {A v } w = i i .,., n _i of the rational function

Y- A _ n:-,'(A-A,) ^

A - a,; ~

a(A)

'

W

238 where a(A) = n ! L a ( ^

—a

0 - We remark that for what follows it is pertinent t o view these

as the zeroes of the lower off-diagonal element in the matrix ./V(A) given by (1.9). The canonically conjugate momenta {C»}»=l,.,,n-1 1

are

given by:

"

c» = 2-£ A„ - <Xj so the canonical 1-form 9 on TmSn-1

(2.2)

has the form n-l

0=J2(vd\v.

(2.3)

It is more convenient to work with a polynomial in A instead of A - 1 , so we set £(A) = A—'1(A) .

(2.4)

Since the Hamiltonian flow is of the isospectral form (1.1), we have an invariant spectral curve, defined by the characteristic equation: dei(L(X) - zl) = j(w2 - 0(A)) = 0

(2.5)

.a-a^+ivSWgj^)

(2.6)

where

and 0(A) = - 8 a ( A ) P(A) .

(2.7)

The coefficients {Pi} of the invariant polynomial P(A) = A " " 1 + P 1 A " - 2 + ■•• + .?>„_!

(2.8)

define our complete set of integrals. In particular, the Hamiltonian (1.13) is essentially the first coefficient: P j = H + const.

(2.9)

T h e main point is that the momentum £„ can be explicitly expressed in terms of the corresponding ellipsoidal coordinate A„ and the invariant spectral d a t a alone:

1 y/P(K)

C" _ly/P(K) = " 44 «(A„) a(A„)

C

(21Q)

(2.10) (2 10) '

239 (cf.

[GHHW], [AHP], [AHH2] for further details). It therefore follows, if we restrict to

the invariant Liouville-Arnold torus T n _ 1 = {(A„, £„) | Pi = const.}, that a generating function 5 ( A j , . . . A n _ 1 , P 1 ) . . . P„_i) exists (denned within a tubular neighborhood of T n _ 1 ) such that 60 | |Tjni -. -1 i==e i SdS{X\,. (A1,.■ ..A P nn_- 1l )) .. n_ ■A P1i,,.-.. .. P n -1 i, ,P

(2.11)

Substitution of (2.10) into (2.3) and integration gives S in the completely separated form: Substitution of (2.10) into (2.3) and integration gives S in the completely separated form: Passing to t h e coordinates

*aS *aS

(213) (213)

"^J.7*max e - S / 73w"

(214)

Passing to the coordinates conjugate t o the P ; , we obtain conjugate t o the P ; , we obtain

n_1

/•A,

\n-j-l

(2.14) fA, \ n - i - l conjugate to the P; , we obtain These are "completely ignorable coordinates'', in terms of which the flow is linear for all n_1 n_1

Q

[K,

\n-i-l

(214)

These are "completely ignorable coordinates'', terms ofappropriate which the flow linear for all the Hamiltonians generated by { P i , . . . P „ - i } . in Choosing linearis combinations thet hHamiltonians generated by {these, P i , . . . together P „ - i } . Choosing appropriate linear of e {Qi} to ensure periodicity with the corresponding dual combinations linear combi­ These are "completely ignorable coordinates'', inwith terms of which the flow is linear linear combi­ for all of t h e {Qi} to {Pi}, ensureprovide periodicity these, together the corresponding dual nation of the the action-angle variables . For the Hamiltonian (1.13), (2.9), the Hamiltonians by action-angle { P i , . . . P „ - i }variables . Choosing appropriate linear combinations nation of the {Pi}, generated provide the . For the Hamiltonian (1.13), (2.9), Hamilton's equations give us: f)TT of the {Qi} to ensure periodicity these, together with the corresponding dual linear combiHamilton's equations give us: a i rvariables . For the Hamiltonian (1.13), (2.9), nation of the {Pi}, provide the action-angle i=through 6ia the hyperelliptic system (215) and hence the flow is obtained implicitly Hamilton's equations give us: air and hence the flow is obtained- implicitly through the hyperelliptic system 1 ,M«) (215) — ' "=M*AM ' & = AWi = «i,i< ■ and hence the flow is obtained implicitly through the hyperelliptic system y/QW

Q wr '

Sl«-5TO- M -

(2 16)

-

We note that since Q(X) is a polynomial of degree (2n — 1) , allowing (X,w)

to

take complex values in equation (2.5) defines a complex curve X of genus g = n — 1, and hence the left hand side of (2.16) consists of abelian integrals over the g linearly independant holomorphic differentials of the first kind. Up to a change of basis, (2.14) may be interpreted as the Abel map: A : SgX -+ J(X) = C'/T

,

T = ( period lattice of X) ,

240 taking the unordered set of points on X with local coordinates {A„, tw(A„)} to their image in the Jacobi variety J(X).

Thus, (2.16) implies that the corresponding flow in J{X)

linear, and allows us to identify the invariant torus T"* -1 with a real subtorus of

is

J(X).

This was an entirely classical calculation, making use of the Lax representation (1.1) only as an aid in determining our complete set of integrals. The key point consists of choosing the ellipsoidal coordinates {A„} given by (2.1), in which the conjugate momen­ t u m z„ can be expressed in terms of A„ alone, together with the invariants { P i . . . , P „ - i } determining the spectral polynomial P(A) . The completely separated form (2.12) for the Liouville generating function then implies that its derivatives (2.13) are given by holomorphic abelian integrals. These results are based on the fact that when the spectral curve X defined by eq. (2.5) is hyperelliptic, Hamilton's equations, expressed in ellipsoidal coordi­ nates, have the form of a Weierstrass (or Jacobi) system of differential equations associated to this curve. The hyperellipticity is a consequence of the fact that the Lax equation in­ volves 2 x 2 matrices - or more precisely, the underlying loop algebra is u(2). Numerous other systems, such as the sine-Gordon, NLS and MKdV equations, can be given a 2 x 2 Lax form ([AbS], [FNR]) and integrated in a similar manner, using the hyperellipticity of the spectral curve. The question is, how can such a procedure be extended to apply to Lax systems (1.2a,b) of the type denned by (1.17), (1.20), (1.21), governing the CNLS equation? In this case the matrices are 3 x 3 , the spectral curve is of the form d e t ( i ( A ) - zld) = -z3

+ za(\)P(\)

+ a2(X)R(X)

= 0 ,

(2.17)

which is trigonal, not hyperelliptic, and ellipsoidal coordinates do not naturally arise. Is there a generalization of ellipsoidal coordinates that will do the same work of separation of variables for the case of trigonal, or more general spectral curves arising from Lax equations of higher rank; i.e., having the same relationship to more general curves that ellipsoidal coordinates have to hyperelliptic curves? The answer is: yes, and such coordinate systems, which will be described in the following section, arise naturally from the study of integrable systems within the framework of loop algebras by means of the algebro-geometric methods developed in recent years ([Kr], [Du], [KrN], [AvM], [vMM], [RS], [AHHl]). Earlier work by Dickey [D] showed how a Darboux coordinate system suitable for applying the Liouville integration method could be deduced for "stationary flows" within the context of the

241 Krichever-Novikov approach ([KrN], but did not develop the relationship to the coadjoint orbit structure in loop algebras, which will be central to the approach described here.

3.

