Nonlinear Evolution Equations and Dynamical Systems: Proceedings of the Icm2002 Satellite Conference, Yellow Mountains, China 15-18 August 2002

Since Q is arbitrary, we have Ut-Vx

+ [U, V] = 0.

(5)

That is, zero-curvature equation holds. In what follows, for the sake of application convenience, we only consider Lie algebra gl(2, C).

A Direct Extension of Lie Algebra An-\

35

2. Hierarchy of Evolution Equations Associated with Lie Algebra gl(2, C) Consider Lie algebra Al=gl(2,C),

<6)

Q=(*_°i)>

here A is a spectral parameter. Let ei

= (-A o) ' 62 = (A O) ' C3 = (o A) ' We find that

[ei,e2] = - 2 e 3 , [ei, e3] = -2e 2 , [e2,e3] = -2ei, [ei, Q] = [e2, Q] = [e3, Q] = 0. (7) We define ei(n) = ei ® A 2 n + 1 = eiA 2 n + 1 ,e 2 (n) = e 2 A 2 n + 1 ,e 3 (n) = e3A2".

(8)

It is easy to find that from (2) [ei(m),e2(n)]=-2e 3 (m+n+l), [ei(ra), e 3 (n)]=-2e 2 (m+n), !

[e2(m),e3(n)]=-2ei(m.+n), degei(n)=2n+l, dege2(n)=2n+l, dege3(n)=2n. (9)

Thus a new loop algebra A* is presented with a set of basis (8). Set (px = UQ
U=l

V = \ Yl ( a mei(-m) + h m e 2 (-m) + c m e 3 (-m))

\m>o

I

yu

(10)

cA (a + 6) A2 2 6)A cX a

+

solving the auxiliary equation VX = [U,V]

(11)

36

Fukui Guo, Yufeng Zhang & Qingyou

Yan

yields that Q"mx

=

4"m+l

2 f C m + 2sbm,

Omx = 2aTO_|_i — 2qCm +

2sam,

Cmx = -2
- m) + bme2(n - m) + cme3(n - m)), V_i

=

m=0

X2nV - Vln\

then (11) can be written as -V$> + [U, Vin)] = V™ - [U, V[n)].

(13)

The terms of the left-hand side in (13) are of degree> 0, while the terms of the right-hand side are of degree< 1. Therefore, the terms of both sides in (13) are of degree 0,1. Hence -V^ + [U, V^n)] -2a„+ie 2 (0) - 2 6 n + i e i ( 0 ) + 2g6 n+ ie 3 (0) 2ran+1e3(0). Note again V^ = V+ , then the zero-curvature equation Ut - V^n) + [U, V{n)] = 0

(14)

determines the Lax integrable system utn=lr\

= \sj

t

2anH \-2qbn¥i + 2ranHJ

= - 1 0 0 2bnii = J 2bn+1 , \ 0 0 f / \ 2cn J \ 2cn J (15)

where J is a Hamiltonian operator. Prom (12), it is easy to find that /-2o n + A / -s + qd^qd 2bn+i = \{d-2rd~lqd) V 2c„ / V -d'lqd

^(d+2qd-1rd) -s-rd~lrd -d^rd

qd~lsd -rd^sd -d~lsd

Therefore, (15) can be rewritten as utn = ( r |

=JLn\

2ar

I.

(16)

A Direct Extension

of Lie Algebra A n _ i

37

We consider a reduction case of the system (16). Taking s = 0 in (16), we present a formalism of the well-known AKNS hierarchy ,1 _ M W ,

- f 0 l \ f 59_1^ ^y-lOjKUd^rd-'qd)

\{d+2qd-'rd)\n -rd-lrd )

(-2aq\ \ 2ar J '

l

''

Again taking n = 1, a = 1 in (17) leads to the following nonlinear evolution equations qn =Qx + r(r2 - q2), rtl =rx+q{r2 - q2). Taking n = 2 in (17)) yields a generalized nonlinear Schrodinger equations

{

«ta = f rxx + f (q(r2 - q2))x + ^fr{r2 - q2)2 + ar{rqx - qrx), rt2 = f qxx + f (r(r 2 - q2))x + ^-q(r2 - q2)2 + aq(rqx - rxq).

It is easy to verify that JL = L*J. Therefore, the system (16) is integrable in the Liouville sense. A direct caculation shows that f (V, f ) = -2aX\ (V, f > = 2bX\ (V, fg> = 2cX2, ]^(V,^)=6cX3 + {Arb-Aqa)X3 + 2scX.

V

;

Substitution of (19) into trace identity reads that — U6c + 4rb-4qa)X3

+ 2scX} = X-i—A7

5u

dX 2n+1

Comparing the coefficients of A

/-2aA4\ 26A4 .

(20)

V 2cA2 /

in (20) yields

-2
Su \ where ffn = 3cn+a+2rfcn+^-2qa„+a+»c„

2

\ ^ &re

' Hamiltonian functions.

(22)

38

Fukui Guo, Yufeng Zhang & Qingyou

Yan

Until now, we obtain the Hamiltonian structure of the system (16) as follows

-*•-"• ( " J ) - ' ^ - ' f ^

<23>

where K = JL. It is easy to verify t h a t J = aJ + bK is a Hamiltonian operator, i.e. {J, K} is a operator pair. Therefore, (23) is bi-Hamiltonian structure of the system (16) References 1. Guizhang Tu. J. Math. Phys. 1989, 30(2):330. 2. Wenxiu Ma. Some aspects of nonlinear integrable systems, Ph.D. dissertation, Academia Sinica, Beijing, China, 1990. 3. Xingbiao Hu. J. Phys. A: Math. Gen. 1997, 30:619. 4. Fukui Guo. Acta. Math. Sinica. 1997, 40(6):801.(in Chinese) 5. Engui Fan. J. Math. Phys. 2000, 41(11):7769. 6. Chaohao Gu, et al. Soliton Theory and Its Application. Zhejiang Publishing House of Science and Technology, 1990. (in Chinese) 7. Fukui Guo. A class of Lie algebras. Reprinted from Journal of Shandong University of Science and Technology, 2002.(in Chinese)

ON GAUGE-GRAVITY CORRESPONDENCE OPEN-CLOSED STRING DUALITY

AND

SENHU Department of Mathematics and Inter-disciplinary Center for Theoretical Studies University of Science and Technology of China Hefei, Anhui 230026, P.R. China

XIAO-JUN WANG Inter-disciplinary Center for Theoretical Studies University of Science and Technology of China Hefei, Anhui 230026, P.R. China

In this paper we discuss some aspects of gauge-gravity correspondence as part of open-closed string duality. In such a correspondence people find a supergravity dual of a super-symmetric gauge theory. Some physically interesting theories such as SUSY Yang-Mills admit such a SUGRA dual. It then gives very useful information on the side of gauge theory, even in the regime of strong coupling.

1. Introduction It is now very well accepted that the matter world are governed by gauge theory. Three interactions, with gravity an only exception, are unified in the Standard Model which is a generalization of Yang-Mills theory. There are also strong evidence that TV = 1 Super Yang-Mills theory be the right theory for matter world if we go to smaller scale a little bit further. The coincidence of coupling constants of three interactions from TV = 1 Super Yang-Mills model is a strong evidence that it is most likely a physical theory. And it is interesting to get more information about the theory. Yang-Mills theory and TV = 1 SYM as well, exhibits many interesting properties. Classically they are gauge invariant. When we are in four dimension, they are also conformal invariant. Quantum mechanically the conformal symmetry is broken and it exhibits asymptotic free property. This property allows us to introduce hierarchy of scales which is needed to make contact with experiments. Asymptotic free property also suggests 39

40

Sen Hu & Xiao-Jun

Wang

that the perturbative version of quantum theory is well defined. However the lower energy limit is extremely difficult although it is very much needed to explain some fundamental phenomenons such as quark confinement, chiral symmetry breaking and even the calculation of masses of particles. It corresponds to the regime of strong coupling of the quantum field theory which is not even defined. The recent developments of string theory provide an effective way to deal with the strong coupling regime of quantum field theory. Thanks to the invention of D-branes people found various kinds of dualities in string theory. In other words, one theory can be equivalent to another theory with a proper mapping. Among them open-closed string duality is particularly interesting. One knows that gauge theory arises naturally from open string theory and gravity is described by closed string theory. The duality of open-closed string induces gauge-gravity correspondence. Some gauge theories can be described by a suitable gravity theory (We called it a super-gravity dual)! And the dual gravity theory provides information of strong coupling which helps us to deal with problems of confinement, chiral symmetry breaking etc. In this note we shall introduce some of the models. In particular we shall consider the following two models: 1) Maldacena-Nunez solution: It corresponds to a large number of D5branes wrapped on a super-symmetric two cycle inside a Calabi-Yau manifold. It is dual to d=4, M = 1 SU(N) SYM. 2) Klebanov-Strassler solution: It describes the geometry of the warped deformed conifold when one places N D3-branes and M fractional D3-branes on the apex of the conifold. It is dual to a Af = 1 SYM with gauge group SU{N + M) xSU(M). 2. Large N and Strings 't Hooft introduced another parameter to gauge theory. It is the rank of the gauge group. We may consider SU(N) as the gauge group for Yang-Mills and then we consider large iV expansions. What he found is that in the expansion planar diagrams dominate 1. This lead him to conjecture that 4D SU(N) quantum field theory is equivalent to a string theory of coupling constant 1/N. In 2 't Hooft introduced a notion of holography, i.e. the real degree of freedom of some theories are only boundary fields. There are some evidences to support this. . Gross-Taylor 3 consider a two dimensional toy model. They verified a two dimensional string theory is equivalent to a two dimensional gauge

On Gauge-Gravity

Correspondence

and Open-Closed String Duality

41

theory. . Witten 4 considers a string theory which is dual to Chern- Simons theory. He 5 also show that Chern-Simons theory is dual to a conformal field theory which is a realization of holography. . An exciting conjecture was made by Maldacena who proposed that N = 4SU(N) SYM is equivalent to a type IIB Super string theory on AdSs x S5 with Ramond-Ramond charge N. In 1995 Polchinski 6 introduced a notion of D-branes which sparks the second revolution of string theory. A D-brane is a sub-space in space-time where an open string ends. A D-brane can also be considered as a state in closed string theory. Closed strings propagate among D-branes. Polchinski found some interesting D-branes which carries half of the super-symmetry and it is a BPS state. Its central charge can be calculated from SUSY algebra. It couples to Ramond-Ramond field and the charge can be considered as the number of D-branes. If one looks at D-branes from the open string point of view the lower energy limit gives gauge theory. When one puts number of N D-branes together one gets an U(N) gauge theory. This opens a possibility that a gauge theory is dual to a string theory. It really depends on how you look at the theory, as an open string theory or as a closed string theory. It also provides information of strong coupling. Strominger and Vafa 7 applied it to study black hole and they got a microscopic derivation of extremal black hole entropy. Maldacena makes the above open-closed string duality in a precise conjecture 8 . He considered N D3 branes in type IIB string theory. He found the background super-gravity solution which reads: ds2 = f-i(r)(-dt2

+ dx\ + dx\ + dx\) + / * (r){dr2 + r2dQ25)

l+^-,R4=4irg3a,2N

f(r) =

F5 = (1 + *)dt A dxi A dx2 A dxz A d/ _ 1 . Near the horizon, i.e. r is nearly zero, we have: o4

f(r) = -r,N=

r

/

F6.

42 Sen Hu & Xiao-Jun Wang

And the metric is AdS^. Maldacena made the following striking conjecture: The Type IIB string theory on AdSs x S5 with string coupling gs and string tension T is exactly equivalent to the Af — 4 SYM in 4D with coupling constant QYM and number of colors N. Recall that T = ±yJgYMN = ±V\ = R2/(2ird)A*9. = gYM,R being the radius of the AdS space. To make it easier to verify the conjecture one often takes some limits. If we let gs —> 0, with T-fixed, then the corresponding gauge theory is in the regime gyM —* 0, A = 9yMN fixed, it is the large N limit of the gauge theory where only planar diagrams are kept. If one takes gs —+ 0 with T —> oo the string theory reduces to type IIB supergravity in AdSs x S5 and the gauge theory side is Af = SYM in 4D at strong t'Hooft coupling, i.e. gyM ~* 0 with A —> oo. The correspondence is made clear by 9 and 10 . We called it Witten's Ansatz. Let 0(x) be an observable in the boundary conformal field theory. Let (j> be a field in the bulk which couples to the boundary field operator. Let S be the supergravity action of classical solutions of the proposed super gravity dual with boundary values couples to the boundary operators. Then we have:

It is useful to observe that the left hand side is purely classical and the action of classical solutions of supergravity satisfies Hamilton-Jacobi equations. The right hand is a quantum field theory and it satisfies renormalization group equation. The above observation helps us to verify Maldacena's conjecture. E. Verlinde suggested that those two equations concides u . We really need to see whether they satisfy the same equations. The original conjecture has been pretty well understood, thanks to works of Freedman, Verlinde, etc. For a recent review see D'Hoker and D. Freedman 12 . In particular they derived holographic renormalization equations which is the Hamilton-Jacobi equation for the supergravity viewed as a Hamilton systems. Af = 4 SYM is a conformally invariant theory so the /? function is nothing but zero! However a physical theory has to involve a scale. And we are more interested in asymptotic free gauge theories. To make the above correspondence more useful we have to consider SYM with less supersymmetries, i.e. Af — 2 and Af = 1.

On Gauge-Gravity

Correspondence

and Open-Closed String Duality

43

3. G a u g e - G r a v i t y C o r r e s p o n d e n c e for Less Supersymmetries We are mainly interested in getting supergravity dual for M = 2 and J\f = 1 SYM. We have examined two proposals for such a gauge-gravity correspondence. 1). Maldacena-Nurez solution 13 : The MN solution is obtained by finding a domain wall solution of a seven dimension gauge supergravity after truncating its 5 0 ( 4 ) gauge group to SU(2). The metric ansatz for the domain-wall solution is (see also 19 ) ds2 = e2^r\dxlz

+ dr2) +

r2e2^dn2,

dfl2, = d62 + sin2 8d2, (0 < 6 < TT, 0 < < 2TT). One can compactify the 7D SUGRA on S2 and get a 5D effective action: f d4xdre2k{'idvkdl/k

S5=r]5

+

- 2dvhdvh -

dvxdvx

4{ )2 2{ )2 { )2 V{x h)}

t - t -t - > -

with h = g-f,k = 3 / 2 / + g, V(x, h) = - r ^ 2 ( 4 + 2e-2h The domain wall solution is given by: e 2fc+x =

ze2z

-2x

=

1

_ 1 +

l/2e~ih-2x)

Ce~2Z

2z In

20 21

- , we consider a dimension four operator: O = Tr(pv$+Dv*

+ 2$AD^A

- 1/4F„M A *Fvlt).