Liouville G e n e r a t i n g F u n c t i o n for Higher R a n k S y s t e m s

a) A L G E B R O - G E O M E T R I C INTEGRATION M E T H O D

Before proceeding to the question of suitable Darboux coordinates and separation of variables for more general systems, let us recall the, by now, standard approach to integration of polynomial Lax equations in terms of line bundles over spectral curves (cf. e.g. [RS], [AvM], [AHH1]). Given a Lax equation of the form ^ ( A ) = [iZ(A-1,I(A))+,i(A)],

(3.1)

where JZ(A_1, z) is a polynomial in its arguments and L(A) = LoA"- 1 + • ■ • + I „ - i

(3.2)

is an r x r matricial polynomial (note that we have set £(A) = A n _ 1 L(A) as in (2.4)), the invariant spectral curve X0 C C 2 is determined by its defining equation: P(A, z) = det(Z(A) - *I) = 0 .

(3.3)

This will generally be r-sheeted over C P 1 , corresponding to the r eigenvalues 2i(A) for any given A. A compactification Xo «-» X which allows us to treat the points at A = oo on an equal footing may be denned by regarding such curves as embedded in a surface T , defined as the total space of the fine bundle 0(n — 1) —» C P 1 , with (A, z) viewed as base and fibre coordinates over one of two affine neighborhoods: U0 = C P 1 — {oo} , U0 = C P 1 — {0}, providing a Leray covering of T (cf.

[AHH1]). (A further desingularization of X may

be required at points where the spectrum of L(X) is nonsimple, but we suppress such complications in this discussion; see [AHH1] for details). We define a line bundle E —> X which is the dual of the eigenvector line bundle for LT(X); i.e. the fibre of E, at a generic point (A, z) G X, is the dual of the 1-dimensional eigenspace {„(A, z) e C r | LT(X)v(A, z) = z(X)v(X, z)} .

(3.4)

242 Denoting by t —* L(X,t)

the flow of matricial polynomials satisfying (3.1), we get

a corresponding flow t —* Et of line bundles. Under suitable genericity assumptions one shows ([RS], [AHH1] that the degree of E, is: degE, = g + r - 1 ,

(3.5)

where g is the (arithmetic) genus of X . Expressing Et in terms of its initial value Et = E0 ® Ft ,

(3.6)

where Ft is of degree zero, we find that the transition function on Uo 0 U\ defining Ft is etR(\~

,z)^ ^ j

nence

f _> pt defines a linear flow on the Jacobi variety J(X)

. The inverse

problem, i.e. determination of L(A,i) from Et, is solved by noting t h a t , relative to a chosen trivialization of Eg, and a basis of sections for Et —* X determined by a normalization at A = oo, L(X, t) is determined as (iW L(X,t)

= tp-1\

•..

V where { « J ( A ) } I = I „

r

\ U .

(3.7)

*W

are the eigenvalues of L(X) at A and the (r x r ) matrix tp(X) is the

so-called generalized Baker - Akhiezer matrix. The entries of «/' are defined by evaluating sections of Et, represented on U\ by meromorphic functions on S with zeros at r

1

points over A = co, poles permitted at a fixed divisor A of degree g, and an essential singularity at A = 0 of the form etR^x {p,(A)}i=i

r

'*' . Let the r points of the curve over A be denoted

and let S{ : Ui —* E\ut be a section vanishing at {py(oo}j/i. Then relative

to the trivialization on U\ we define ij>ij{\) = s;(py(A)). Such functions may, for divisors A in general position, be expressed by standard means (see e.g. [Du], [Kr]) in terms of ©-functions and integrals of Abelian differentials of the second kind having the same polar part at A = oo as

dR(\~l,z).

This method of solution, although it effectively produces a formula that expresses L(X,t)

in terms of theta functions, and hence solves the Lax equation (3.1), bears no

apparent relation to the underlying Hamiltonian dynamics of the system.

However, a

careful examination of the underlying procedure reveals that it may also be obtained from a generalization of the Liouville generating function and the Darboux coordinates

243 discussed for the byperelliptic case in the previous section. Rill details of this analysis will b e provided elsewhere [AHH2], but we give here a summary of the key results.

b ) L A G R A N G I A N F O L I A T I O N AND D A E B O U X C O O R D I N A T E S

T h e line bundle E denned above is determined by the zero divisor of any of its sections. Of the g + r — 1 points forming such a divisor, it is possible to choose r — 1 of t h e m over A = oo. (Such sections are exactly what define the rows of the Baker - Akhiezer matrix ip.) There remains a set of g points, with coordinates {A„, •*„}„-!

g

on X0.

In

computational terms, let Z(A, z) be the classical adjoint (matrix of cofactors, transposed) of (1(A) - z l ) ; then Z(A, z)(L( A) - zl) = V(A, z ) I .

(3.8)

Choosing a fixed vector Vb G C , {A„, z v } I / = i ) ... l J may be chosen as the roots of the system of polynomial equations Z(A,z)V 0 = 0 ,

(3.9)

which, from the construction above, give the zeroes of the section defined by tpVo. Let Of!(x) denote the coadjoint orbit in ( § + ) * containing N(X), a rational element in gl(r)

N{x)=

* of the form

t^i=l

We consider, as before, L(\)

A

(3.10) a

'

and £(A) defined by:

i(A) =

\-na{\)N{\)

= L0 + i i A " 1 + i 2 A " 2 + ■ • • + X ^ A - " - 1 " 1 J

£(A) = A— L(A) .

(3.11a) (3.11b)

T h e group G with Lie algebra g acts on 0^(\)

by conjugation and this action, which

is Hamiltonian, has Lo, the leading term in L(A), as moment map [AHP]. It follows that L0 is constant under the AKS Hamiltonian flow for any $ 6 ^(fl*)- We may therefore make a Hamiltonian reduction 0N(A) => ONMIto=««.

-

S = (0NWU„=C.O/GLO ,

(3.12)

244 where GL0 C G is the stabilizer of £o- On a generic set, (3.12) defines a smooth fibration and the base space inherits a reduced symplectic form wred from the Kostant

Kirillov

form on 0]^{\\. This reduced symplectic manifold 5 has precisely dimension 2g. In order that the functions {A„, zv}v=i,...,g

on Ojv(A)Uo=conji. be projectable to S, they must be G L 0 invari­

ant. It is clear from their definition as roots of equation (3.9) that a sufficient condition for this is that the group Gj, 0 leave the eigenspace [V0J invariant. Thus, we require L 0 and VQ to be chosen compatibly in the sense that Gi„ C G[v0], where G[v0j C G/(r) is the subgroup stabilizing the one dimensional subspace [Vb] C Cr- This is satisfied if V0 is chosen as an eigenvector of La. Let K be the isospectral foliation of C?N(A). Since K is Gl(r) invariant, it projects to define a foliation of S, which we denote KsC = zv/a(\v)

,

Define (3.13)

where, as before, n

a(A) = n ( A - « .

(3-14)

i=i

THEOREM 3.1. There exists an open dense subset, So C 5 , with spectral curves of constant genus g, on which the projection 0Ar(A)|io=c<m»<. —> 5 is a fibration. The foliation Ks is a Lagrangian foliation, with leaves identified with J(X) — T>x, where T>x is a codimension 1 subvariety (essentially the Q-divisor). {A„, £„}„ = i

s

On the open dense submanifold So C S the functions

project to define a Darboux coordinate system, a WrerfU. = J ^ d C A d\u

i.e.

.