We calculate < 0(x) > from the side of supergravity and we get: CO

< 0(x)

>= J^AU=*O.*O=O =

r^rise2*0/^.

It agrees with calculation from QFT calculation:

< 0(x) >= r g T t e ^ ^ e - 4 eG Xxp pi { - 1 ^22 - } . 4TT 2

Ng ?

We also calculate two point functions and we get

YM

44 Sen Hu & Xiao-Jun Wang

< 0(p)0(q) >= - 4 ^ f ( e - 2 - 6 r o 2 ^ V ( p + ?). By comparing it with QFT calculations we get

e

-2

.9

a -2^2

— 6r 0 — = A . Ai

And we get /? function of the renormalization flow: N

W M )

=

a

/,

+

—Z-Z9YM(\ 07r

8TT2

L

6ex

P i ~ ju„2

^9YM

,

^,

ex

16?r2

1N

> + °( P{-T772—»• ^9YM

The leading order term agrees with renormalization flow equation. In the sub-leading terms we get non-perturbative information which also agrees with QFT calculations 17-18. For J\f = 1 case we consider gaugino bilinear operator 03(x) = Trijj{x)i>{x). Similar calculations are performed in 21 and we found that:

iP >SUGRA=

J

Zl^-^—e

%&.

9YM

It agrees with QFT calculation if we put IQ3 = A 3 . We get /3 function from SUGRA side:

^9YM)

= - ^ ^9YM{1+ M ( 1 16^

QFT calculation gives

n(n

+

'-^ IGTT2

+

0(9YM))-

17 18

x_

' : 3Ar

„3

N n

9YM,-I

It is a puzzle for us why the sub-leading terms does not agree. 2) Klebanov-Strassler solution 14 , 15 , 16 The KST solution is realized by placing M D3-branes and N fractional D3-branes on the conifold. The solution is proposed to be dual of pure SYM with gauge group SU(N + M) x SU(M). One may reduce the rank of the

On Gauge-Gravity

Correspondence

and Open-Closed String Duality

45

gauge group by a chain of duality cascades. And when M is a multiple of N we get Af = 1 SYM of gauge group SU(N) at end of the duality cascades. The ten dimensional metric in the string frame is: dsj0 - h~1/2(T)dxndxn

+

h1/2(r)dsl,

^ = l/2 e 4 / 3 X(r)(-^(dr 2 + (55)2) + cosh2 C-^Uig')2

K{T)=

+sinh 2 (J)Sf = 1 ( f f i ) 2 ),

(sinh(2r)-2r)1/3 2V3sinh'

4 = -l/4Tr(FltvFliV)

+

TrtfLhp)

Calculations from SUGRA gives: < e>4(p)04(g) > = (gYMN)2(a0A4

+ alP2A2

+ a2p4 log A2 + ...)5\p + q).

We get radial/energy scale relation: log/i/M = ^ + l o g r + C. We also get /3 function:

P(9YM)

= ~J^9YM(1

~ ^9YM

+

0{gYM)).

Again the leading order term agrees with QFT and the sub-leading order term does not. There could be much more correspondence of such kinds in string theory and in M-theory. It often involves special geometry to preserve SUSY when we do compactification from a theory of higher dimensions. It is interesting to explore them 24>25>26.

46

Sen Hu & Xiao-Jun

Wang

4. Confinement In particle physics there is a fascinating phenomenon called quark confinement. We know that all elementary particles consist of quarks but we never detect quarks in experiments. Theorists proposed that quarks are confined. For example if we have two quarks with opposite charge q, q, there shall be an electric tube connecting two quarks. It is proposed that the tension between q and q is proportional to the length of the flux tube. As a result when we go to lower energy limit the tension increases enormously and the two quarks cannot be separated. So in experiments we only observe composites of quarks which give elementary particles. To explain such a phenomenon we have to go to the lower energy limit which is the opposite regime of asymptotic free. It is in the regime of strong coupling. Theorists usually consider this problem by going to lattice gauge theory. String theory provides another effective way of explanation. The key point to explain confinement is the following: If we consider quarks to propagate along a loop C, then it is proposed that the Wilson loop observable exhibits area law. In other words, let C bounds a minimal surface E and let A be its area. Then the amplitude of observing C is exp(—kA(E)). Remarkably string theory helps us to go to the regime of strong coupling. In fact, the picture of confinement is modified. In string theory we are in a higher dimensional space. One of the proper space-time to consider is the AdSs space. We may consider the world we are living is a brane. Quarks and anti-quarks exist in our brane world. The brane is part of the space-time where gravity lives. It is proposed then the above flux line exists outside the brane. If we consider a loop lying on the brane we may consider a surface bounding C in AdSs space. We wish to see whether we have area law in the new setting. It the area law were true it also gives an explanation of confinement for the dual gauge theory on branes. Confinement for TV = 2 supersymmetric Yang-Mills are derived by the celebrated work of Seiberg and Witten 22 . They derived it via a weak-strong

duality (S duality) which is a another source of inspiration for the second string revolution. In 23 we consider circular Wilson loop of TV" = 1 SYM from the KST supergravity solution. The Wilson loop exhibits area law at long distance and logarithmic law at short distance. Therefore, TV = 1 SYM lies in confinement phase at low energy scale and be asymptotic free at high energy scale. Our work also suggest a phase transition between low energy scale

On Gauge-Gravity Correspondence and Open-Closed String Duality 47 and high energy scale.

References 1. G. 't Hooft, Planar diagram field theories, in "Under the spell of the gauge principle" by G. 't Hooft, World Scientific, 1994. 2. G. 't Hooft, Dimensional Reduction in Quantum Gravity, gr-qc/9310026. 3. D. Gross and J. Taylor, Two Dimensional QCD is a String Theory, Nucl.Phys. B400 (1993) 181-210, hep-th/9301068. 4. E. Witten, Chern-Simons Gauge Theory As A String Theory, Prog.Math. 133 (1995) 637-678, hep-th/9207094. 5. E. Witten, Quantum field theory and the Jones polynomials, Communication of Mathematical Physics, 1989. 6. J. Polchinski, Dirichlet-Branes and Ramond-Ramond Charges, Phys. Rev. Lett. 75 (1995) 4724-4727, hep-th/9510017. 7. A. Strominger, C. Vafa, Microscopic Origin of the Bekenstein-Hawking Entropy, Phys. Lett. B379 (1996) 99-104, hep-th/9601029. 8. J. Maldacena, The large N limit of supreconformal field theories and supergravity, Adv. Theor. Math. Phys., 2 (1998) 231. 9. S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett., B428(1998), 105. 10. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2(1998) 253. 11. de Boer, E. Verlinde, H. Verlinde, On the Holographic Renormalization Group, JHEP 0008 (2000) 003, hep-th/9912012. 12. E. D'Hoker and D. Freedman, Supersymmetric Gauge Theories and the AdS/CFT Correspondence, hep-th/0201253. 13. J. Maldacena and C. Nunez, Towards the large N limit of M = 1 super Yang-Mills, Phys. Rev. Lett., 86(2001) 588. 14. I. Klebanov and E. Witten, Superconformal field theory on threebranes at a Calabi-Yau singularity, Nucl.Phys. B536 (1999), 199. 15. I. R. Klebanov and A. Tseytlin, Gravity duals of Super symmetric SU(N) x SU(N + M) gauge theories, Nuclear Physics B574 (2000) 123. 16. I. R. Klebanov and M. J. Strassler, Supergravity and a confining gauge theory: duality cascades and xSB resolution of naked singularities, JHEP 08 (2000) 052. 17. V. Novikov, M. Shifman, A. Vainstein and V. Zakharov, Exact Gell-MannLow function of supersymmetric Yang-Mills theories from instanton calculations, Nucl. Phys. B229(1986); ibid., Beta function in supersymmetric gauge theories: Instanton versus traditional approach, Phys. Lett. B166(1986) 329. 18. N. Seiberg, Supersymmetry and nonperturbative beta functions, Phys. Lett. B 206(1988) 75. 19. P. Di Vecchia, A. Lerda and P. Merlatti, N = 2 and Af = 1 Super Yang-Mills theories from wrapped branes, Nuclear Physics, B 6 4 6 (2002) 43.

48 Sen Hu & Xiao-Jun Wang 20. X. J. Wang and S. Hu, Gauge/gravity duality, Green functions of M = 2 SYM and radial/energy-scale relation, JHEP 10(2002) 005, hep-th/0207145. 21. X. J. Wang and S. Hu, Green functions of Af = 1 SYM and radial/energyscale relation, Phys. Review D., hep-th/0210041. 22. N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in M = 2 supresymmetric YAng-Mills theory, Nucl. Phys. B 426 (1994) 19. 23. X. J. Wang and S. Hu, Confinement of M = 1 super Yang-Mills from supergravity, hep-th/0303141. 24. P. Di Vecchia, A. Liccardo, R. Marotta, F. Pezzella, Gauge/Gravity Correspondence from Open/Closed String Duality, hep-th/0305061 25. M. Bertolini, Four Lectures On The Gauge/Gravity Correspondence, hepth/0303160 26. S. Cucu, H. Lu and J. F. Vazquez-Poritz, Interpolating from AdSu-2 x S to AdSD, hep-th/0304022.

I N T E G R A B L E SEMI-DISCRETIZATIONS OF T H E A K N S EQUATION A N D T H E HIROTA-SATSUMA EQUATION*

XING-BIAO HU State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080, P.R. China

HON-WAH TAM Department of Computer Science, Hong Kong Baptist University, Kowloon Tong, Hong Kong, P.R. China

Integrable semi-discretizations of two model equations for shallow water waves are investigated. As a result, one integrable differential-difference version for the AKNS equation and three integrable differential-difference versions for the Hirota-Satsuma equation are found. These four differential-difference versions are transformed into bilinear forms. Backlund transformations, soliton solutions and Lax pairs for these differential-difference equations are presented.

1. Introduction The purpose of this paper is to consider integrable semi-discretizations of the following two model equations for shallow water waves: /*0O

ut - uxxt - 4:uut + 2ux /

utdx' + ux = 0

(1)

JX

and /•OO

ut - uxxt - 3uut + 2>ux I utdx' + ux = 0. (2) Jx As a result, one integrable differential-difference version for (1) and three integrable differential-difference versions for (2) are found. The method "This work was supported by Hong Kong RGC grant HKBU2065/01P, the National Natural Science Foundation of China (grant no. 10171100 ) and the knowledge innovation program of AMSS, Chinese Academy of Sciences. 49

50

Xing-Biao

Hu & Hon-Wah

Tarn

used here is Hirota's bilinear formalism with emphasis on bilinear Backlund transformations. It is noted that bilinear Backlund transformations for the KdV equation etc. were first introduced by Hirota in Ref.l. Equation (1) is referred to as the AKNS equation 2 while equation (2) is called the Hirota-Satsuma equation 3 . Concerning (1) and (2), much research has been conducted citeHU4. By setting v = — ^ J°° u dx', equation (1) and equation (2) can be rewritten as ut - uxxt - Zuut - Auxvt - 2uvxt + ux = 0, u = 2vx,

, .

Vxt — Vxxxt - Qvxvtx - 6vxxvt + vxx = 0,

(4)

{6)

and

respectively. For the sake of convenience in calculations, in sections 4 and 5, we will just consider equation (4) without linear terms vxt and Wxxxt + GWxWtx + 6wxxWt

= 0.

(5)

It is noted that equation (6) can be obtained from equation (4) by a simple transformation v = w + \{x + t). 2. Integrable Semi-discretization of the A K N S Equation In this section, we will consider an integrable semi-discretization of the AKNS system (3) and (4). First of all, we propose the following differentialdifference system ln(l + a2u(n)) = a(v(n + a) — v(n — a)),

0- + ^)lT^Sb + U
The AKNS Equation and the Hirota-Satsuma Equation 51

system (6) into the bilinear form 1 [2aDz smh(-aDn)

1 3 + cosh(-a£>„) - c o s h ( - a D n ) ] / ( n ) . f(n) = 0, (7)

1 [2a2Dt smh(-aDn)

3 1 - Dt sinh(-a£> n ) - 2aDzDt cosh(-aDn)

a(cosh(-aD n ) - cosh{^aDn))

+ 7Dtsmh(-aDn)}f{n).

+

f(n) = 0. (8)

By the dependent variable transformation u(n)

/ ( n + 2a)/(n - a)

f(n + a)f(n)

1 / n+ a i;(n) = - In ,

1 ,

where 2 is an auxiliary variable and the Hirota's bilinear differential operator D™D\ and the bilinear difference operator exp(SDn) are defined byRef.5 ^9 dt

'?*••'-(£-& exp(5Dn)a(n)

»b(n) = exp

\ dn

_d_ dt'

a(y,t)b(y',t')\y,=ytt>=t,

a(n)b(n )\n'=n = a(n + 5)b(n — 5).

dn' J

Concerning bilinear equations (7) and (8), we have the following result: Proposition 2.1: The bilinear equations (7) and (8) have the Backlund transformation eaD»f(n).g(n)

+ fie-aD»f(n).g(n),

= \f(n)g(n)

(2aDz + \e-aD»

+ 7 ) / ( n ) . g(n) = 0,

[(2a2 + 8 + —)Dt - \Dte~aD" -a\e-aDn

(9)

+6}f(n)*g(n)

+

(10)

-DteaD" (11)

= 0,

where A, fi, 7 and 6 are arbitrary (10) constants. Starting from the bilinear BT (9)- (11), we can derive a Lax pair for the differential-difference AKNS system (6). Firstly, set il>{n) = f(n)/g(n),u(n) v(n) = - In a

' g(n)

= -~ az

g(n + 2a)g(n — a) g(n + a)g(n)

1

52

Xing-Biao

Hu & Hon-Wah

Tarn

in (9)- (11). Then from the bilinear BT (9)- (11) and after some calculations, we can obtain the following Lax pair for (6): iP(n + a) = \ip{n)eav{n-a)-av(n) + ^{n - a), 2 2 A A (2a2 + 2— + 8)ipt(n) + 2a—vt(n - a)ip(n)

(12)

+2\atP{n - a ) e » W - » ( » - « ) [ „ ( ( „ ) + Vt(n _ a) - h + Oil>(n) = 0.(13) We have checked that the compatibility condition of (12) and (13) yields the AKNS system (6). To sum up, we have Proposition 2.2: Equations (12) and (13) constitute a Lax pair for (6). In the following, we shall simply denote, without confusion, f(n, t, z) = f(n) or / . Concerning BT (9)-(ll), we have the following result: Proposition 2.3: Let /o be a solution of (7) and (8). Suppose that fi(i = 1,2) are solutions of (7) and (8), which are related to /o under the BT (9)—(11) with parameters (Xi,fJ.i,-fi,9i) and Xj ^ 0,[ij ^ 0, (j = 1,2), fj ^ 0 (j = 0,1, 2). Then f12 defined by e x p ( - - a £ » n ) / 0 . / i 2 = c[Ai exp(--a£>„) - A 2 e x p ( - a D n ) ] / i . / 2

(14)

is a new solution related to f\ and fa under the BT (9)-(11) with parameters (A2,/i2,72,02)> (Ai,/ni,7i,#i), respectively. Here c is a nonzero constant. As an application of the nonlinear superposition formula (14), we construct soliton solutions of (7) and (8). Choose, for example, /o = l , c = , \ . It is easily verified that

l + e^1

(^2,^2, 72,02)

l + e^

(Ai,j*i,7i.0i)

where _1 -12

, Aie-^-A2cr)1 Ai - A2

|

Ax-Aae-^^ Ai — A2

{

A i e ~ ^ - A oe 1 Ai — A2

ap2

g r ?l+'?2

(15)

The AKNS Equation and the Hirota-Satsuma

Equation

53

with rn =Pin+-

1 . , , N asiah(api) , _„_. n , v y smh(aPi)z + \ z t + rtf, A( = 1 + e op «, a 2 cosn(apj) - a — 2

Hi = - e - ° p S 7 i = - ( 1 + e - o w ) . ^ = a ( l + e~° Pi )In general, along this line, we can obtain multi-soliton solutions for (7) and (8) step by step. 3. An Integrable Differential-Difference Equation for the Hirota-Satsuma Equation (4) Firstly, it is noted that equation (4) is reduced to the following equation by integration: vt ~ Vxxt - Qvxvt + vx = 0.

(16)

In the following, we will focus on the Hirota-Satsuma equation (16). Firstly, we propose the following integrable differential-difference equation for the Hirota-Satsuma equation (16): v(n + a) - v(n) - (v(n + a) — v(n))xx -3(v(n + a) - v(n))(vx(n

— (v(n + a) — v(n))3

+ a) + vx(n))

+ -a[(v(n + a) + v{n))x + {v(n + a) - v{n))2] = 0.

(17)

It is shown that in the continuum limit as a —> 0, the equation (17) is reduced to the Hirota-Satsuma equation (16). Next, we will show that (17) is integrable in the sense of having a Backlund transformation and Lax pair. To this end, we first transform the equation (17) into the bilinear form [Dx sinh(^aDn)

- D\ s i n h ^ a A , ) + ^aD2x cosh(|a£> n )]/(n). f(n) = 0

(18) by the dependent variable transformation v(n) = ( l n / ( n ) ) x . Concerning bilinear equation (18), we have the following result: Proposition 3.1: The bilinear equation (18) has a Backlund transformation DxeiaD"f(n),g(n) [\Dxe~iaD" [Dx -D3x-^aD2x

= +fieiaDn

+ {-\ii+\a\)e-±aD»)f{n).g{n),

(19)

+ 3fxD2x + (-3/i 2 + a^)Dx + 7 ] / ( n ) . g(n) = 0, (20)

54

Xing-Biao

Hu & Hon-Wah

Tarn

where A, [i and 7 are arbitrary constants. In the following, we are going to derive a Lax pair for equation (17). For this purpose, we set ip(n) = g(n)/f(n),v(n) = ( l n / ( n ) ) x . Then from the bilinear BT (19) and (17), we have a Lax pair for (20): [Xtp(n + a) - tp(n)]x = (v(n) - v(n + a))[\ip(n + a) + ip(n)] +fj,rp{n) + (-A/x 4- ^a\)ip(n + a), ipxxx(n) + (3/i - -a)ipxx(n)

(21)

+ (6vx(n) + 3/j,2 - a/x -

l)ipx(n)

+ [(6/x - a)vx(n) + 7]V>(n) = 0.

(22)

4. A n Integrable Differential-Difference Equation for (5) In this section, we will report the following new integrable differentialdifference equation for the Hirota-Satsuma equation (5): uxxxx(n

+ 1) - uxxxx(n) +3(uxxx(n

+ 3(u x (n + 1) - ux(n))2(uxx{n

+ 1) + uxxx(n))(ux(n

+3(u x x (n + 1) + uxx(ri))(uxx(n = -(«x(n + 2) - u x ( n ) ) e --(Ux(n

+ 1) -

uxx(n))

+ 1) - ux(n)) + 1) -

uxx(n))

u(n+2)+u(n) 2u(n+1)

-

+ 1) - Ux(n - l)) e «("+2)+«(n-l)-2 U (n) -

(23)

In fact, this new equation is derived from the following coupled bilinear system [DZDX + Dx sinh(D n )]/(n). f(n) = 0,

(24)

[Dz smh(±Dn)

(25)

+ Dl 8 i n h ( | D n ) ] / ( n ) . f(n) = 0

by the dependent variable transformation u(n) = l n / ( n ) , where z in (23) and (25) is an auxiliary variable. We now consider the continuous analogue of the equation (23). Setting U(ne) = u(n) and ne = t, we have u(n + 1) = U(t + e). We now expand u(n + 1) as u(n+l) = U +

d e2 d2 e3 d3 e-U+-W2U+^W3U

Mn + D = Ux + e£-U

+

^

U

+

+ ...,

^

U

(26) +

... (27)

The AKNS Equation and the Hirota-Satsuma

Equation

55

Substituting (26),(27), and etc. into (23) and neglecting higher-order terms of e, we obtain Uxxxxt + 6UxxxUxt + 6UxxUxxt = 0.

(28)

Obviously equation (28) becomes the Hirota-Satsuma equation (5) by setting w = Ux. In the following, we want to show that the equation (23) is integrable in the sense of having Backlund transformation and Lax pair. In fact, we have the following results: P r o p o s i t i o n 4 . 1 : The bilinear equations (24),(25) have the Backlund transformation (Dxe?D"

- XDxe-iD" 3

2

- /j,e?Dn + A / / e - ^ » ) / ( n ) . 5 ( n ) = 0,

(29)

2

(30)

(Dz + D X- 3fiD x + 3n Dx + 7 ) / ( n ) . g(n) = 0, D

(Dz + ±e "

-

l

D

-\e- « + w)f{n).g(n)

= 0,

(31)

where A, //, 7 and ui are arbitrary constants. This result can be proved by using Hirota's bilinear operator identities. We omit the details of the proof. Instead we are going to construct soliton solutions of (24) and (25) by using the BT (29)-(31). Firstly, by applying the BT (29)-(31) to the trivial solution f(n) — 1, we can obtain the 1soliton solution g(n) = 1 + exp (pn + sinh 3 (p)x — sinh(p),z + if) , where p and rf are constants, for the parameters A = l , 7 = o> = 0, and fi = — sinh 5 (p). Furthermore, by applying the BT (29)-(31) to the 1-soliton solution f(n) = 1 + exp(77i), we can deduce the following 2-soliton solution g(n) = 1 + Aiem

+ eV2 + A 2 e r)1+r ' 2 ,

where rji = pi-n + sinh 3 (pi)x - sinh(pj)z + rft, sinh 3 (pi) + sinh 3 (p2) M = j J , - sinh 3 (pi) + sinh 3 (p2) sinh(|(pi-p2)) sinh(i(p! +P2))' with pi and 770 being constants for the set of parameters A = l , 7 = u; = 0 and fi = - s i n h 3 ( p 2 ) - Besides, by using MATHEMATICA, we can show

56

Xing-Biao

Hu & Hon- Wah Tarn

that the system (24) and (25) has the 3-soliton solutions f=l+em+e^+eT>*+A12er>1+r>i+A13er!i+rl3+A23em+rl3+AnA23A13e^+r>2+r>3, where Vi = Pi™ + sinh^ (pi)x - sinh(pj)z + rft, ^ sink* fa) - sigh* fa) sinh (\(pj -p,)) sinh* fa) + sinh*( Pj ) sinh (\{Pi + Pj))' Next, we are going to derive a Lax pair for equation (23) from the BT (29)-(31). For this purpose, we set ip(n) = g(n)/f(n),u(n) — l n / ( n ) . Then from the bilinear BT (29)-(31), we have

(A^(n + 1) - VH)* = (u(n) - u(n + l))x(ip(n) + Xip(n + 1)) - /i(A-0(n + 1)) - ip(n)), (32) _\-leu{n+l)+u(n-l)-2u{n) _

,/

_ ^\

1 Ae u(n+l)+„(n-l)-2 U (n)^ n

+

y

+

^

( n )

+ 3fiipxx(n) -f 6/j,uxx{n)ip(n) + tpxxx(n) + 6uxx{n)ipx(n)

+ 2>^2tj)x{n) - 7r/>(n) = 0.

(33)

We can show the following result: Proposition 4.2: Eqns. (32) and (33) constitute a Lax pair for (23). 5. Another Integrable Differential-Difference Equation for (5) In this section we will report another integrable differential-difference equation for the Hirota-Satsuma equation (6): uxxx(n + 1) - uxxx(n - 1) + (u(n + 1) - u{n)) - (u(n) - u(n - 1)) + -u(n + 1) - u(n) + -u(n - 1) + 6ux{n + l)(u(n + 1) - u{n))2 + 3ul(n + 1) - 3u x (n - 1) - 6ux{n - l)(u(n) - w(n - l)) 2 + 4uxx(n + l)(u(n + 1) - u(n)) + 4uXx(n - l)(u(n) - u(n - 1)) + 2u xx (n)(u(n + 1) - u(n - 1)) = 0, (34) which can be transformed into the following coupled bilinear system by the dependent variable transformation u(n) = ( l n / ( n ) ) x : [2DZDX sinh 2 (iD„) + DZDX - ^Dxsmh(^Dn)]f(n). {Dzsinh(±Dn)+D3xSmh(lDn)}f(n).f(n)

= 0,

f(n) = 0, (35) (36)

The AKNS Equation and the Hirota-Satsuma

Equation

57

where z is an auxiliary variable. We now consider the continuous analogue of the equation (34). Setting U{ne) = u{n) and ne = t, we have u(n + l) = U(t + e). We expand u(n+ 1) as e2 d2

d

„(„ + i ) - t , + e _ [ , +

T

_t;

e3 d3

(37)

!, + ..

+

*n+»-u-+'mu+TZB?u+TE&u+•••

<38>

Substituting (37),(37), and etc. into (34) and neglecting higher-order terms of e , we obtain Uxxxt + 6UxxUt + QUxUxt = 0.

(39)

Obviously equation (39) is just the Hirota-Satsuma equation (5) by setting w = U. In the following, we will present a Backlund transformation for (35),(36) and a Lax pair for (34). The result is as follows: Proposition 5.1: The bilinear equations (35) and (36) have the Backlund transformation [£> z e-5 D " + \DzevD» D

\Dxei "

- (A - \i)e*D»

0

D

- X^Dxe-i "

(D2 +Dl+

l

X

Dn

+ ue^ " - \- uje-^ ]f{n) 2

2

3LUD

+ je-$D"}f(n),g(n)

+ 3u Dx)f(n).g(n)

= 0,

(40)

,g(n) = 0,

(41)

= 0,

(42)

where A, 7 and w are arbitrary constants. This result can be proved by using Hirota's bilinear operator identities. We omit the details of the proof. Instead we are going to construct soliton solutions of (35)) and (36) by using the BT (40)-(42). Firstly, by applying the BT (40)-(42) to the trivial solution f(n) = 1, we can obtain the 1soliton solution g(n) = 1 + exp ( pn - ( - tanh (^PJJ

x + - tanh ( -p\ z + if J ,

where p and 770 are constants, for the parameters A = 1,7 = \ and UJ = - ( i t a n h {\p)Y• Furthermore, by using MATHEMATICA, we can show that the system (35)) and (36) has the 3-soliton solutions f=l+er>l+em+er'3+A12em+,i2+A13e,n+r>z+A23er>2+r>3+A12A23A13er>1+r»+ri3,

58

Xing-Biao

Hu & Hon- Wah Tarn

where r)i=pnA

=

( - tanh (-pA

J

x + - tanh (-pA

z + rft,

tanhs ( | P i ) - tanhs (i P j -)

sinh ( | ( P i -p-,))

tanh 5 ( i P i ) + tanhs ( i P j )

sinh (\{pt +

Pj))'

In the following, we are going to derive a Lax pair for (34) from the BT (40)-(42). For this purpose, we set tp(n) = g(n)/f(n),u(n) = (ln/(n))x. Then from the bilinear BT (40)-(42), we have (\-1rl>(n + l)-il>(n))x

=

(u(n) - u{n + l))(V»(n) + A" V ( " + 1)) + w ( A _ V ( n + 1)) - V ' H ) , (43) {ip(n) + A - V ( " + l))xxx - 3w(^(n) + A"lip(n + l))xx + [uxx(n + 1) - uxx(n) + 3(u x (n + 1) + ux(n))(u(n

+ 1) - u(n))

+(u(n + 1) - u(n))3 + ^](A"V(n + 1) - V(n)) + [-3w(u x (n + 1) + ux{n)) + 3(u(n) - u(n + l))(ux(n

+ 1) -

+3{ux{n + 1) + u x (n) + u2)(4>(n) + A - V ( n + 1))* = 0.

ux(n))

(44)

We can also show the following result: Proposition 5.2. Eqns. (43)-(44) constitute a Lax pair for (34). 6. Conclusion and Discussion The search for new integrable equations is a difficult and challenging problem in soliton theory. Compared to continuous case, less work has been done in discrete case. Recently the problem of integrable discretizations of integrable systems has become one of the focal points in this direction. By doing so, more and more discrete integrable systems have been found. In this paper, we have considered integrable semi-discretizations of two model equations for shallow water waves. As a result, one integrable differentialdifference version for the AKNS equation and three integrable differentialdifference versions for the Hirota-Satsuma equation are found. These four differential-difference versions are transformed into bilinear forms which belong to the classes of the following generalized bilinear equation F(Dx,sinh(a1Dn),---

,smh{aiDn))f{n).f(n)

=0

The AKNS Equation and the Hirota-Satsuma

Equation

59

or coupled generalized bilinear equations F1(Dx,Dz,smh(a1Dn),-

• • ,sinh(aiDn))f(n)»f(n)

=0,

F2(DX, Dz, sinh(ai£) n ), • • • , sinh(a;L> n ))/(n) • / ( n ) = 0, where F and Fi(i = 1,2) are even order polynomials in DX,DZ, sinh(aiD„), • • •, sinh(af£)„), and I is a given positive integer; the a , , i = 1,2, • • • , I are I different constants, and F 4 (0,0,---,0) = 0. It is noted that several new integrable differential-difference equations of these types have been found in the literature by testing bilinear Backlund transformations. Besides, Backlund transformations, soliton solutions and Lax pairs for these four differential-difference equations are presented. Acknowledgements One of the authors (XBHU) would like to express his sincere thanks to the organizers Profs. Cheng Yi, Hu Sen, Li Yishen, Ye Xiangdong and Zhang Pu for their invitation and support to attend the meeting. References 1. Hirota, R.: A New Form of Backlund Transformations and Its Relation to the Inverse Scattering Problem, Prog. Theor. Phys., 52, 1498-1512 (1974). 2. Ablowitz, M.J., Kaup, D.J., Newell, A.C. and Segur, H.: The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math. 53(4), 249-315 (1974). 3. Hirota, R. and Satsuma, J.: N-soliton solutions of model equations for shallow water waves, J. Phys. Soc. Japan 40, 611-12 (1976). 4. Musette, M. and Conte,R.: Algorithmic method for deriving Lax pairs from the invariant Painleve analysis of nonlinear partial differential equations, J. Math. Phys. 32(6), 1450-57 (1991); Hu, X.B. and Li, Y.: Nonlinear superposition formulae of the Ito equation and a model equation for shallow water waves, J. Phys. A: Math. Gen. 24, 1979-86 (1991); Quispel, G.R.W., Nijhoff, F.W. and Capel, H.W.: Backlund transformations and singular integral equations, Physica A 123(2-3), 319-359 (1984); Clarkson, P.A. and Mansfield, E.L.: On a shallow water wave equation, Nonlinearity 7(3), 975-1000 (1994); Leble, S.B. and Ustinov, N.V.: 3rd-order spectral problems-reductions and Darboux transformations, Inverse Prob. 10(3), 617-633 (1994); Conte, R., Musette, M., Grundland, A.M.: Backlund transformation of partial differential equations from the Painleve-Gambier classification II. Tzitzeica equation, J. Math. Phys. 40(4), 2092-2106 (1999); Estevez, P.G., Conde, E. and Gordoa, P.R.: Unified

60 Xing-Biao Hu & Hon-Wah Tarn approach to Miura Backlund and Darboux transformations for nonlinear partial differential equations, J. Nonlinear Math. Phys. 5(1), 82-114 (1998). 5. Hirota, R.: Direct Methods in Soliton Theory, In: Solitons ed. R.K. Bullough and P. J. Caudrey, Springer, Berlin, 1980.