(3.15)

i/=i

The full proof of this result will be detailed elsewhere [AHH2]; the following is a brief sketch of the main steps involved. The identification of the isospectral leaves of Ks with J(X)

is shown, e.g., in [AHHl]. It may be seen from the structure (3.10) of elements of

ON that, for n sufficiently large, the G t 0

action is free and proper on an open dense

subset, where projection to the quotient becomes a fibration. Defining M(A,C) = ^ - C I ,

(3.16)

245 the spectral curve Xo determined by eq. (3.3) is equivalently denned by: detM(A,C) = 0 ,

(3.17)

z =
(3.18)

with the identification

Let M(A, Q denote the classical adjoint of M(A, Q. Thus Z( A, C) = [a( A)]"" 1 A/(A,C)

(3.19)

M(A„,C„)Vo = 0 .

(3.20)

W(X,0 = M(\,C)V0,

(3.21)

and eq. (3.9) implies

Define r

viewed as a two parameter family of C - valued functions on On, whose components all vanish at the parameter values {A = A„,£ = £„}: m(A»,C») = 0 .

(3.22)

Since we are assuming that L(X) has a nondegenerate spectrum, JW(A„, ("„) has rank 1. It follows from the Lie-Poisson bracket relations satisfied by the matrix elements of JV(A),

{NWii,N(a)u) = * » , ,

_ 'KVbt-m*)^

A — tj

,

(3.23)

A —a

that the Poisson brackets of the functions W{ A, Q, evaluated at pairs of divisor coordinates

{(VC(«)> (*»>&<)}

ale

{Wi(A,C),W0(«=C.} = 6*JCti •

(3'25)

where XftsdrtF£ / 31V;

fjj = I aivj \

9A

(3.26) dW: \

aWj I d< /{A=A„,<=0,}

(3.27)

246 Locally, the functions (\V(N),£V(N))

are determined by any two of the relations

(3.22) that are functionally independant, say, those for i = 1,2. Applying the chain rule to the evaluation of Poisson brackets of the divisor coordinates, and differentiating the two determining relations implicitly along the orbit ON gives:

({*„K} {A„c,}\ w ( A , o , w 1 ( » , ,0} o } {Wi(A,C),Wa(0 w ( A , o , w i ( » , o n r FF. ^. Tr r1i f{A„,A„} {VCr}> . , p , r l f.i/{Wi(A,C),Wi(a = o%r 12 J {W 2(A,C),W2(<7 ,0} UCwM (C, (3.28) (3.28)

,«r

r

evaluated at {A = AM,£ = ^,a

= A„,£ = £„}. Combining with (3.25) gives the result:

{A„,A„} = {0.,C*} = 0,

{Ap,C»)««K»'

(3-29)

This does not yet prove that the functions ( A ^ C ) give a Darboux coordinate system for the reduced manifold, since they do not Poisson commute with all the constraints determining 0jv|£ o sconj< - only those generating G t 0 . To verify that the projections of these functions to the reduced symplectic manifold 5 nevertheless satisfy the same Poisson bracket relations as (3.29) with respect to the reduced symplectic structure, we proceed in the usual way for constrained Hamiltonian systems. That is, we modify the functions (A M ,£„) by adding terms proportional to the constraints, chosen so t h a t the Hamiltonian vector fields of the modified functions be tangential to the constrained submanifold CAT|L 0 =C»(- K w e choose a generic reduction condition with diagonal LQ: (L0)ij

= ptSij ,

(3.30)

where the eigenvalues {fit} are chosen distinct, the modified divisor coordinates are:

. E u .{A„,(£io)ji} .«( _- ^Pi1 ( I o ) o

A„ = A„-

Mi

C, c, == cC,> - E

«

(3.31a)

1*'

W ~N

J '-{L»)a (£»)o-■

( 331b )

Evaluating their Poisson brackets, it follows that on the constrained manifold, these modified functions satisfy the same relations (3.29) as do (A„,C„). T h e 2 j projected functions therefore do, indeed, provide a Darboux coordinate system for t h e reduced manifold 5 , with reduced symplectic structure given by eq. (3.15). For nongeneric reduction conditions (i.e. i 0 having a degenerate spectrum, as for the CNLS example, or a nondiagonal

247 Jordan normal form) the constraining procedure must be modified accordingly, but the result remains valid. We now have, at least on the reduced space So, a Darboux coordinate system that generalizes the ellipsoidal coordinates denned by eq. (2.1) in the case of hyperelliptic X. (Note t h a t equation (3.9), with Vo — ( 1 , 0 ) T does indeed reduce to equation (2.1) plus the spectral equation (2.5) for this case, and, analogously, for all cases with r = 2.) The generalization of formula (2.12) defining the Liouville generating function follows similarly. Denning the canonical 1-form

9 = V (¥dX, = V -%-rdK ,

(3.32)

we have that the restriction 6\Ks=dS(X1,...,Xl,P1,...,P!l)

(3.33)

is exact. Note t h a t we have introduced precisely g spectral invariants {Pi}i=i,... l9 that parametrize the spectral curves X whose Jacobi varieties J(X) foliation of So-

define the Lagrangian

Since So is 2g dimensional, there must be g free parameters defining

the permissible curves X C T obtained by compactification of the Xo's arising from the characteristic equations. An explicit parametrization of these curves is given in [AHH1], but the general result will not be needed here. Integrating (3.33) from some initial point on the invariant torus (X fixed) with local coordinates {Aj,z°}, along a p a t h within the torus, we have: g(A1,...,A>,tt,...,p,) = j r / ^ *

( A

'*;:"'

p

')dA,

(3.34)

determined u p to elements of the period lattice. As in the r = 2 case, the coordinates z„ = z{\v,P\,...

,Pg)

are define d implicitly by the spectral equation (3.3) and hence

depend only on the spectral invariants {Pi,...,P}}

and the single corresponding base

coordinate A„ entering in the divisor 9

T> =1£/p(\v,zv)

.

B=l

where p(A,z) E X denotes the point with coordinates (A, 2).

(3.35)

248 Applying now t h e standard procedure from canonical transformation theory, we define the new coordinates {Qi} conjugate to t h e {Pi} by : n

0,_

dS

A

[x>

Ri{\,z,Pu..

ap-^yAS

$

■'P9hx,

(3.36)

where

R

^-ix)W^p--

(3 37)

•'•>■

-

It follows from the structure (3.10), (3.11) of L(A) that iij is a polynomial in b o t h A and z. For any Hamiltonian H(PU ..., Pg) in t h e ring generated by t h e {Pi}'a we then have a linea r flow in terms of the new coordinates: Qi = Q°i+ <M

(3.38a)

Pi = const.

(3.39b)

where a; = ^ | r . The rational functions defining the integrand in (3.37) have the form of Poincare residues ([GH], [AHH2]) with respect t o the embedding X ■—» T

If we knew that the

polynomials iZ,(A,«s,Pj,,...,P g ) have the correct structure to imply that the differentials wj a

^Ri(X,z,P ( A ^ . P ^u..- . - . PP , , ) ^ ' ^d\

( 3 4Q)

(3.40)

dz

are holomorphic and independent on X, the relation (3.36) would, up t o a change of basis, define the Abel m a p A : S9X J(X)

—> J(X),

with ( Q i , . . . ,Qg) G C ' linear coordinates on

= C 3 / r . That is, (3.36), (3.38a) would define the linearization of flow on the Jacobi

variety J(X).

T h e next theorem, which is the last main result of this section, tells us that

this is precisely the case.