DIFFERENTIAL EQUATIONS A N D CONFORMAL FIELD THEORIES*

YI-ZHI HUANG Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019 USA yzhuang&math.rutgers. edu

We discuss the recent results of the author on the existence of systems of differential equations for chiral genus-zero and genus-one correlation functions in conformal field theories.

1.

Introduction

Two-dimensional conformal field theories form a particular class of nontopological q u a n t u m field theories which have now been formulated and studied rigorously using various methods from different branches of m a t h ematics. In physics, these theories describe perturbative string theory and also critical phenomena in condensed m a t t e r physics. T h e y are also used to describe such phenomena as disorder in condensed m a t t e r physics and to construct nonperturbative objects such as D-branes in string theory. In mathematics, they are closely related to infinite-dimensional Lie algebras, infinite-dimensional integrable systems, t h e Monster (the largest finite sporadic simple group), modular functions and modular forms, Riem a n n surfaces and algebraic curves, knot and three-manifold invariants, Calabi-Yau manifolds and mirror symmetry, and many other branches of mathematics. We also expect t h a t many mathematical problems can be solved by constructing and studying the corresponding conformal field theories. Moreover, the study of such theories might provide hints to possible deep connections among these different branches of m a t h e m a t i c s and will probably shed light on the construction and study of higher-dimensional nontopological q u a n t u m field theories. Mathematically, a geometric formulation of conformal field theory was first given around 1987 by Segal 21 > 22 - 23 and Kontsevich. Segal 2 2 ' 2 3 further 'This research is supported in part by NSF grant DMS-0070800. 61

62

Yi-Zhi

Huang

introduced the important notions of modular functor and weakly conformal field theory which describe mathematically the subtle and deep chiral structures in conformal field theories. One urgent problem is to give a construction of (chiral) conformal field theories in this sense. To construct conformal field theories in this sense and to study these conformal field theories, it is necessary to construct and study chiral correlation functions on Riemann surfaces. For chiral correlation functions on genus-zero Riemann surfaces (or simply called chiral genus-zero correlation functions) associated to lowest weight vectors in minimal models 1 and in Wess-Zumino-Novikov-Witten models 25 , Belavin-Polyakov-Zamolodchikov and Knizhnik-Zamolodchikov found in their seminal works 1 ' 16 , respectively, that these functions actually satisfy certain systems of differential equations of regular singular points (now called the BPZ equations and the KZ equations, respectively). In the case of Wess-Zumino-Novikov-Witten models, it is also known from the works of Tsuchiya-Ueno-Yamada 24 and Bernard 2 ' 3 that chiral correlation functions on higher-genus Riemann surfaces (or simply called chiral higher-genus correlation functions) satisfy systems of differential equations of KZ type. These equations play fundamental roles in the construction and study of the minimal models and Wess-Zumino-NovikovWitten models. A natural question is whether for general conformal field theories satisfying natural conditions, there exist systems of differential equations of regular singular points satisfied by chiral genus-zero correlation functions. More generally, we are interested in whether there exist systems of differential equations for chiral higher-genus correlation functions. The existence of such equations will allow us to study chiral correlation functions using the theory of differential equations and to construct chiral conformal field theories using these correlation functions. Recently, the author 13,15 established the existence of such differential equations in the genus-zero and genus-one cases under suitable natural conditions and applied these equations to the construction of genus-zero and genus-one chiral theories. In the present paper, after a brief discussion of the notion of conformal field theories in the sense of Segal and Kontsevich, we give an overview of these differential equations. For details, see Refs.13, 15. For a recent exposition on conformal field theories in the sense of Segal and Kontsevich and the author's program of constructing such theories from representations of vertex operator algebras, see Ref.14. In the next section, we recall roughly what a conformal field theory is in the sense of Segal 21 ' 22 ' 23 and Kontsevich and what a weakly conformal

Differential

Equations

and Conformal Field Theories

63

field theory is in the sense of Segal 22>23. We discuss systems of differential equations for chiral genus-zero and genus-one correlation functions in Sections 3 and 4, respectively. 2. Conformal Field Theories Consider the following geometric category: The objects are ordered finite sets of copies of S1. The morphisms are conformal equivalence classes of Riemann surfaces whose boundary components are analytically parametrized by the copies of S1 in their domains and codomains. The compositions of morphisms are given using the boundary parametrizations in the obvious way. This category has a symmetric monoidal category structure for which the monoidal structure is defined by disjoint unions of objects and morphisms. Roughly speaking, a conformal field theory is a projective linear representation of this category, that is, a locally convex topological vector space H (called the state space) with a nondegenerate bilinear form and a projective functor from this category to the symmetric monoidal category with traces generated by H (that is, the category whose objects are tensor powers of H and morphisms are trace-class maps), satisfying some natural conditions. Conformal field theories in general have holomorphic (or chiral) and antiholomorphic (or antichiral) parts. Both parts also satisfy an axiom system which defines weakly conformal field theories. Roughly speaking, weakly conformal field theories are representations of geometric categories obtained from holomorphic vector bundles over the moduli space of Riemann surfaces with parametrized boundaries. Our strategy is to construct chiral or antichiral genus-zero and genusone parts of conformal field theories first and then using these to construct the full conformal field theories. In the remaining part of this paper, we shall discuss only chiral genus-zero and genus-one theories. 3. Differential Equations and Chiral Genus-Zero Correlation Functions We first explain the main ingredients of chiral or antichiral genus-zero theories. The chiral or antichiral parts of genus-zero theories have been shown to be essentially equivalent to algebras of intertwining operators among modules for suitable vertex operator algebras (see Refs.7, 8, 9, 10, 12). So

64

Yi-Zhi

Huang

here we briefly describe vertex operator algebras, modules and intertwining operators. A vertex operator algebra is a Z-graded vector space V = LLez ^(n) equipped with a vertex operator map Y : V ® V —+ ^[[z,,? - 1 ]], a vacuum 1 G V and a conformal element w &V, satisfying a number of axioms. One version of the main axiom is the following: For u\,U2,v G V, v' G V = U n g z ^ n ) ' the series (v',Y(ui,zi)Y(u2,z2)v) (v',Y(u2,z2)Y(u1,zi)v) (v\ Y(Y{ui,zi

- z2)u2,

z2)v)

are absolutely convergent in the regions \z\\ > \z2\ > 0, \z2\ > |zi| > 0 and \z2\ > \zi — z2\ > 0, respectively, to a common rational function in z\ and z2 with the only possible poles at zi, z2 = 0 and z\ = z2. Other axioms are: dim V(n) < co, V(n) = 0 when n is sufficiently negative (these are called the grading-restriction conditions); for u,v G V, Y(u,z)v contain only finitely many negative power terms; for u G V,

Y(l,z) = l,

l\mY(u,z)l

= u;

z—>0

let L(n) : V -> V be defined by

Y(LJ,Z)

= Enez-kW*-""2'

then

[L{m), L(n)] = (m - n)L(m + n) + -rz(m3 - m)5m+b,o (c is called the central charge of V), ^ - F ( u , z) = Y ( L ( - l ) u , z) az

for u G V

and L(0)u = nu

for u £ V(n)

(n is called the weight of u and is denoted wt u). For u G V, we write y(w,z) = E „ e z w n ' 2 _ r i _ 1 where un G End V. A V-module is an C-graded vector space W = U „ e c W(n) equipped with a vertex operator map Yw '• V ®W —> ^ [ [ z , z - 1 ]] satisfying all those axioms for V which still make sense. Let W\, W2 and W3 be T^-modules. An intertwining operators of type (w w) 1S a n n e a r m a P 3^ : Wi ® Wb -*

Differential

Equations and Conformal Field Theories

65

W^i-2}) where W^jz} is the space of all series in complex powers of z with coefficients in W3, satisfying all those axioms for V which still make sense. That is, for wi e Wi and w2 S W2, the real parts of the powers of z in nonzero terms in the series y(w\,z2)w2 have a lower bound; for u G V, w1€W1,w2G W2 and v/3 e W£ = Unec(W^U' (w'3, Yw3 {u,

zi)y{wi,z2)v)2)

(w'3, y{wi, z2)YW2 (u, z1)w2) {w'3,y(YWl(u,zi

-

z2)wi,z2)w2)

are absolutely convergent in the regions \z\\ > \z2\ > 0, \z2\ > \z\\ > 0 and |-22| > \z\ — z2\ > 0, respectively, to a common (multivalued) analytic function in z\ and z2 with the only possible singularities (branch points) at z\,z2 = 0 and z\ = z2; also

^-zy(wuz)

=

Y(L(-l)wuz).

For more details on basic notions and properties in the theory of vertex operator algebras, see Refs.5, 4. We need the following notions to state the result on differential equations in the genus-zero case: Let K b e a vertex operator algebra and W a Vmodule. Let Ci(W) be the subspace of W spanned by elements of the form u-\w for u £ V+ = LI„>o ^(n) a n d w G W. If dim W/C\(W) < 00, we say that W is Ci-cofinite or W satisfies the Ci-cofiniteness condition. For chiral genus-zero theories, the main objects we want to construct and study are chiral genus-zero correlation functions. Let Wi for i = 0 , . . . , n+1 and Wi for i = 1 , . . . , n— 1 be V-modules and let ^ i , y2,..., yn-i, yn be intertwining operators of types ( ^ i j , ( ^ J , . . . , {^"-[J, (w#Ci)' respectively. Let uij € Wi for i = 0 , . . . ,n + 1. Formally, chiral genus-zero correlation functions are given by series of the form ( w o J l K , ZX) • • -yn(wn,

Zn)wn+i).

(1)

Theorem 3.1: Let Wi for i = 0 , . . . ,n + 1 be V-modules satisfying the Ci-cofiniteness condition. Then for any Wi € Wi for i = 0 , . . . , n + 1, there exist Ok, i(zi,. ..,zn)e

Clzf1,...,

z*1, {zi-z2)~l,

(Z1-Z3)-1,...,

(zn-i-zn)~l\,

for k = 1 , . . . , m and I = 1 , . . . , n, such that for any V-modules W for i = 1 ,n — 1, any intertwining operators ^i,3^2, • • • , ^ n - 1 , X i , of types

66

Yi-Zhi

Huang

(wx^,)' ( v S 2 ) ' • • •' ( w ^ w L i ) ' ( v ^ n + i ) ' respectively, the series (1) satisfy the expansions of the system of differential equations d"V ^ . •Q-^ + 2^akj{zu...,zn)-—^ oz

i

fc=i

m k

sd

~

l,...,n

dz

i

in the region \zi\ > •••\zn\ > 0. Moreover, there exist ak, i(zi,... ,zn) for k = 1 , . . . , m and I = 1 , . . . , n such that the singular points of the corresponding system are regular. Similar systems of differential equations have also been obtained by Nagatomo and Tsuchiya in Ref.20. Using these equations and other results on vertex operator algebras, modules and intertwining operators, chiral genus-zero weakly conformal field theories in the sense of Segal have been constructed. In particular, the direct sum of a complete set of inequivalent irreducible modules for a suitable vertex operator algebra has a natural structure of an intertwining operator algebra. Thus for such a vertex operator algebra, (1) is absolutely convergent in the region \z\\ > • • • \zn\ > 0 and associativity and commutativity for intertwining operators hold. For more details on intertwining operator algebras and chiral genus-zero weakly conformal field theories, see Refs.6-14. 4. Differential Equations and Chiral Genus-One Correlation Functions The second logical step is to construct chiral genus-one theories, that is, to construct maps associated to genus-one surfaces and prove the axioms which make sense for genus-one surfaces. Assume we have a weakly conformal field theory in the sense of Segal. Then for given elements in the state space of the theory, these maps give certain functions on the moduli space of genus-one surfaces with punctures (the space of conformal equivalence classes of such surfaces). They can be viewed as (multivalued) functions of z\,..., zn e C

and r 6 H (the upper half plane). Here as usual, r corresponds to a torus given by the parallelogram with vertices 0,1,r and 1 + r and zi,...,zn correspond to points on the torus. But functions of zi,...,zn and T are in general only functions on the Teichmuller space, not functions on the moduli space. To construct genusone theories, we do need to construct mathematical objects (vector bundles and holomorphic sections) on the moduli space, not the Teichmuller space.

Differential Equations and Conformal Field Theories 67

The moduli space is the quotient of the Teichmiiller space HI by the modular group SL(2, Z). So we have to construct SL(2, Z)-invariant spaces of functions of the form above. These functions in the S'L(2,Z)-invariant spaces are called chiral genus-one correlation functions. The first result in this step was obtained by Zhu 26 . He constructed chiral genus-one correlation functions associated to elements of a suitable vertex operator algebra V. Using his method, Miyamoto 19 constructed chiral genus-one correlation functions associated to elements of F-modules among which at most one is not isomorphic to V. But Zhu's method cannot be generalized to construct chiral genus-one correlation functions associated to elements of V-modules among which at least two are not isomorphic to V, because he used a recurrence formula which cannot be generalized to this general case. In Ref.15, the author solved completely the problem of constructing chiral genus-one correlation functions from chiral genus-zero correlation functions. As in the genus-zero case, one of the main tools is systems of differential equations. To construct these functions from representaions of a vertex operator algebra, we need some conditions on the vertex operator algebra and its modules. We first need some concepts: Let V be a vertex operator algebra and W a V-module. Let C2(W) be the subspace of W spanned by elements of the form U-2W for u,w £ W. Then we say that W is C^-cofinite or satisfies the Cz-cofiniteness condition if dim W/C2(W) < oo. It is easy to see that if V(n) = 0 when n < 0 and V(0) = C I , a V-module W is C2-cofinite implies W is Ci-cofinite. We now assume that the vertex operator algebra V satisfies the condition that (1) is absolutely convergent in the region \zi\ > •••\zn\ > 0 and associativity and commutativity for intertwining operators hold. We shall use the notation qz = e2niz for z £ C. Let W*, Wi be V-modules, and Wi £ Wjjoi i = 1 , . . . , n. For any intertwining operators yi,i = l,...,n,oi *yP es ( w - ^ ) ' respectively, let -FVi

yn(wi> • • •. wn, 21, • • •, zn;r)

= TTWny1(U(qZl)wuqZl)---yn(U(qzJwn,qZn)q^°)-"\

(2)

where c is the central charge of V and for z £ C \ {0}, U{qz) = eV°sM+i*rSqz)L(0)e-L+(A)^

Q

<

<

^

68

Yi-Zhi Huang

L+(A) = J2j>i AjL{j)

and Aj for j > 1 are determined by

-— log(l + 2-niw) = I exp I ^

^4j U ) J + l

d

|

| yj_

Let x

1

-^ = - 4-

v--

/

\

I

1

1 1

L

(kT +

l)2)'

be the Weierstrass zeta function and the Weierstrass p-function, respectively, and let p m ( z ; r ) for m > 2 be the elliptic functions denned recursively by pm+l{z,T)

=

—

pm(Z;T).

m oz We also need the Eisenstein series mez\{o} iez

(m,i)^(0,0)

V

v

;

'

See, for example 17,18 , for detailed discussions on these elliptic functions and the Eisenstein series. Let R = C[G4(T),G6(T),p2(Zi

- Zj;T),p3(zi

- Zj\ r ) ] j , j = i,...,„, i<j,

that is, the commutative associative algebra over C generated by the series GA{T), GQ{T), p2(zi - ZJ;T) and p3(zi - zy, T) for i, j = 1 , . . . , n satisfying i < j . For m > 0, let Rm be the subspace of R spanned by elements of the form G4fcl(r)G6fc2(r)p2fe3(^ - ^ r ) ^ - ZJ;T) for ki,k2,k3,k4 > 0 satisfying 4fci + 6fc2 + 2/C3 + 3/wt = m. We introduce, for any a e C , the notation

0,-(a) = 2TTZ— + G 2 (r)a + G2(T) j S , — - $ > i ( * - ^ ^ ) ^ T

Differential

Equations

and Conformal Field Theories

69

for j — 1 , . . . ,n. We shall also use the notation Yl"Li ®(aj) *° denote the ordered product 0(a\) • • • 0(am). Then we have the following result: Theorem 4.1: Let V be a vertex operator algebra satisfying the conditions stated above and let Wt for i = 1 , . . . ,n be F-modules satisfying the Cicofiniteness condition. Then for any homogeneous Wi G Wi (i — 1 , . . . , n), there exist o.P, i(z\,...

, z n ; r ) G Rp,

bPi i(zi, ...,zn\r)

G i?2p

for p = 1 , . . . , m and i = 1 , . . . , n such that for any V-modules Wi (i = 1 , . . . , n) and intertwining operators 3^ of types ( M/ '^.) (i = 1 , . . . , n, Wo = ty„), respectively, the series (2) satisfies the expansion of the system of differential equations

a^r + £ap,i( z i.-". z n; T )^-7s=F = = 0 , az

J J Oi ( ^ fc=l

»

p=l

(3)

^ i

wt u>i + 2(m - fc) j ^

\i=l

/

m

m-p

/ n

+ ^ 6 P i i ( z 1 , . . . , z n ; r ) ] ^ 0* p=l

fc=l

] T wt

\ Wi

+ 2(m - p - fc) 1

\i=l

/

7 = 1 , . . . ,n, in the regions 1 > \qZl\ > ••• > \qZn\ > \qT\ > 0. Moreover, for fixed T G H, the singular points of the (reduced) system (3) are regular. The elliptic functions and the Eisenstein series discussed above have the following modular transformation formulas (see, for example, 1 7 ): For any . , G5L(2,Z),

°2 ( f r r f ) = ^T + 5fG^T) -

Z V7T

OLT +

+ 5 7T + 6

(3\

= (jT +

27ri

7(7T + S),

6)mpm(Z,T),

for k > 2 and m > 1. Using these formulas, it is straightforward to verify the following modular invariance of the system (3)-(4):

70

Yi-Zhi

Huang

Proposition 4.1: Let
\ Wt W\-\

G5L(2,Z),

Wt Wn

1 \ ^T + SJ is also a solution of the system

(3)-(4).

/

/ \cT

(f

,

z\ , , + d '" cT

n

zn , ar + fi + d jT + S

(3)-(4).

Using these systems of differential equations, chiral genus-one correlation functions have been constructed as the analytic extensions of sums of series of the form (2) in the region 1 > \qZl\ > ••• > \qZn\ > \qT\ > 0. Together with other results in the representation theory of vertex operator algebras, it has been proved that for suitable vertex operator algebras, the vector space of these chiral genus-one correlation functions are invariant under a suitable action of the modular group SX(2, Z). See Ref.15 for details. Acknowledgment I am grateful to Sen Hu and Jing Song He for their hospitality during the conference. References 1. A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite conformal symmetries in two-dimensional quantum field theory, Nucl. Phys. B241, 333380 (1984). 2. D. Bernard, On the Wess-Zumino-Witten models on the torus. Nucl. Phys. B303), 77-93 (1988. 3. D. Bernard, On the Wess-Zumino-Witten models on Riemann surfaces. Nucl. Phys. B309 (1988), 145-174. 4. I. B. Prenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, preprint, 1989; Memoirs Amer. Math. Soc. 104, 1993. 5. I. B. Frenkel, J. Lepowsky, and A. Meurman, Vertex operator algebras and the Monster, Pure and Appl. Math., Vol. 134, Academic Press, New York, 1988. 6. Y.-Z. Huang, A theory of tensor products for module categories for a vertex operator algebra, IV, J. Pure Appl. Alg. 100 , 173-216 (1995). 7. Y.-Z. Huang, Two-dimensional conformal geometry and vertex operator algebras, Progress in Mathematics, Vol. 148, Birkhauser, Boston, 1997.

Differential Equations and Conformal Field Theories 71 8. Y.-Z. Huang, Intertwining operator algebras, genus-zero modular functors and genus-zero conformal field theories, in: Operads: Proceedings of Renaissance Conferences, ed. J.-L. Loday, J. Stasheff, and A. A. Voronov, Contemporary Math., Vol. 202, Amer. Math. Soc, Providence, 335-355 (1997). 9. Y.-Z. Huang, Genus-zero modular functors and intertwining operator algebras, Internat. J. Math. 9, 845-863 (1998). 10. Y.-Z. Huang, A functional-analytic theory of vertex (operator) algebras, I, Comm. Math. Phys. 204, 61-84 (1999). 11. Y.-Z. Huang, Generalized rationality and a "Jacobi identity" for intertwining operator algebras, Selecta Math. (N. S.), 6, 225-267 (2000). 12. Y.-Z. Huang, A functional-analytic theory of vertex (operator) algebras, II, to appear; math.QA/0010326. 13. Y.-Z. Huang, Differential equations and intertwining operators, to appear; math. Q A/0206206. 14. Y.-Z. Huang, Riemann surfaces with boundaries and the theory of vertex operator algebras, to appear; math.QA/0212308. 15. Y.-Z. Huang, Differential equations, duality and modular invariance, to appear; math.qA/0303049. 16. V. G. Knizhnik and A. B. Zamolodchikov, Current algebra and Wess-Zumino models in two dimensions, Nucl. Phys. B247, 83-103 (1984). 17. N. Koblitz, Introduction to elliptic curves and modular forms, Second Edition, Graduate Texts in Mathematics, Vol. 97, Springer-Verlag, New York, 1993. 18. S. Lang, Elliptic functions, Graduate Texts in Mathematics, Vol. 112, Springer-Verlag, New York, 1987. 19. M. Miyamoto, Intertwining operators and modular invariance, to appear; math.qA/0010180. 20. K. Nagatomo and A. Tsuchiya, Conformal field theories associated to regular chiral vertex operator algebras I: theories over the projective line, to appear; math.qA/0206223. 21. G. Segal, The definition of conformal field theory, in: Differential geometrical methods in theoretical physics (Como, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 250, Kluwer Acad. Publ., Dordrecht, 165-171 (1988). 22. G. B. Segal, The definition of conformal field theory, preprint, (1988). 23. G. B. Segal, Two-dimensional conformal field theories and modular functors, in: Proceedings of the IXth International Congress on Mathematical Physics, Swansea, 1988, Hilger, Bristol, 22-37 (1989). 24. A. Tsuchiya, K. Ueno and Y. Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, in: Advanced Studies in Pure Math., Vol. 19, Kinokuniya Company Ltd., Tokyo, 459-566 (1989). 25. E. Witten, Non-abelian bosonization in two dimensions, Comm. Math. Phys. 92, 455-472 (1984). 26. Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9,237-307(1996).

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LIE S Y M M E T R I E S FOR LATTICE EQUATIONS

D. LEVI Universitd

Dipartimento di Fisica "E. Amaldi", degli Studi Roma Tre and Sezione INFN, Roma Via della Vasca Navale 84, 00146 Roma, Italy

Tre,

Lie symmetries has been introduced by Sophus Lie to study differential equations. It has been one of the most efficient way for obtaining exact analytic solution of differential equations. Here we extend this technique to the case of differential difference and difference equations.

1. Introduction The main object of this report is to present an introduction to the field of linear and non-linear difference or differential difference equations 1>2'3<4. By a difference equation we mean a functional relation, linear or nonlinear, between functions calculated at different points of a lattice. Why are these systems important? They appear in many applications. First of all they can be written down as a discretization of a differential equation when one is trying to solve it with a computer. In such a case one reduces the differential equation to a recurrence relation: — = f{x,u)

=>

v(n +

l)=g{n,v(n)).

On the other hand we can consider dynamical systems defined on a lattice, i.e. systems where the real independent fields depend on a set of independents variables which vary partly on the integers and partly on the reals. For example we can have: d2u(n, t) ——Y— = F(t, u(n, t), u(n -l,t),..,

u(n - a,t), u(n + 1, t),.., u(n + b, t))

These kind of equations can appear in many different setting; among all, those which are more interesting from a scientific point of view, are associated to the evolution of many body problems, to the study of crystals, to biological systems, etc.. 73

74

D. Levi

As an example of possible applications we consider the problem of the transmission of energy in one dimensional molecular systems, problem which is of particular relevance for understanding the functioning of physical systems of biological interest 5 . This is a particularly hot topic as it has been argued that some relevant biological processes require the transport of energy with low dispersion along essentially one dimensional chains or substructures of macromolecular systems that can be regarded as quasi one dimensional chains (such as the spines in an a helix 6 ) . A mechanism for the nondispersive transport of vibrational energy along hydrogenon bonded chains was proposed by Davydov and its continuos limit for small lattice spacing gave rise to a Non-linear Schroedinger equation with soliton solutions 7 . If such soliton like solutions are valid also at biological temperatures is an open problem. In the case of diatomic non-linear lattices we can describe such systems by the following dynamical systems: Mi'xn - ki(yn - xn) + k2(xn - j/„_i) - e(3i{yn - xn)2 +e/32(xn - Vn-if M2yn + ki{yn - xn) - k2(xn+i

= 0

- yn) + e/3i{yn -e(32{xn+i

(1)

- yn)

2

xnf = 0

where M\ and M2 are the different values of the two atomic masses, e is a small parameter while k\, k2, 0\ and (32 are four constants of order 1. When k2
Lie Symmetries

for Lattice Equations

75

doing a continuum limit. The numerical results (see Figs. 1-3) show clearly the relevance of the non-linear terms in this non-dispersive energy transport. The analytic calculations, hint to the result but, as involve a continuum limit, are not sufficient. To conclude I would like to comment on the third point concerning the richness of the world of the differential difference equations with respect to that of partial differential equations by presenting a result recently obtained by MacKay and Aubry 8 . In a theorem contained in their work MacKay and Aubry showed that dynamical chains are much richer than their continuum conterparts; almost any Hamiltonian network of weakly coupled oscillators has a 'breather' solution while the existance of breathers for a nonlinear wave equation is rare. This implies that the discrete world can be richer of interesting solutions and thus worthwhile studying by itself. Lie groups have long been used to study differential equations. As a matter of fact, they originated in that context 9 ' 1 0 . They have been put to good use to solve differential equations, to classify them, and to establish properties of their solution spaces n . Applications of Lie group theory to discrete equations, like difference equations, differential-difference equations, or ^-difference equations are much more recent 17 . References 1. W.G. Kelley and A.C. Peterson, Difference equations: an introduction with applications, (Academic Press, New York 1991). 2. R.P. Agarwal, Difference equations and inequalities: theory, methods and applications, (Dekker, New York 1992). 3. M. Toda, Theory of Nonlinear Lattice (Springer, New York 1988). 4. M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, (SIAM, Providence 1981). 5. A. Campa, A. Giansanti, A. Tenenbaum, D. Levi, and O. Ragnisco, Phys. Rev. 48, 10168 (1993). 6. A.S. Davydov, Phys. Scr.20, 387 (1979); Biology and Quantum Mechanics (Pergamon, New York 1982). 7. A.C. Scott, Phys. Rep. 217, 1 (1992). 8. R.S. MacKay and S. Aubry, Nonlinearity 7, 623 (1994). 9. S. Lie, Klassifikation und Integration von Gewohnlichen Differentialgleichungen zwischen x,y die eine Gruppe von Transformationen gestatten, Math. Ann. 32, 213 (1888). 10. S. Lie, Theorie der Transformationgruppen (B.G. Teubner, Leipzig, 1888, 1890, 1893).