THEOREM 3.2. The differentials

{a>j};=i

g

axe holomorphic

on X and provide a basis

for H°(X, KX) (where KX is the canonical bundle on X). In fact, a more general version of this theorem is valid allowing, for example, polyno­ mial orbits or degenerate spectra at a finite number of points. T h e resulting curves acquire

249 singularities at such degenerate points, requiring the curves to be desingularized by blowing up a suitable number of times. The genus formula must be modified accordingly for these cases (cf. [AHHl]), but otherwise the same procedure remains valid. Again, the full proof of Theorem 3.2 may be found in [AHH2], but we indicate here the key ideas behind it. Since the family of curves given by equation (3.2) all pass through the same r points over each A = a,- and over A = oo , the structure of the polynomials in the differentials (3.39) coincides with the explicit realization of holomorphic 1-forms obtained by evaluation of Poincare residues of 2-forms on T with simple poles along X (cf.[GH], [AHH2]). The polynomials ft,(A, z) defined by equation (3.37) arise from infinitesimal deformations of the spectral curves within the class embedded in T via the spectral condition (3.3). Denoting by /if the normal bundle relative to the embedding X •—* T, the constraints implying the passage of the curve through the fixed points over A = <*;, oo, imply that the space of infinitesimal deformations within this class of spectral curves consists of sections of M that vanish at these points. These deformations may therefore be interpreted as sections of JV(—n — 1) = JV ® C(—n — 1), where O(l) is the line bundle obtained by pulling back 0(1) —* CP1 to T and restricting to X. An explicit computation of the transition functions [AHHl] shows that 0(—n — 1) is identical to the restriction KT\X °f the canonical sheaf for T to X. Hence Af(—n — 1) ~ M ® KX — KXi the latter following from the identification KT\X = A T*(T)\x

=

K

X ® A/-* (where, as

usual, sheaves of sections are denoted by the same symbol as the line bundles). Explicitly, a section of A/"(—n — 1) is expressed by the vector field

r(z,A) d 9£ dz ' 8x where r(z,A) is a polynomial of the appropriate degree, corresponding to sections of C(r(n — 1) — 1) = Af(—1) that vanish over A = a,-. given by:

r(z,X) r(»,A) d9d _ r(z,X) r(»,A) , . r(*,A) BP dz %«» Bz

The map A/"(—n — 1) —» KS is

'

which is the result obtained via Poincare residues, with r(z, A) any linear combination of the polynomials iZ<(A, z) given by equation (3.37). This interpretation also allows an intrinsic definition of the symplectic structure. The tangent space to the leaves of the isospectral Lagrangian foliation may be identified with

250 the space ^(X,

Ox)

of infinitesimal deformations of the line bundles over X, and the

deformations of the curves correspond to elements of 'H 0 (X, KX)- The embedding X —» T gives rise to a splitting of the tangent space of So by extension of line bundles to the first formal neighborhood of any given spectral curve. This allows an identification of the tangent space to So at any point with the direct sum W 1 (X, Ox) + "H°(X, KX), upon which a natural symplectic form is defined in view of the Serre duality. This may then be shown equivalent t o the reduced orbital symplectic structure. (See [AHH2] for full details.)

c) S I N G U L A R C U R V E S AND M I S S I N G D A R B O U X C O O R D I N A T E S

In order to apply the above procedure to the integration of the CNLS example and similar problems, two further difficulties must be resolved. First, the functions u and v, which are related to L(X) by equations (1.16b) are not invariant under G L „ , and hence are not defined on the reduced space S. Second, from (1.16a), the spectrum of £(A) at A = co is degenerate, giving rise to additional singularities in the spectral curve over A = oo that must be resolved. This implies a reduction of the genus g of the spectral curve involved and hence, seemingly, to an incomplete set of Darboux coordinates. These problems are resolved by noting that, rather than fixing a level set of LQ and quotienting by GL0, we may pick a symplectic submanifold of ON0, invariant under the flows in question on which all the constraints implicit in the special form (1.16ac) are satisfied.

The Darboux coordinates may be completed by adding to the divisor

coordinates {A„, z v } v =i,..., s a sufficient number of further spectral coordinates "at infinity" i.e., determined by the leading terms in the series for L(A) about A = oo. The resulting abelian integrals include some with poles at oo, but this poses no obstacle to evaluating the solution in terms of ©-functions. The procedure is quite general, but we illustrate it here only for the case of the CNLS equation. First, we note that because of the special form (1.21) assumed for L(X) the spectral curve takes the form (2.17). where n-2

P(A)=]£P,A'

(3.41a)

l=o n-3

fl(A)=£fl,A'

(3.41b)

251 are invariant polynomials. If we consider the curves for which the leading terms of L( A) are given by equation (1.16a-c), we see that, besides the fact that there are double points at (A = a,-, z = 0), there is also a double point over A = oo, with the intersecting branches tan­ gential to order three. Resolving these singularities gives rise to a curve of genus g = 2n—8, which is the number of (finite) pairs {A„, zv}v=il...,3

satisfying eq. (3.9). The full coadjoint

orbit, however, is of dimension 4n, and hence we must impose a certain number of symplectic constraints, compatible with the CNLS flows, and add some further coordinates to obtain a complete Darboux system. The constraints are defined as follows. First, rather t h a n requiring the specific form for L0 given by (1.16a), we just set the off-diagonal terms equal to zero. This gives six constraints that define a symplectic submanifold only away from the points where the eigenvalues are degenerate. However, we now impose six addi­ tional constraints that define a symplectic submanifold even at the points of degeneracy. This consists of requiring that for three pairs of coordinates, say {Ai,2(}j =1|2l 3 (counted with multiplicity), we have M(\i,zi)

= 0.

(3.42)

When LQ has multiple eigenvalues, one of the Aj's is oo. The CNLS curves correspond to the case where the three points { A ; , Z J } coincide at Aj = oo. Eq. (3.42) is equivalent to ker(L(A;) — z;I) having dimension greater than one. Thus, at each such point, L(A;) has a multiple eigenvalue, with at least two independent eigenvectors.

An equivalent

characterization of these constraints is to require that for £(A) near the CNLS curve (where it is diagonalizable near oo, with a double eigenvalue osculating to order two in A - 1 ) , L(X) be diagonalizable, even at the neighbouring double eigenvalue points. Note also that the spectral curves are singular at these points, and, generically, each singularity reduces the genus by one unit. Thus, on the constrained submanifolds, the spectral curves generically have genus g — 2n — 8. The resulting submanifold of the orbit ON is symplectic, but of dimension 4n — 12, and hence the set {A (/ ,Zi,}i/=i I ... l 2n-8 still does not provide a complete coordinate system. The "missing" set of remaining Darboux coordinates are easy to identify however. We may choose them as {qt + ^ln(pi — p 2 ) , q i + ^ln(pi — P2),Pi,P2}, where ?i = ln(u), Pi = (£0)22, u = (£i)i2,

q2 = ln(u)

(3.43a)

P2 = (£0)33

(3.43b)

v=(Li)t3.

(3.43c)

252 The orbital symplectic form, restricted to this 4n—12 dimensional symplectic submanifold has the form: d\v A dzv

E

t

.

,

1

+ dgi A dpi + dq2 A dp2 + ... ,

/ a

...