76

D. Levi

11. P.J. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1993). 12. N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics (Reidel, Boston, 1985). 13. L.V. Ovsiannikov, Group Analysis of Differential Equations (Academic, New York, 1982). 14. G.W. Bluman and S. Kumei, Symmetries and Differential Equations (Springer, Berlin, 1989). 15. G. Gaeta, Nonlinear Symmetries and Nonlinear Equations (Kluwer, Dordrecht, 1994). 16. P. Winternitz, Group theory and exact solutions of nonlinear partial differential equations, In Integrable Systems, Quantum Groups and Quantum Field Theories, 429-495, Kluwer, Dordrecht, 1993. 17. S. Maeda, Canonical structure and symmetries for discrete systems, Math. Japan 25, 405 (1980). 18. D. Levi and P. Winternitz, Continuous symmetries of discrete equations, Phys. Lett. A 152, 335 (1991). 19. D. Levi and P. Winternitz, Symmetries and conditional symmetries of differential-difference equations, J. Math. Phys. 34, 3713 (1993). 20. D. Levi and P. Winternitz, Symmetries of discrete dynamical systems, J. Math. Phys. 37, 5551 (1996). 21. D. Levi, L. Vinet, and P. Winternitz, Lie group formalism for difference equations, J. Phys. A: Math. Gen. 30, 663 (1997). 22. R. Hernandez Heredero, D. Levi, and P. Winternitz, Symmetries of the discrete Burgers equation, J. Phys. A Math. Gen. 32, 2685 (1999). 23. D. Gomez-Ullate, S. Lafortune, and P. Winternitz, Symmetries of discrete dynamical systems involving two species, J. Math. Phys. 40, 2782 (1999). 24. S. Lafortune, L. Martina and P. Winternitz, Point symmetries of generalized Toda field theories, J. Phys. A: Math. Gen. 33, 2419 (2000). 25. R. Hernandez Heredero, D. Levi, M.A. Rodriguez and P. Winternitz P, Lie algebra contractions and symmetries of the Toda hierarchy, J. Phys. A: Math. Gen. (in press). 26. D. Levi and R. Yamilov, Conditions for the existence of higher symmetries of evolutionary equations on the lattice, J. Math. Phys. 38, 6648 (1997). 27. D. Levi and R. Yamilov, Non-point integrable symmetries for equations on the lattice, J. Phys. A: Math. Gen. (in press). 28. D. Levi, M.A. Rodriguez, Symmetry group of partial differential equations and of differential-difference equations: the Toda lattice vs the Korteweg-de Vries equations, J. Phys. A: Math. Gen. 25, 975 (1992). 29. D. Levi, R. Yamilov, Dilatation symmetries and equations on the lattice, J. Phys. A: Math. Gen. 32, 8317 (1999). 30. D. Levi, M.A. Rodriguez, Lie symmetries for integrable equations on the lattice, J. Phys. A: Math. Gen. 32, 8303 (1999). 31. R. Floreanini, J. Negro, L.M. Nieto and L. Vinet, Symmetries of the heat equation on a lattice, Lett. Math. Phys. 36, 351 (1996). 32. R. Floreanini and L. Vinet, Lie symmetries of finite-difference equations, J.

Lie Symmetries for Lattice Equations 77 Math. Phys. 36, 7024 (1995). 33. G.R.W. Quispel, H.W. Capel, and R. Sahadevan, Continuous symmetries of difference equations; the Kac-van Moerbeke equation and Painleve reduction, Phys. Lett. A 170, 379 (1992). 34. G.R.W. Quispel and R. Sahadevan, Lie symmetries and integration of difference equations, Phys. Lett. A 184, 64 (1993). 35. V.A. Dorodnitsyn, Transformation groups in a space of difference variables, in VINITI Acad. Sci. USSR, Itogi Nauki i Techniki, 34, 149-190 (1989), (in Russian), see English translation in J. Sov. Math. 55, 1490 (1991). 36. W.F. Ames, R.L. Anderson, V.A. Dorodnitsyn, E.V. Ferapontov, R.K. Gazizov, N.H. Ibragimov and S.R. Svirshchevskii, CRC Hand-book of Lie Group Analysis of Differential Equations, ed. by N.Ibragimov, Volume I: Symmetries, Exact Solutions and Conservation Laws, CRC Press, 1994. 37. V.A. Dorodnitsyn, Finite-difference models entirely inheriting continuous symmetry of original differential equations. Int. J. Mod. Phys. C, (Phys. Comp.), 5, 723 (1994). 38. V. Dorodnitsyn, Continuous symmetries of finite-difference evolution equations and grids, in Symmetries and Integrability of Difference Equations, CRM Proceedings and Lecture Notes, Vol. 9, AMS, Providence, R.L, 103112, 1996, Ed. by D.Levi, L.Vinet, and P.Winternitz, see also V.Dorodnitsyn, Invariant discrete model for the Korteweg-de Vries equation, Preprint CRM2187, Montreal, 1994. 39. M. Bakirova, V. Dorodnitsyn, and R. Kozlov, Invariant difference schemes for heat transfer equations with a source, J. Phys. A: Math.Gen., 30, 8139 (1997) see also V. Dorodnitsyn, R. Kozlov, The complete set of symmetry preserving discrete versions of a heat transfer equation with a source, Preprint of NTNU, NUMERICS NO. 4/1997, Trondheim, Norway, 1997. 40. V. Dorodnitsyn, Finite-difference models entirely inheriting symmetry of original differential equations Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics (Kluwer Academic Publishers), 191, 1993. 41. V.A. Dorodnitsyn, Finite-difference analog of the Noether theorem, Dokl. Akad. Nauk, 328, 678, (1993) (in Russian). V. Dorodnitsyn, Noether-type theorems for difference equation, IHES/M/98/27, Bures-sur-Yvette, 1998. 42. V. Dorodnitsyn and P. Winternitz, Lie point symmetry preserving discretizations for variable coefficient Korteweg - de Vries equations, CRM-2607, Universite de Montreal, 1999 ; to appear in Nonlinear Dynamics, Kluwer Academic Publisher, 1999. 43. V. Dorodnitsyn, R. Kozlov and P. Winternitz P, Lie group classification of second order ordinary difference equations, J. Math. Phys. 41, 480 (2000). 44. D. Levi, L. Vinet, and P. Winternitz (editors), Symmetries and Integrability of Difference Equations, CRM Proceedings and Lecture Notes vol. 9, (AMS, Providence, R.L, 1996). 45. P A . Clarkson and F.W. Nijhoff (editors), Symmetries and Integrability of Difference equations (Cambridge University Press, Cambridge, UK, 1999).

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MULTI-SCALE R E D U C T I O N FOR D I F F E R E N T I A L D I F F E R E N C E EQUATIONS A N D I N T E G R A B I L I T Y D. LEVI Dipartimento di Fisica "E. Amaldi", Universitd degli Studi Roma Tre and Sezione INFN, Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy

Perturbation methods has proved very successful in analyzing classes of partial differential equations, pinning out integrable cases and providing informations on their long time behavior. Here we present the state of art of this approach in the case of differential equations defined on a lattice. In particular we present the results of Leon and Manna on the multiscale analysis of discrete nonlinear evolution equations and show that their resulting equation has not the same integrability property as Ablowitz and Ladik Discrete Nonlinear Schrodinger equation.

1. Introduction Dynamical systems denned on a lattice or Differential Difference Equations (DDE's), are systems where the dependent fields are functions of a set of independents variables which vary partly on the integers and partly on the reals, like for example the DDE d?u(n t) ——-^— = F(t,u(n,t),u(n-

1,t),..,u(n - a,t),u(n

+ l,t),..,u(n

+ b,t)).

DDE's can appear in many different setting. Among them, those which are more interesting from a scientific point of view, are associated to the evolution of many body problems, to the study of crystals, to the analysis of biological systems, etc.. As an example in the last case, let us consider the problem of the transmission of energy in one dimensional molecular systems, problem which is of particular relevance for understanding the functioning of physical systems of biological origin 1 . This is a particularly hot topic as many important biological processes require the transport of energy with low dispersion along essentially one dimensional chains as, for example, the spines in an a helix 2 . Davydov proposed a mechanism for the nondispersive transport of energy along hydrogen bonded chains. Its continuous limit for small lattice spacing gave rise to the Nonlinear Schrodinger equation (NLS) which has 79

80

D. Levi

soliton solutions 3 . If such soliton like solutions are valid also at biological temperatures is an open problem. In the case of diatomic nonlinear lattices we can describe such systems by the following dynamical system: Mi±n - ki{yn - xn) + k2(xn - yn_i) - e/3i(yn - xn)2

(1)

2

+e/32{xn - y„_!) = 0 M2jjn + kl(Vn ~ Xn) - k2{xn+l

~ Vn) + C/9l(j/n ~

-ef32(xn+i

Xn)2

- yn)2 = 0

where M\ and M2 are the different values of the two atomic masses, e is a small parameter while fci, k2, /?i and (32 are four constants of order 1. When k2
Multi-scale Reduction for Differential

Difference Equations and Integrability

81

ential Equations (PDE's). In Section 3 we expose the results of Leon and Manna on the multiscale analysis of the Toda Lattice and the derivation of a New Discrete NonLinear Schrodinger equation (NDNLS). Section 4 is devoted to the analysis of the NDNLS and the prove that this equation has not the same integrability properties as Ablowitz and Ladik Discrete NonLinear Schrodinger equation (DNLS). Section 5 is devoted to some conclusive remarks. 2. Multiscale reduction for P D E Usually a physical problem is described by a complicate equation which, according to the problem at study, can be either a PDE or a DDE. Through the multiscale analysis of the system 5 ' 6 , 7 one can deduce from the basic model a simplified equation which preserves the longtime features of the starting problem. For PDE's the multiscale analysis consists in the introduction of - coarse grained space variables - slow time variables and provides the long-time behavior of the system. To simplify our exposition we will consider a class of PDE's for one dependent complex variable in two independent real variables u(x, t), which we can write as Du = eF(u, utX, utXX, ...;u*,u*x,...)

(2)

where u* denotes the complex conjugate of u and e is a small parameter. The linear dispersive differential operator D is given by B

M

— ) , Lj=J2amkm, ameR, °x m=o F represent the nonlinear part of the PDE, such that D = dt+iuj(-i

M > 1.

(3)

oo

F(eu, eu,x, eu,xx,...;

eu*, eu*x,...) = ] T

m {m) e

F (u,uiX,u,xx,...;u*,u*x,...),

m=1

(4) where F^ are homogeneous polynomials of order m. Du = 0 is the linearized equation whose solution reads oo

/

•oo

dku{k,t)Skx-"(kW.

(5)

82

D. Levi

The corresponding group velocity is given by dw(ife) (6) dk We propose a solution of the general nonlinear equation (2), which involves the coupling between the various modes admissible in the linear equation (5), of the form: 9

oo

«(*,*)= £

e^e^ntf.T),

(7)

n=—oo

z = kx - u>(k)t, Z = ep[x-vgt],

7„ > 0, T = eH,

7„ € Z+ (p,q)> (0,0),

(p,q) € R+

where (£, T) are slow varying variables and ijjn are slowly varying functions which describe the modulation of the linear wave. Let us notice that the sum contained in eq.(7) is an asymptotic sum and (p, q, 7„) are parameters to be chosen in such a way that as e vanishes a finite, nontrivial, consistent result emerges from eq.(2). £ provides a rescaled dilatated new space variable while T is the new slow time. We insert the ansatz (7) into eq.(2) and look for a nontrivial PDE for V>i(£, T). TO do so we - equate the coefficients of the different powers of e*z, - make appropriate choices for p, q and 7„ with 71 = 0, - consider the limit e —> 0. As a result we get contributions from all orders of e, but only the lowest ones are relevant. Moreover all higher modes corresponding to n ^ 1 as well as the mode n = 0, are coupled to the first one (n = 1) and give the modulation functions ipn in terms of ^1 = il>As a result of this reductive procedure, we get usually an equation of NLS type

#, T = m/\« + m2i>

(8)

which is integrable if a and /3 are real constant. The essence of the perturbative reductive technique, outlined above, consists in a sequence of change of variables and limiting processes. As such it cannot change the integrability properties of the system at study 7 . Moreover to solve a reduced equation cannot be more difficult than to solve the original problem. In fact we can apply the reduction technique at the level of the known solution of the initial integrable problem and we will

Multi-scale Reduction for Differential

Difference Equations

and Integrability

83

obtain a solution of the reduced one. Zakharov and Kuznetzov 8 showed that one can apply this technique also at the integration mechanism level and obtain the integration mechanism of the reduced equation. 3. Multiscale reduction for D D E In 1999 Jerome Leon and Manuel Manna 9 proposed a set of tools to perform multiscale analysis on a DDE which describe the propagation of a signal along a nonlinear lattice. They considered the case of a dependent varable depending on a discrete lattice variable n and a continuous real variable t. To carry out the reduction at the discrete level they defined - a large scale grid which transform a lattice into a new lattice, - a transformed time depending on the lattice position, - a slow variation hypotesis. As solutions of the linearized DDE one considers discrete waves of the form oo

/•OO

dfcu(fc,i)ei[fcn
(9)

/ -oo •oo where a is the lattice spacing. The large scale grid is defined in term of a large integer number N and consequently the small parameter of the theory is given by

<10>

^h-

The rescaled discrete variable £„, the coarse grained variable, is given by in = e2n, while the new slow time rn(t) reads . rur. rn=e[t+-],

dcj(k) v, = -±L.

(11) , . (12)

Let us notice that if £ n = m is an integer, than it will corrispond to values of n multiples of N such that n = mN. Following the case of the PDE, presented in Section 2, we consider the following ansatz oo

un{t) = 5 3 e « [ w - w < f c > * W £ > ( T n ( t ) ) , i=—oo where jj (J ^ 1) are constants greater than zero and 71 = 0.

(13)

84 D. Levi We write a given D D E for un(t)

A „ u n = un+1

where by I

-u„_i;

in t e r m s of discrete derivatives

A*un

= 5 ^ ( - l ) ' ( , J un+k_2i

(14)

I we m e a n t h e usual binomial coefficient. T h e transformation

(11) will define a discrete derivative in t h e m variables of t h e function un AmUn

= Un+N - Un-AT = A m U m = U m + l - U m _ l =

_[*pr9(2q

+ i)\(q + iy.

^

( 9 -Z)!(2Z + l ) ! ^ "

2I+1

""•

Leon and M a n n a t h a n introduce t h e following slow variation \Akn+1un\=e2\Aknun\

(15)

hypothesis

+ 0(ei).

(16)

Under this hypothesis one gets = e2Amum

Anun

+ 0{ti).