(3.44)

a A

„=i

\ v)

when evaluated on the submanifold where the three singular points {A,, Zj},=i,2,3 coincide at oo. The remaining terms only involve {pi, p2) and hence just give rise to a shift in the integration constant when the Liouville integration method is applied. Applying the Liouville method, we express the generating function for a canonical transformation to the linearizing coordinates as S = S(\a,q1,q2,Pi,Rj,pi,P2) where {Pi}j=o

n-3, {Rj}j=o,..,n-7y

,

(3.45)

Pi, Pi are the independent integrals of motion and

{A„}„=i,.. l 2n-8, 4i = hiu, q2 = lnti provide coordinates on the invariant Lagrangian man­ ifold C. Then on C, dS=y2 -Tr\dX" + Pid9i + P^i

(3-46)

and, integrating

5

=Y, j " $udX+Piqi+P2q2 ■

(3 47)

-

Taking into account the constraints on the admissible curves, one can only deform the CNLS curve into curves of the same genus, i.e., into curves with the same number of singularities. In concrete terms, this allows one to express i i n _ 8 , f l n _ 5 , i i n _ 4 in terms of the other integrals of motion. The leading terms, on the other hand, can be written as functions of pi,P2 Pn-2 = pl+pl+

P1P2

Rn-3 = P?P2 + plpi .

(3.48a) (3.48b)

Since the Hamiltonians for the x and t flows are just Hx = -Pn-3

(3.49a)

At = Pn-4 ,

(3.49b)

253 the canonically conjugate coordinates dS

sr> -

{Qj

\

= W/^-'"-*'

sc

-

dS

{Sk

undergo linear flow:

\

} =0

7

= a^ * '-"- '

I-

dS

{qi

i

= a^ }i=1 ' 2

(3.50)

Qj = Cj + Sj>n-3X + Sj^n-it

(3.51a)

Sk = dk

(3.51b) (3.51c)

9i = «i •

Evaluating the derivatives in (3.50), taking into account the constraints which determine P„_ 2 ,.R„_3, ..,Rn-e

in terms of the other constants of motion, and inserting ;he values

P! = p 2 = — i/3 corresponding to the CNLS curve, we obtain: 2

W

'

V

+

\ZX"~2

f^

hiJo

3 a W f t > - 5 * " " 6 + «y.n-4(A"- 5 - ajA"" 6 )]

-3** + a(A)P(A)

i 3

+

ia(A)« > , n _ 3 (A"- - a i A n - 4 + (a? -
>»-» /A, S k

~ ^ L

a (A)A*

(3.52b)

- ^ + a(A)P(A)rfAl

j = 0 , . . . , n - 3,

(3.52a)

& = 0 , . . . , ra - 7 ,

9i = lnu + 7 + fcj J

(3.53a)

g2 = lnu + 7 + fc27 ,

(3.53b)

where J, J are defined by

/=

2A

2

3

4

2 + + i°W[-*"~ A ■' VV1 " / /*' ^ [ [^ " """ 3 a ( A ^ A " ~ 3 _ -0 l °i^"~ " ~ 4 + +( Q(gf l _ ^! X [ -3*» + a(A)P(A)

^7o

I

| a ( A ) ( - a ? + 2a!a 2 - a 3 ) A " - 6 ' dA -3z2+a(A)P(A)

+

7 r-v" t '-«£/

-3*» + a(A)P(A)

5 -a 2°2)A"~ )A""5l ]

a A A 6 ( ) "" n - 3 ;+: ( A)P(A)^-

(3.54a) 4b (3.54b)

^ )

T h e constants a!,a2,a3,fci,fc 2 are defined by: n ai = ^ a i , ■=1 ■=i

n a2 = 2_jaiak, Mi Mi

n aiCijak , a3 = J j <*;<*><** M . ' i tfc i*i*k

(3.55)

254 Jfc! + jfc2 = -(coefficient of A 2 " - 5 in a(A)P(A)) , 3

9

n 1

Jfcjjfcj = -^(coefficient of A "" in (iA - a(A)P(A) - 3a?(\)R(\)) 3

. (3.56b)

T h e g = In — 8 Abelian integrals entering in the expressions for {Qj\j—a {<S'*}*=o

(3.56a)

n-a and

n-7 are all regular and linearly independent, and hence (3.52a,b) defines, up to

a change of basis, the Abel map: A : S'X

-» C» .

The two remaining abelian integrals, I and J in (3.53a,b) are singular with simple poles in their arguments over A = oo. Using reciprocity theorems relating abelian integrals of the first and second kinds ([GH], [AHH2]), eq. (3.53a,b) may be solved for ln(u) and ln(t>) in terms of {Sj,

Qk}.

One can then, in the usual manner, invert the Abel map, and express u, t; in terms of i and ( by using 0-functions. The resulting solution for u, v is of the form ([AHH1]):

.«*+•* B&AI + W + xV) u(x v ,<)= --Kie u(x,t)-K Q{Al+tU e(A! +tU ++xV)xV) «„,+*„ &{Ai 0 ( ^ 3++W« / ++XV)xV) v(x rt -=- K2 Koe""^" v(x ,*) &(A1 + tU +xV) v{x,t)-K e Q y ' 2

where the vectors U, V € C

{Al

+ iU + xV

(3"57a) (3 57b^

( 3 - 57b )

are determined by applying to the x and t coefficients in

eq. (3.51a) the linear transformation taking the integrals (3.52) to a normalized basis of abelian integrals of the first kind with respect to a standard homology basis. The additional integration constants appearing in the exponential terms in (3.57a,b) are similarly determined in terms of the transformation of (3.54a,b) to normalized abelian integrals of the second kind. The remaining constants A%,A%fA3 appearing in the 0 - function arguments and the overall normalization constants Ki,K% may be linearly expressed in terms of the integration constants in eq. (3.51a-c), the Abel m a p evaluated at the three points at infinity and the Riemann constant.

255 REFERENCES

[AbS] Ablowitz, M.J.aad Segur, H., Soliiona and the Inverse Scattering

Transform, SIAM

Studies in Applied Mathematics 4, Philadelphia 1981. [Ad] Adler, M., "On a Trace Functional for Formal Pseudo-Differential Operators and the Symplectic Structure of the Korteweg- de Vries Equation", Invent.

Math. 50 (1979),

219-248. [ A H H l ] Adams, M., H a m a d , J. and Hurtubise, J., "Isospectral Hamiltonian Flows in Finite and Infinite Dimensions II. Integration of Flows", Comm. Math. Phys. (to appear, 1990) [ A H H 2 ] Adams, M., Harnad, J ; and Hurtubise, J., "Darboux Coordinates and LiouvilleArnold Integration in Loop Algebras", (CRM preprint, in preparation). [ A H P ] Adams, M., Harnad, J. and Previato, E., "Isospectral Hamiltonian Flows in Finite and Infinite Dimensions I. Generalised Moser Systems and Moment Maps into Loop Algebras", Commun.

Math. Phys. 117, 451-500 (1988).

[ A v M ] Adler, M. and van Moerbeke, P., "Completely Integrable Systems, Euclidean Lie Algebras, and Curves", Adv. Math. 3 8 , 267-317 (1980). [B] Beauville, A., "Jacobiennes des courbes spectrales et systemes Hamiltoniens completement integrables", (Preprint, 1 989). [D] Dickey, L.A., "Integrable Nonlinear Equations and Liouville's Theorem I, II", Com­ mun. Math. Phys., 82, 345-360; ibid. 82, 360-375 (1981). [Du] Dubrovin, B.A., "Theta Functions and Nonlinear Equations", Russ. Math.

Surveys

3 5 , 11-92 (1981). [ F N R ] Flaschka, H., Newell, A.C. and Ratiu, T., "Kac-Moody Algebras and Soliton Equa­ tions II. Lax Equations Associated to A^h',

Physica 9 D , 300 (1983).