(17)

Consequently Un+1(Tn+1)

= Un+i(Tn + — ) =

(18)

V

9

= um(r)

H

um,T{j)

+ -^[-^Um^TT 0 V*

2 2 + — [-Ju m ,TT + w m + i ( r ) - um(r)]

+ 3 — (um+i Vg

T

- U m , T )] +

+

0(e4)

where r „ = rm = r . A similar formula can be written down for w „ _ i ( r n _ i ) . As an example of application of this theory we consider t h e reduction of t h e Toda Lattice 9 •j.U\

_ gXn + lM-XnW _ gI„(t)-X„_l(t)

Qg\

Denning V n (i) = exn(t) and B n ( t ) = e ( e *»(*)-*»-i(*) - 1) we can write eq.(19) as a first order system of equations with quadratic nonlinearity

Bn(t) = [1 + eBn(t)][Vn(t) - Vn-i(t)] Vn(t) = Bn+1(t) - Bn(t), (Bn(t), Vn(t)) € R

(20)

Multi-scale Reduction for Differential Difference Equations and Integrability 85

The dispersion relation is u(k) = 4sin 2 (^f) and consequently eq.(13) reads oo

Bn(t)=

£

e7Vj[fc"*-"(fc>V%n,Tn(t)),

V _ j =V' 0 >

j=—oo oo

vn(t)= £ ^eii[kmT-uWt]U)^n,Tn(t)),

(21)

r j = U)*

j=-oo

where £„ and rn{t) are given by eqs.(ll, 12) and ~fj are constant greather than zero with 71 = 0 and j-j — 7j. Defining <j>m (T) = r)m(r), introducing the ansatz (21) into eq.(20) and solving the equations for j = 0,1,2 at the lowest order in e, we get that V'm(T) an< ^
(22)

where a = t a n 2 ( ^ ) and j3 = sin(fc(r). 4.

Study of the N D N L S

It is well known that both the Toda Lattice and the Discrete NLS 12 ' 13 have infinite classes of commuting local generalized symmetries 10'u,12,13 independent from n and t. We should expect that similar results could be obtained also for the NDNLS given by eq.(22). Let's study the integrability of eq.(22). As r}m(T) is a complex function, we can rewrite eq.(22) as a system. As we are interested in showing the existence or non existence of higher order local symmetries or commuting flows we can take r complex and, with no loss of generality, write eq.(22) as um = um+i - u m _x + u2mvm, % = -{Vm+l

- Vm-i)

(23)

2

+ V mUm.

Instead of considering eq.(23), we can write down the following class of coupled DDE's Um = Um+1 + $ ( « m - l , V m - l ) + *(ll f f l , Vm) = / , Vm = - V r o + 1 + * ( « m - l , » m - l ) + V(um,Vm)

OUm

ovm

dum

= g,

dvm

(24)

86 D. Levi

which have the same integrability properties as (23). In eq.(25) a, /3, 7 and 8 are arbitrary nonzero constants. Denning

/um\ W„

(26)

\VmJ we can rewrite (24) as a first order matrix equation (27)

where

(28)

Vm )

f \g The symmetries for eq.(27) can be defined as Wm>\ = G(Wm+n, W m + n _ i , . . . , Wm+n>).

(29)

Having written the system in evolutionary form we can consider the formal symmetries14 and look for the existence of a common set of solutions of (27) and (29). This is equivalent to look for an approximate solution L of the Lax equation L,t = [F.,L],

(30)

where F* is the Prechet derivative of F and is given by F. = F&T + J*°> + FfrVT-1,

(31)

where T is the shift operator Tfm = / m + i and Fm are rank 4 matrix obtained as partial derivatives of F, given by Fm = a^F • For convenience, it is simpler to write them as 2 x 2 blocks, 0

E

eSPo (4 where E is the unit matrix, and the matrices Q*W m' (i = —1,0,1) are denned as

00)

J,Um+i

9,um+i

J,Vm+i

\

9,vm+i)

Multi-scale Reduction for Differential

Difference Equations

and Integrability 87

i.e. ft(-l)

_

/'*.«m-l *,«m-l \

ft(0)

_ (®,Um $,Vm\

The solution of eq.(30) can be written as a formal serie in decreasing powers of T, whose coefficients are matrices of rank 4 L

= L(n)Tn+i(n-l)Tn-l+

;

(32)

with £,(") 7^ 0 if a solution of order n exists. Defining Q(L) = Lt-[F.,L]

(33)

we find that, if L is of order n than £l(L) is of order n + 1, i.e. £l(L) = fi(n+i)Tn+i + Q(n)Tn + a n d t h e s e r i e s representation of L is a formal symmetry of order n and lenght I if the highest / coefficients flW, with i = n + 1 , n, . . . , n — Z + 2 are zero. In such a case we say that L is an approximate solution of (30) with I exact coefficients. If we multiply formal symmetries among themselves we get again formal symmetries, i.e. il(LL) = Q(L)L + Lfl(L) and the existence of formal symmetries implies the existence of symmetries. Consequently, if we are not able to construct formal symmetries, symmetries will not exist. If the NDNLS is to be as integrable as the Toda Lattice, from which it has been derived, or the DNLS of Ablowitz and Ladik, than it should possess local symmetries of high order, as high as one wants. We limit ourselves to consider local symmetries which do not dependent explicitly from (m,T), as the Toda Lattice and DNLS equation have nonlocal (m, r ) dependent high order symmetries. Under this ansatz a first order linear difference equation has only constant solution. We can state the following theorem: Theorem 1. The system (27) doesn't have local symmetries (29) for n > 4. The proof of this Theorem can be found in 15 and is obtained by looking for the existence of formal symmetries of order n > 4. This Theorem is a consequence of the fact that in this case we are not able to calculate the first four coefficients of L.

88

D. Levi

5. Conclusions We have shown here the great importance of being able to apply the reductive perturbation technique to DDE. Unfortunately the way proposed by Leon and Manna 9 doesn't produce a convincing result as the equation one obtain by reducing the Toda Lattice equation has not the necessary integrability properties. As the reductive perturbation technique should reduce integrable equation into integrable equations, a flaw in the derivation must exist. From this point of view a few points can be presented for future consideration: - the dispersion relation of the Toda Lattice is degenerate - the ansatz (16) gives undetermined results as soon as k > 3. This suggest that a new slow variation approximation ansatz has to be found. Work on this is in progress. References 1. A. Campa, A. Giansanti, A. Tenenbaum, D. Levi, and O. Ragnisco, Quasisolitons on a diatomic chain at room temperature,Phys. Rev. 48, 10168-10182 (1993). 2. A.S. Davydov,Biology and Quantum Mechanics (Pergamon, New York 1982). 3. A.C. Scott, Davydov's soliton, Phys. Rep. 217, 1-67 (1992). 4. R.S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 7,16231643 (1994). 5. T. Taniuti, Reductive perturbation method and far fields of wave equations, Prog. Theor. Phys. Suppl. 55 (1974) 1-35 6. F. Calogero and W. Eckhaus, Nonlinear evolution equations, rescaling, model PDEs and their integrability, Inverse Problems 3 (1987) 229-263 7. A. Degasperis, Multiscale expansion and integrability of dispersive wave equations, Lectures given at the Euro Summer School What is Integrability?, 13-24 August 2001, Isaac Newton Institute, Cambridge, U.K. 8. V.E. Zakharov and E.A. Kuznetsov, Multicale expansion in the theory of systems integrable by the inverse scattering transform, Physica 18D (1986) 455-463 9. J. Leon and M. Manna, Multiscale analysis of discrete nonlinear evolution equations, J. Phys. A: Math. Gen. 32 (1999) 2845-2869 10. R. Hernandez Heredero, D. Levi, MA. Rodriguez and P. Winternitz P, Lie algebra contractions and symmetries of the Toda hierarchy, J. Phys. A: Math. Gen. 33 5025-5040. 11. D. Levi, M.A. Rodriguez, Lie symmetries for integrable equations on the lattice, J. Phys. A: Math. Gen. 32, 8303 (1999)

Multi-scale Reduction for Differential Difference Equations and Integrability 89 12. M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations, J. Math. Phys.16 (1975) 598-603 13. Hernandez Heredero R H, Levi D, and Winternitz P, 1999, Symmetries of the Discrete Burgers Equation J. Phys. A: Math. Gen. 32 2685-2695. 14. A.V. Mikhailov, A. B. Shabat and R. I. Yamilov, The symmetry approach to the classification of nonlinear equations. Complete lists of integrable systems. Russian Math. Surveys 42 (1987) 1-63 15. D. Levi and R. Yamilov, Dilation symmetries and equations on the lattice, J. Phys. A:Math. Gen. 32 (1999) 83178323

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p - H A R M O N I C M A P S W I T H APPLICATIONS*

SHIHSHU WALTER WEI Department of Mathematics University of Oklahoma Norman, Oklahoma 73019 USA [email protected]

The purpose of this note is to give a general overview of the theory of pharmonic maps, and its applications.

1. Introduction The use of the p-harmonic maps has contributed to our understanding of mathematics and the universe. A p-harmonic map (u : M —> N between Riemannian manifolds) is a critical point of the p-energy functional {Ep(u) = \JM \du\p dx with respect to any compactly supported variations in a Sobolev space L\{M, N)). Examples of p-harmonic maps include harmonic maps (in which p = 2), geodesies, minimal varieties, conformal maps between n-manifolds such as Stereographic projections (in which p = n 4 2 ' 5 4 ), and p-harmonic morphisms. They appear in a broad spectrum of contexts for solving various problems. For instance, Bombieri-De Giorgi-Giusti5 construct a 1-harmonic function u : R2m -> R whose zero-level set u _ 1 (0) = { i £ R2m : x\ + • • • + a;2n = x m+i "I 1~ x2m} i s a n area-minimizing cone over the product of (m — 1)spheres {x € R2m : x2 + - • -+x2m = x2n+1 + - • -Vx\m = \) in R2m for m > 4. This cone provides the first counter-example to interior regularity for solutions to the co-dimension 1 Plateau problem. Constructing a smooth 1harmonic function in hyperbolic space, S.P. Wang and the author have settled Bernstein Conjecture in hyperbolic space Hn, Hn x Hn, Hn x SO(n, 1) and many other associated spaces by their counter-examples 64 . In particular, these constructions give the first set of examples of complete, smooth, ' S u p p o r t in part by OU Research Award and Presidential International Travel Fellowship. 91

92

Shihshu Walter

Wei

embedded, minimal (hyper-)surfaces in hyperbolic space in all dimensions. Since then many works on this topic were developed by M. Anderson (in his Berkeley thesis) 2 , R.M. Hardt and F.H. Lin 26 , and recently by others. In the case dimM = 1, it is well known that the study of geodesies leads to many beautiful discoveries in Riemannian Geometry (such as the Cartan-Hadamard, Bonnet-Meyer, Synge, Rauch Comparison Theorems), Classical Morse Theory (where it plays an important role in solving the higher dimensional Poincare conjecture53) and Morse Theory on infinite dimensional spaces such as Banach manifolds 43 ' 47 ' 52 . The purpose of this note is to give a general overview of the theory of p-harmonic maps, and its applications. Examples are also provided showing that the curvature hypothesis (3) on a compact manifold N is essential for the Sphere Theorem 7.4. In fact, in the absence of the curvature hypothesis one can have iTi(N) ^ 0, for all 1 < i < [p]. 2. Existence Theory We assume that Mn is a compact Riemannian manifold with possibly non-empty boundary, and Nk is isometrically immersed in Rq. LP(M,N) q denotes the set of maps u : M —> R whose component functions have first weak derivatives in Lp and u(x) G N a.e. on M. The p-energy for u G L\(M, N) is given by

Ep{u) = - f \du\p dx where du denotes the differential of it, dx is the volume element of M and 1 < p < oo. A map u G LP(M, N) is said to be p-harmonic if it is a weak solution to the Euler-Lagrange equation for Ep on L\{M,N). u is called p-stable (or p-minimizing) if u is a local (resp. global) minimum of the p-energy functional Ep on L\(M, N) having the same trace on dM. u is said to be p-unstable if u is not p-stable. In order to state the general existence theory of p-harmonic maps, it is necessary to define the following map which is an analog of a tangent cone (obtained by a blow-up technique) in the theory of minimal varieties: Definition 2.1: A map u : i? J ' +1 —> N is said to be a p-minimizing tangent map (p-MTM) if u is p-minimizing on every compact subset of R?+l and is a homogeneous extension of u : S*' —> N of degree-zero.

p-Harmonic

Maps with Applications

93

Generalizing the work of Eells-Sampson19, Schoen-Yau50, and Burstall 4 which treats the case p = 2, the author proves the existence of p-harmonic maps, representing components of the space C°(M,N) n L\(M,N) by p-harmonic maps: Theorem 2.1: (Wei55) Let M be a complete Riemannian n-manifold and N be a compact Riemannian manifold with a contractible universal cover N and assume that N has no non-trivial p-minimizing tangent map of Re for I < n. Then any continuous (or more generally L\—) map from M into N of finite p-energy can be deformed to a C 1 '" p-harmonic map minimizing p-energy in the homotopy class, where 1 < p < oo. Our method employed is the direct method in the Calculus of Variations 4 ' 62 and the regularity theory 25,66 . Furthermore, the work of Bishop-O'Neill, Eberlein, Burns and Sacks-Uhlenbeck7'18'8>48(cf. also Theorem 2.10 57 ), shows that there are various classes of manifolds TV which satisfy the above geometric measure theoretic condition on N. In particular, we have Corollary 2.1: Every continuous map on a complete Riemannian manifold M to a compact Riemannian manifold N of finite p-energy is homotopic to a C 1 , Q p-minimizer, provided The sectional curvature SecN of N is nonpositive, orthe universal cover N of N has no focal point, or or N is a surface such that its universal cover N has no conjugate point, or More generally, N supports a strictly convex function. This result in the case p = 2 and SecN < 0 is a pioneering theorem of Eells-Sampson19 where the heat flow method was used. In representing m-th homotopy classes as a sum of various m-harmonic maps (not necessarily as a single one), the author obtains Theorem 2.2: (Wei56) Let Nm be compact without boundary or compact with convex boundary, where m > 1. Then for every C 1 map <j> : Sm —> N, there exists C1,a m-harmonic maps Uj : Sm —> N,j = 1 , . . . , s, such that TV, j = 1 , . . . , s. The case m = 2 is a pioneering theorem of Sachs-Uhlenbeck48. On the other hand, the author also proves the following existence theorem under the assumption of vanishing irn(N) in which the case n = 2 and dM = 0, is due to Sacks-Uhlenbeck48, Schoen-Yau51, and Lemaire 37 . This result is also due to Jost 36 by a different approach.

94

Shihshu

Walter

Wei

Theorem 2.3: (Wei57) Every continuous L" map <j> from a compact Riemannian n-manifold M without boundary (resp. with boundary dM, where boundary trace <J)\QM is Lipschitz) into a compact Riemannian manifold JV with nn(N) = 0 can be deformed into an n-harmonic map whose n-energy is an absolute minimum in its homotopy class (resp. homotopy class relative to dM).