[ G H ] Griffiths, P. and Harris, J., Principles of Algebraic Geometry, Wiley, New York, 1978. [ G H H W ] Gagnon, L., Harnad, J., Hurtubise, J. and Winternitz, P., "Abelian Integrals and the Reduction Method for an Integrable Hamiltonian System", J. Math. Phys. 2 3 , 1248-1277 (1985).

256 [K] Kostant, B., "The Solution to a Generalized Toda Lattice and Representation The­ ory", Adv. Math, 3 4 (1979), 195-338. [Kr] Krichever, I.M., "Methods of Algebraic Geometry in the Theory of Nonlinear Equa­ tions", Russ. Math. Surveys 32, 185-213 (1977). [ K r N ] Krichever, I.M. and Novikov, S.P., "Holomorphic Bundles over Algebraic Curves and Nonlinear Equations", Runs. Math. Surveys 3 5 , 53 (1980). [RS] Reiman, A.G., and Semenov-Tian-Shansky, M.A., "Reduction of Hamiltonian sys­ tems, Affine Lie algebras and Lax Equations I, II", Invent.

Math. 54, 81-100 (1979);

ibid. 63,423-432 (1981). [S] Symes, W., "Systems of Tode Tupe, Inverse Spectral Problems and Representation Theory", Invent.

Math., 13 - 59 (1980).

[ v M M ] van Moerbeke, P. and Mumford, D., "The Spectrum of Difference Operators and Algebraic Curves" Ada Math. 143, 93-154 (1979).

257 A LOOP ALGEBRA DECOMPOSITION FOR KORTEWEG-DE VRIES EQUATIONS

Randolph James Schilling Department of Mathematics Louisiana State Univertsity Baton Rouge, LA 70803-4913

ABSTRACT Matrix Lax equations for the hierarchy of isospectral deformations of an n" 1 order scalar differential operator are formulated in terms of the Kirrillov-Poisson bracket on a rather unusual representation of the dual of the Lie algebra of polynomial loops over gl(n, d). TABLE O F C O N T E N T S (§1) (§2) (§3) (§4)

INTRODUCTION LIE ALGEBRAIC PRELIMINARIES THE NATURAL CONVERSION THE LOOP ALGEBRA DECOMPOSITION BIBLIOGRAPHY

258 (§1) I N T R O D U C T I O N In his 1968 papei l , P. D. Lax showed that the Korteweg-de Vries equation: «i + t i : » + ««i = 0,

(1.1«)

is equivalent to the scalar differential opeiatoi equation ■a

where

L = d2 + u/6,

B = 24d3 + 3(ud + du)

d = —. (1.1b) ox The idea of an isospectial deformation was introduced to mathematics. Many remarkable features of the Korteweg-de Vries equation such as the existence of infinitely many conservation laws 2 and the existence of an exact integration procedure 3 have been related or attributed to the isospectrality of L. C. S. Gardner 4 showed that the Korteweg-de Vries equation is Hamiltonian with respect to the Poisson bracket over the space of smooth periodic functions u of x given by Lt = [B,L]

{/,»}(■) = f

and

V / O ) • I VjHu) dx

(1.2a)

JM

where M = Ht/Z?, t = d and V / is defined in terms of the usual Li inner-product. It is given by the Euler expression for the gradient of a functional f:

/(«) = jM P(», .», -. .)d* => v / M = - = g M ) > ^gj.

(1.2b)

The system (11a) was shown to be a completely integrable Hamiltonian system. Analogous results were obtained by V. E. Zakhorov and L. D. Faddeev 5 with M being a space of smooth initial data that decays rapidly as | x |—• oo. I. M. Gel'fand and L. A. Dikii e considered differential operators L of the form n-> n->

L-L =--dann + ^ u y ■A*)*"-' (x)3n-J

(1.3a) (1.3a)

j3=1 =i

where «i = 0. They used the fractional powers of L to generalize the Lax equation (1.1a). They showed that the generalized Korteweg-de Vries equations are Hamiltonian with respect to a Poisson bracket given by (1.2a) with an appropriate choice of a skew adjoint matrix differential operator I. Hamilton's equations have the form Btu = t -

(1.3b)

where u is a vector consisting of the coefficients of L. M. Adler 7 discovered the Lie theoretical structure of the Lax representions of the generalized Korteweg-de Vries equations. Let £((»;)) denote the algebra consisting of all formal symbols ai(x)Tl' where the a, are smooth functions of x and m is an arbitrary integer. Let 71 denote the space of polynomial symbols. Then 11 is the dual, with respect to the Adler/Manin generalized trace form (), of the subalgebra M consisting of all symbols of the form J2 =_ aj(x)v'■ The Kirrillov-Poisson bracket is given for £ S 01 and functions f and g on 91 by this formula:

{/,y}(« = «o[v/(0,v,({)])).

(1.4)

259 Gradients are defined using (). Hamilton's equations, when expressed in terms of the coeficients of £, were shown to coincide with the Gel'fand/Dikii equations. B. Kupershmidt and G. Wilson s (See also 10)described the Hamiltonian formalism underlying the generalized Korteweg-de Vries equations of a differential operator of the form (1.3a) where the Uj are I x I matrices. The phase space is the differential algebra generated by the entries of the Uj. Their Poisson bracket, SP (1.6) IP, «}(■)= Su JQ ou acts not on functionals as in (1.2) but instead on their densities. Hamilton's equations are given by (1.3b). Of course the point of the paper s is to explain in terms of Miura transformations the existence of two compatible Hamiltonian structures t\ and 11, the Gel'fand/Dikii brackets. Either of these may be used to formulate the generalized Korteweg-de Vries equations. Wilson 9 defined a generalized AKNS spectral problem for any simple Lie algebra 0. A generalized AKNS spectral problem is given by £U = 0 where

£ = d - (11(1) + XF),

(1.6)

F is a regular element of g and u lies in the span of the root vectors of the unique Cartan subalgebra containing F. Wilson's Hamiltonian formalism uses differential algebras. Flaschka/Newell/Ratiu (FNR) 1X described the usual AKNS hierarchy in the following way. Let 0 = «/(2,<E) and let L(0) denote its loop algebra. Then £(0) = 91 © ft where 01 is the subalgebra of polynomial loops and ft is the subalgebra consisting of loops in strictly negative powers of A. The AKNS hierarchy was shown to be Hamiltonian with respect to the Kirillov-Poisson brackets on the dual of 91. This dual was identified with the annihilator ft of ft in L(0) using a generalized trace form anologous to Adler's trace form. The Poisson bracket is given by (1.4) for £ S ft . The relationship between these two papers may be described in the following way. Let £ be an operator of AKNS type. Wilson ( 9 , lemma (2.4)) described a system of generators of the centralizer of £ in the algebra of formal series over 0 in powers of A and A - 1 . If Z is a generator then one has Z€ft° and u + \F = *m{XZ). (1.7) The spectral problem is related to its centralizer by a projection coming from a Lie algebra decom­ position. This observation lies at the heart of the this paper. V. G. Drinfel'd and V. V. Sokolov 12 developed a matrix formalism to describe scalar Lax equations. Let L be a differential operator as above but with scalar coefficients. There are of course many ways of converting the scalar spectral problem Lij) = Xi/> to an equivalent first order matrix problem. Any two conversion £1 and £2 are related by a gauge transformation: £ 2 = Six)'1

£.!S{x)

(1.8)

for some lower triangular matrix S with ones on its diagonal. The standard conversion is defined by UJ+i = d'ip and the natural conversionis defined by W3+i = Ljip where Lj is the Gel'fand/Dikii fractional power of L of order j and ij> is an eigenfunction of L. Let M. denote the set of gauge classes of smooth operators £. Let us illustrate the Drinfel'd/Sokolov approach with a 2 x 2 example. Let 0 = sl(2,
260 in their paper to an arbitrary simple Lie algebra. Let us consider the first order 2 x 2 matrix differential operator given by £ = d - B where B = (A + q),

A =

0 1 A 0

] a n d q = [ g " j] .

(1.9.)

The following two spectral problems are equivalent SU = 0 <^=> Li> = \il> where L = d2 - u and « = q + v2 - vx. The conversion used in (1.9a) is referred to in consider the centralizer of £ in L(p): C(£)=lz

=

~°

*

u

(1.9b)

as the general element of the gauge class. We

= 0oiZ,

€L(e):[£,Z]

=

[B,Z]\.

The entries of Z g C(£) satisfy this system of equations: ax = (X + q)b + c,

bx = 2a-2vb,

cx = 2vc + 2(A + q)a.

(1.9c)

These equations have a solution of the form a(l.Od)

v\~l

- - ( u x + 2v«)A _ i + — (euux - u „ x - 2vt«XI + 6i/u')A"a + • • •, 4 16

A-,-iuA-3+i(3u2-ux)A-3 + --, 2 8 c = 1 + - {q - vx - i/ 3 )A _1 + -(uxx + iuux - u2 + 4t/2u)A"J + ■ ■ •. 2 8

6=

We let oy denote the coefficient of A - ' in a and so on for b and c. The operator Lax equation, 6V£ = [^''(A), £] where dr = -£-

(1.9e)

and £' r ' is the polynomial part of \TZ, was shown in 12 to be compatible or consistent with the structure of £. This equation comes down to a, b and c as dTq = 2o r + i

and- dtv = - 6 r + i

(l.Of)

and therefore one has dTu = 2dbT+l.

(l.Og)

For instance, if r—2 then (1.9g) is just a rescaled version of the Koiteweg-de Vries equation (1.1a): 9lU

=

j(6uu* ~ « « i ) -

(1.8h)

One result in 12, Proposition (3.18), applied to this example, implies that the bT are polynomials in u and its x-derivatives; one may always eliminate v and q in favor of u. This fact is not at all clear from the a,b,c equations (1.9d). One may also conclude ( 1J , Proposition (3.18)) that the equations (1.9g) constitute the Korteweg-de Vries hierarchy.

261 A functional on M. is a gauge invariant functional on the space of smooth matrix-valued functions q. The following are Poisson brackets on the space T of gauge invariant functionals on M: {/. j}l(q) = «»•(V/(q)[e2ll , V 9 (q)])

and

{/, g} 2 (q) =
Gradients are defined with respect to the pairing («, v) = f tr(uv) dx. The Lax equation (1.9e) was shown to be Hamiltonian with respect to each of the symplectic structures (1.6). The space A4 is isomorphic to the space of smooth scalar operators L and, as is shown in 12, under this isomorphism {, }i and {, } 2 correspond to the first and second Gel'fand/Dikii brackets. It is pointed out in 12 , and this is easy to check by a computation, that (1.9e) is not in general compatible with a particular conversion such as the standard conversion. This problem was handled (12, Lemma (3.8)) by adding a correction to Z^T'. In this paper we shall deal directly with the natural conversion. We use results in 12 to describe the centralizer of £. We shall show that there exists a subalgebra A of L((j) such that L(g) = 91 © A where 91 is the subalgebra of polynomial loops, Z belongs to a translate ( + &° and B = ir„(AZ). Thus, as in the AKNS setup of Wilson and FNR, the spectral problem is related to its centralizer by a projection coming from a Lie algegra decomposition. The annihilator of A is defined with respect to a generalized trace form. The Lax equation (1.6) is shown to be equivalent to a Hamiltonian system o n f + S with respect to its Kirrillov-Poisson structure; J?° having been identified with the dual of the Lie Algebra 91. Our decomposition is unusual in that it does not seem to be related to a Lie algebra automor­ phism like many of the examples in 12 See also 13 ° and 9 . In our study of the centralizer of £ we obtain some interesting relationships between the algebraic formalism of Drinfel'd/Sokolov and the Baker function formalism of Cherednik. See (§3). This might be viewed as a weakness in this paper. We had to go outside the confinements of algebra and use Baker function analysis in proving some of our results. There are presently some other shortcomings which we hope to work out in M- It is not at all clear where to look for the Gel'fand/Dikii Hamiltonian structures. One should expect a decomposition generalizing ours for any Kac-Moody algebra. We do not even have a coordinate-free description of ^ for L(sl(2,
262 Acknowledgements. This paper arose out of a discussion between H. Flaschka and myself during a midnight ride from Tucson, Arizona to Laramie, Wyoming around the Fourth of July, 1984. The next day Flaschka worked out the n = 2 version of our loop algebra decomposition. This work was supported by a grant from the Louisianna Education Quality Support Fund(86LBR(16)-016-04).

263 (§2) LIE ALGEBRAIC PRELIMINARIES A Poisson structure or Hamiltonian structure on a finite dimensional C°°- manifold M is an anti­ symmetric bilinear form: {,} : C°°(M) x C°°(M) - C°°(M) such that for all f,g,h G C°°(M) one has {f,gh}

= {f,g}h + g{f,h}

and

{/, {g, h}} = { { / , , } , h} + {g, {/, h}}.

The second of these formulas tells us that C°°(Af) is a Lie algebra with with Lie bracket {,}. The first formula tells us that the mapping: h G C°°(M) -> Dh : f G C°°(M) - Dk(f)

= {/, h} G C°°(M)

is a map of C°°(M) into the set of vector fields on M; D\ is called the Hamiltonian vector field ofh. We recall that the gradient of a function / G C°°(M) is the section of the cotangent bundle T'(M) given for p G M and £ G TT(M) by these formulas: df(p) = Trf:Tr(M)^<E,

(d/(p),0 = jtfh(t))

|,=o

where y is a curve leaving p with velocity £. We shall often use the notation Vf in place of df. We shall be concerned with the Kirrillov-Poisson structure (g*,{, }g») on the dual g* of a Lie algebra g. This bracket is given for / , g G C°°(g") and a G g* by this formula: {/■9}g-(«) = («.[V/(a),V 5 (a)]). We have made these standard identifications: T^(g") ~ g"" ~ g. The Hamiltonian vectorfield of h G C°°(g*) is given at a G g* by this formula: Dh(a) = - a d v M a ) ( a ) . Suppose our Lie algebra comes with a symmetric nondegenerate bilinear form K : g x g —> <E that is associative in the sense that for all (,t),p G g one has K([£,rj\,p) = K((, [v,p])- We shall identify the dual of g with g itself using K in the usual way. Suppose there is given a direct sum decomposition of g by subalgebras R and 91 ; g = 91 ® A. Let fiP denote the annihilator of .ft with respect to K. Then fi." is a subspace of g that is not, in general, a Lie subalgebra unless M is an ideal. If e G g then there is a natural linear isomorphism: i/>, : e + i?° -* 91*. The pull back {,}, of the Hamiltonian structure {, }«YJ- is given for / , g G C°°(e + £°) and £ G -ft0 by this formula: {/, 9}<(< + 1 ) = *(«■ [ - „ « V/(« + *), »OT o Vg(e + ()]).

(2.1a)

The Hamiltonian vector field corresponding to an h.in C°°(e + M°) is given by this formula:

DhU + 0 = *# [»„ ° Vfc(« + (), <]. A function / G C°°(g) is said to be invariant if it satisfies [V/(£),£] = 0 for all £ G g.

(2.1b)

264 (2.2) Theorem ( A d l e r / K o s t a n t / S y m e s ) : Let us suppose that e belongs to 91° n [A, &]°. (2.2a) The Hamiltonian vector field of the restriction to c + R° of an invariant h € C°°(g) is given by this formula: &h{* + Q = -[*sl<>Vh{t + £),« + «] = [irgxVfc(e + e),« + « ] * « » I U G * ° . (2.2b) The set consisting of all testictions to e + .ft0 of invariant functions on 0 is involutive with respect to {, } e . Proof: If f and g are restrictions to e + A of invariant functions and { £ fir then one has

{/, »}.(« + t) = m, [*„ ° V/(e + 0. *„ ° V«,(e + {)]) = if («= + «, [ r ^ o Vf{* + t), xm o V,(« + ()]) = K(t + £, [ ^ o V/(
JM« + fl = *jPt«R«wk(«+fl.fl oV/i(e + £), e + fl = * J P £*v%+«),e+fl = - [ r < « V M i + ().« + fl = tr„aV*(« + f),«+*fl. All the equalities except the fourth one follow from reasons cited above. Using the associativity of K, one may show that the commutator in the fourth equation already belongs to A .

* There is an often used alternative version of Theorem (2.2) in which, roughly speaking, ev­ erything is translated back to ft . We would like to state this version just to avoid confusion. (2.3) Theorem: Let us suppose that e belongs to <3l° n [M, R\°- Let T e = {/ o 5 e o i : / is an invariant function in C°°(p)} where 5, denotes translation in p by e and i : A —» p is the inclusion map. (2.3a) The Hamiltonian vector field of h o 5, o < is given for ( in )i° by this formula: Dh0sM0

= - k f f o Vfc(e + £}, e + *] = [ » „ o V/«(f + (), f + <].

(2.3b) The set T, is involutive with respect to {, } 0 . Proof. This version follows from the formula Vh o 5, o t(£) = V/i(€ + £) and the arguements used in the previous proof.

265 We would like now to relate the previous discussion to Hamiltonian group actions and momen­ tum mappings. Let G be a Lie group with Lie algebra 0. Let {M,w) be a symplectic manifold. We shall use {,} to denote the Poisson bracket associated to u> and we shall denote the Hamiltonian vector field of an H in C°°(M) by XR. Suppose there is given a Lie group action $ : G X M —> M. The G-action is said to be Hamiltonian if the following three conditions are satisfied: (2.4a) For each g £ G the mapping ig : M —> M, the projection of* |{s}xM o n *° * n e second argument, is a canonical transformation; that is, $*u> = ui. (2.4b) Let $ l P ,p 6 M denote the projection of $ IGX{J>} onto the first factor. For each { € 0 the induced vector field £M(p) = Tt$iP ■£ is Hamiltonian with respect to ID; that is, there exists an F( € C°°(M) such that w(£„, ■) = dF(. (2.4c) The mapping £ —> F( is an homomorphism of Lie algebras that is equivariant in the sense that for all f S £1 and all g € G one has i^Ad.lf) = -Ff ° * s . A momentum mapping associated to a Hamiltonian G-action is a mapping J : M -> a* such that {J{p),i) = F({p) for all p e M and £ 6 0*. Any momentum mapping is equivariant with respect to the coadjoint action of G on p* and the G-action on M; that is, J oi} = Ad' , o J for all g 6 G. This property follows from (2.4c). A collective motion is an integral curve of a Hamiltonian vector field on M with Hamiltonian of the form hoj where h 6 C°°((f). (2.5) Theorem (Guillemin/Sternberg l s ) : Let there be given a Hamiltonian G-action on M and suppose that J : M —» 0* is an Ad*-equivariant momentum mapping. Then J is a canonical mapping of Poisson manifolds; that is, U

w

( ; ( p ) ) = Dk(J(p)) for all p € M and h e C°°(0*).

Equivalently, the induced mapping J* : C"^')

-* C°°(M) is a homomorphism of Lie algebras,

{/, h}s. o J = { / o J, h o J} for all / , h € C°°(a*). Proof: If £ e 0 then let ^(denote the corresponding element of 0** C C°°(Q"). infinitesimal action of G on M and on 0* is given by In = XLt*J One has

(2.5a)

and

(2.5b) Then the

f 0- = Otli.

( « « ) ( % ) ) = To(/oexp(<€)-p) = T0(exp(tf) • J(p)) =

(B-(HP));

hence, we have proven this linear version of (2.5a): J.XL,0j

= DLt.

Given h 6 C°°(0*) and p € M we let £ = Vh(J{p)). Then from the formula

l*,(v) = xhpJ(p)

(2.6c)

266 it is cleai that exp(t£)> p is a curve leaving p with velocity Xhoj{p)- This proves that one may replace L( by h in (2.5c) which proves (2.5a). Now to prove (2.5b) we observe, using (2.5a), that (Dkh(f))(J( •-= U,h} == (D (f))(Hp)) P)) (J(P)) {/.*}»■ s.(J( P)) = =

[KXtoiWHm) j(f))(Hp))

( ■ / . * * .

= (X , h <■/}(P) 0J}(p) ( - ^ h( 1oj{foJ)){p) 0 ./ ( / < »/))(p) == = {/
for a l l / , / i e C ° ° ( g - ) .

♦ This theorem has some immediate consequences. The collective motion p(t), p(0)=p, of h o J is given by p(<) = s ( * ) p =*,(?(*)) where g(t) is satisfies g(0)=e, g{t) =

TtRs{l)Vh(J(g(t)-p))

and R is the right action of G on itself. Thus p(t) lies on the G-orbit of p and if H is any G-invariant Hamiltonian on M then h o J is constant along any integral curve of XH\ hence, {H, h o 7} = 0. The curve J(p(t)) is an integral curve of D^. If h S C co (p") is Ad"-invariant then the Hamiltonian vector field of h is trivial and h is a central element of the Lie algebra ( C ° ° ( B * ) , {, }g-). The collective motion of an invariant h is given by P(t) = 9(t) ■ P where g(t) = exp(t{Vh(J(p)). Let there be given = Hamiltonian G-action on M and suppose that J is an Ad'- equivariant momentum mapping. Let v be a regular element of fl* Then J~l(v) is a submanifold of M. The Ad'-isotropy subgroup of u, denoted G„, acts on 7 _ 1 (i/). Let M„ = J~l(u)/Gv denote the set of orbits of this action. If the action is free and proper then M„ is a manifold and the canonical mapping ir„ ; J~*(v) —> Mv is a submersion ( 18 , Proposition (4.1.23)). Moreover, there exists a unique symplectic form u„ on M„ such that it^uv = i*w where i„ : J~l(v) - . m i s the inclusion mapping. The symplectic manifold (M„,w„) is called the (Marsden/Weinstein) reduced phase space. (§3) THE NATURAL CONVERSION In this section we shall review the paper l2 of Drinfel'd and Sokolov and then we shall be able to explain how our idea fits into their framework. We shall need the following notation. Let ea