3. Uniqueness We find the strong resemblance of p-harmonic maps to geodesies in terms of uniqueness and stability properties. In particular, combining the existence of p-harmonic maps and the second variational formula 57 , the author obtains a Uniqueness Theorem for p-harmonic maps and its corollaries: Theorem 3.1: (Wei57) If UQ and u\ are homotopic p-harmonic maps from M into N with SecN < 0, then they are homotopic through p-harmonic maps us where 0 < s < 1, and the p-energy is constant on any arcwise connected set of p-harmonic maps, i.e. Ep(ug) = EP{UQ) = Ep(ui) for s £ (0,1). Furthermore, each path s — i > us(x) is a geodesic segment with length independent of x S M Corollary 3.1: Every homotopy class of a p-harmonic map from M to N which agree on an nonempty dM with SecN < 0 contains a unique p-harmonic map. Corollary 3.2: / / UQ : M —> N is a p-harmonic map with dM = 0 and SecN < 0. Assume that there is some point of UQ{M) at which SecN < 0. Then uo is unique in its homotopy class unless it is constant or maps M onto a closed geodesic a in N. In the latter case, we have uniqueness up to rotations of a. Corollary 3.3: Every homotopy class of a p-harmonic map from M into N of rank greater than one at some point of M with dM = 0 and SecN < 0 contains a unique p-harmonic map. Corollary 3.4: If SecN < 0 and two homotopic p-harmonic maps uo,ui : M —» N agree at one point, then UQ = u\. Corollary 3.5: (Bochner-Frankel49<1) Let M be a compact, orientable and SecN < 0, but not identically zero. Then its group of isometries is finite and no two elements are homotopic.

p-Harmonic

Maps with Applications

95

These generalize the work of Hartman, where the heat flow method was used 27 . 4. Regularity For p-minimizing map, Hardt-Lin and Luckhaus have shown the following two ground-breaking theorems in the regularity theory that are crucial for their developments. Theorem 4.1: (Hardt-Lin 25 , Theorem 4.5, p.573) Suppose £ is the largest integer such that any p-minimizing tangent map from the unit ball in W into N is a constant map for each j = 1 , . . . ,£. Then the interior singular set of any p-minimizer u £ L 1,p (fi,iV) is empty in case n < £ + 1, is a discrete set in case n = £ +1, and has Hausdorff dimension n — £ — 1 in case n > £ +1 (Where fi is a C 2 bounded open subset of Rn with the Euclidean metric). Theorem 4.2: (Hardt-Lin 25 , Luckhaus 38 ) For p > 1, the Hausdorff dimension of the singular set of a p-minimizing map u € L\{Mn, Nk) in the interior of M cannot exceed n — [p] — 1, in general. If n = [p] + 1, u has at most isolated singularities. If n < [p] + 1 (or if [p] + 1 < n, off the singular set) u is locally Holder continuous up to the boundary and the gradient of u is also locally Holder continuous in the interior of M. Applying Theorem 4.1, we refine the estimates in Theorem 4.2 for (i) Maps into manifolds with nonpositive sectional curvature or a domain of a strictly convex function, (ii) Maps into manifolds of positive Ricci curvature, (iii) Maps into manifolds with boundary. In particular, C. M. Yau and the author prove Theorem 4.3: (Wei-Yau 66 ) Every p-minimizing, L\ map into a manifold of non-positive sectional curvature, or into the domain of a strictly convex function, is C 1 , a . In particular, every p-minimizing L\ map into either a complete noncompact manifold with positive sectional curvature or an open hemisphere is Cl'a. The author has proved in Ref.55, the regularity theorem of a p- energy minimizing L\ map into a /c-dimensional ellipsoid E* or a closed upper half-ellipsoid (E%) + , where (Ek)+ = {(xl,...,xk+l)(ERk+1

:axl+xl

+ --- + xl+1 = 1 and xk+1 > 0} .

96

Shihshu

Walter

Wei

In particular, the author obtains the regularity of p-minimizers when TV is k

a closed upper-hemisphere S+ : Theorem 4.4: (Wei 55 ) Let u:Mn —> S+ be an L\ map which minimizes p-energy on each compact domain of M. Then u is locally Holder continuous up to the boundary and the gradient of u is also locally Holder continuous outside a closed interior singular set Sp of dim(Sp) < n — 3 — [p + 2y/p\. This result (4.4) is also due to Frank Duzaar 16 . The author obtains a regularity theorem for p- energy minimizing L\ maps into a large class of manifolds with positive Ricci curvature, known as p-superstrongly unstable manifolds 55 . Using Hardy Method (compensated compactness) in harmonic analysis, Libin Mou and Paul Yang have established the partial regularity of stationary p-harmonic maps into a sphere, thus generalizing results of L. C. Evans 17 for p = 2. Theorem 4.5: (Mou-Yang 40 ) Let M be a compact n-manifold. For 1 < p < n, a stationary p-harmonic map u £ L\(M, Sk) is C 1 , Q outside a closed singular set Sp with n — p dimensional Hausdorff measure zero. If p = n, then u is Cl>a. More recently, Min-Chun Hong and C.Y. Wang 3 3 have obtained some interesting regularity result for stable-stationary p- harmonic maps into p-superstrongly unstable manifolds (cf. 7.4) in terms of the Hausdorff dimension of the singular set. 5. Solutions to Dirichlet Problem J.

One Dimensional

Dirichlet

Problem:

The interrelationship between p-harmonic maps and geodesies, and between the least p-energy, and the shortest length (cf. 57 Theorem 1.10), and Hopf-Rinow Theorem lead to Theorem 5.1: (Wei 57 ) Given any two preassigned points x\ and X2 in a complete Riemannian manifold M (as boundary data, 0 H-* X\ and 1 H-> x^), there exists a smooth p-harmonic map w : [0,1] —> M with w(0) = xi and u)(l) = x-i which realizes the minimum p-energy -dp among all piecewise smooth maps v : [0,1] —> M with v(0) = x\ and v(l) = X2, in which w is a smooth length-minimizing geodesic and d is the distance between x\ and X2-

p-Harmonic

II.

Higher Dimensional

Dirichlet

Maps with Applications

97

Problem:

Extending Theorem 2.2 to manifolds with boundary, the author solves a higher dimensional Dirichlet Problem for p-harmonic maps, by using techniques in Refs.4, 62, 55, and minimizing p-energy in a class of maps with fixed trace. Theorem 5.2: (Wei 57 ) Let M be a compact Riemannian n-manifold with boundary dM and N be a compact Riemannian manifold with a contractible universal cover TV and assume that TV has no non-trivial pminimizing tangent map of Rl for I < n. Then any u £ Lip(dM, TV) D C° (M, TV) of finite p-energy can be deformed to a p-harmonic map MO G ^'"'{M - dM, TV) n Ca(M, TV) minimizing p-energy in the homotopy class with uo\dM = U\QM , where 1 < p < oo. In particular, every u G C1(M, TV) can be deformed to a C 1 '" p-harmonic map UQ in M — dM minimizing p-energy in the homotopy class with Holder continuous UQ\QM — U\QMIn particular, the author obtains the existence and uniqueness of the Dirichlet problem for p-harmonic maps into nonpositively curved manifolds, generalizing the work of Hamilton 24 : Theorem 5.3: (Wei 57 ) Let SecN < 0. If 0 e Ca(dM,N), then every component of the space of extensions of <j> to C° maps M —> N, with an element of finite p-energy contains a unique p-harmonic representative, which is C1,a in M—dM, Holder continuous up to dM and an 75p-minimum. III. Boundary Data with Image in a Nonconvex Nonempty Boundary

Domain

with

The author also obtains solutions to Dirichlet problem for p- harmonic equation with image in a fc-dimensional ellipsoid E% or closed upper halfellipsoid (JS£), . In particular, he obtains the Dirichlet solution when TV is a closed upper-hemisphere S+ : Theorem 5.4: (Wei 57 ) For any boundary data 77 G L^_x(dM, S+), where p

p < dimM, the associated Dirichlet problem has a C 1 , Q solution u if dim M < 2>+\p+2y/p\, which is a p-energy minimizing map. In general, u has possibly a singular set of Hausdorff dimension at most dim M — 3 — [p+2 y/p\. Regularity properties are described in Theorem 4.4.

98

IV.

Shihshu

Walter

Boundary

Wei

Data with Image in a Geodesically

Small

Disc

Using regularity theorem 4.3, the author has proved the existence of such p-harmonic maps with any preassigned Lipschitz function defined on dM with image in a small disc, generalizing the work of Hildebrandt-KaulWidman 30 in which p = 2. Theorem 5.5: (Wei 61 ) Let M be a compact manifold with smooth boundary dM. Let B^ (y0) be a geodesic ball of a compact manifold TV with center 2/o and radius r < 2JK where SecN < KN and B^(y0) is disjoint from the cut locus of 2/o- Then there exists a Cl B^(yo), minimizing p-energy in the class of maps u € L1,P(M, B^(yo)) with fixed boundary value UQ\QM, where p > 2 Remark 5.1: This theorem is sharp, and the estimate r < J as the nonsmooth equator map x i—> ( A , 0), with r = J .

is optimal

V. Boundary Data with Image in p-Superstrongly Manifolds: e.g. Theorem 7.8.(9), also see Theorem 7.4. in 57.

Unstable

6. n-Energy Identities Generalizing a pioneering theorem of Sachs-Uhlenbeck, the author proves that the n-th homotopy class of a given map from Sn to a compact N is represented as the sum of various n-harmonic maps of Sn to N, and not necessarily as a single one (cf. Theorem 2.4). In fact, the homotopic classes of these blow-up n-harmonic maps are related in an n-Energy Identity: Theorem 6.1: (Wei 57 ) Let N, 0 : Sn -> N, and Uj : Sn - • TV, j = 1,..., s be as in Theorem 2.4. Then the infimum of the n-energy in the homotopy class [] of

inf £„(«) = £

iirf £„(&)•

(1)

Being motivated by both two dimensional approximated harmonic maps and some efforts to understand Hungerbiihler's solutions 32 on the heat flow of m-harmonic maps near their singular points, Changyou Wang and the author consider a class of regular approximated m-harmonic maps which is denned as follows.

p-Harmonic Maps with Applications 99

Definition 6.1: A map u G W1'm(M, N)nC°(M, N) is said to be a regular approximated m-harmonic map if the following three properties hold: (1) (approximated m — harmonicity) the tension field of u, such that -div(\Du\m-2Du)

There is aft.€ L™=i(M,Rk),

= \Du\m-2A(u)(Du,Du)

called

+ h,

in the sense of distributions. (2) (smoothness effect) It satisfies the smoothness assumption: DdDul^Du)

G

L2(M,Rk),

(3) (an priori estimate) It enjoys an eo-priori estimate: there are eo > 0, ao G (0,1), and Co > 0 depending only on M, N ||/i|| -m^. such that if for any ball B2r(x) C M fB2r{x) \Du\m < e0 then u G Ca°(Br(x), N) and [u}cao(Br(x))

< Co-

We are ready to state the following: Theorem 6.2: (Wang-Wei 65 ) For m > 3. Let {un} C W1-m(M,N) be a sequence of approximated m-harmonic maps with their tension fields hn bounded in L»^(M) Assume that un converges weakly in W1,m(M, N) to l,m a map u G W (M, N) and hn converges weakly in L^^(M) to a map h G L"^ r T (M). Assume furthermore that un satisfy: (1) D^Dun^Dun)

G

L]oc(M,Rk),

(2) there exists an eo = eo(M,N) > 0 such that un satisfy the eocontinuity property given by definition 6.2. Then u is an approximated m-harmonic map with its tension field given by h, and there exist 0 < I < oo and nonconstant m-harmonic maps { W J } ' = 1 C Wl'm(Sm,N)!lC1(Sm,N) such that i

lim Em(un,M)

= Em(u,M)

+ Y\Em(uuSm).

(2)

i=l

Here Em(v, E) = fE \Dv\m denotes the m-energy on the set E of a map veW1'm(E,N). Remark 6.1: The case m = 2 is due to Ding-Tian

15

, and C.Y. Wang 6 3 .

100 Shihshu Walter Wei

As a consequence, we have Theorem 6.3: (Wang-Wei 65 ) For m > 3. Let {un} C W1'm(M,N) n l C (M,N) be a sequence of m-harmonic maps and converge weakly in W1'm(M,N) to a map u e W1'm{M,N). Then u e C71(M,iV) and is an m-harmonic map. Moreover, there exist 0 < I < oo and nonconstant inharmonic maps {cJi}\=1 C W1'm(Sm, TV) nC'fS™, TV) such that the energy identity ((2)) holds. Remark 6.2: The case m = 2 is due to Jost

35

, and Parker

44

.

7. Representing Homotopy Classes by p-Harmonic Maps Just as harmonic forms represent cohomology classes in Hodge Theory, stable currents represent homology classes in Geometric Measure Theory 20 , or stable geodesies represent classes in fundamental groups as in Cartan's theorem 9 , so do p-harmonic maps represent homotopy classes 57 . This is one of the features that distinguish p-harmonic maps from harmonic maps in which p = 2: Theorem 7.1: (Wei 57 ) If M is a compact Riemannian manifold, then for any positive integer k, each class in 7Tfe(M) can be represented by a C1,a p-harmonic map UQ from Sk into M minimizing p-energy in its homotopy class for any p > k. The case k = 1 is due to Cartan, and the case k = 2 is due to SachsUhlenbeck. This leads to Theorem 7.2: (A New Generalized Principle of Synge 57 ) Let TV be a compact Riemannian manifold. If there are no nonconstant C 1 , a p-stable maps from Sl into TV for some p > i, then 7T,(TV) = 0. If there are no nonconstant C 1 '" n-harmonic maps from Sn into TV for some integer n > 1, then TT„(TV) = 0. By using an extrinsic average variational method

58

, the author proves

Theorem 7.3: (Topological Vanishing Theorem 5T) Every compact psuperstrongly unstable manifold TV is [p]-connected, i.e. 7Ti(7V) = ••• = 7T b ] (TV)=0.

p-Harmonic

Maps with Applications

101

Recall Definition 7.1: (Wei-Yau66, Hong-Wang33) A Riemannian manifold TV is said to be p-superstrongly unstable (p-SSU) for p > 2 , if there exists an isometric immersion in Rq such that, for every unit tangent vector X to TV at every point y £ N, the following functional is negative vaiued. k 2

FPty{X) = (p - 2)\hx,x\

+ $^(2|/ix,aJ 2 -

hx,x'Kuai)

i=l

where h is the second fundamental form of TV in Rq with standard inner product v»w and Euclidean norm \v\ denned for v,w in Rq; and {c*i,..., ctfc} is an orthonormal frame on TV. Remark 7.1: Recently, M. Ara 3 has extended the above concepts of psuperstrongly unstable manifolds to F-superstrongly unstable manifolds by using the techniques in 57. Theorem 7.4: (Sphere Theorem 57 ) Let TV be a minimal fc-submanifold of a unit Euclidean sphere such that the Ricci curvature Ric of TV satisfies

fc Ric N >

fc where 2 1 (TV is not necessarily minimal), then TV is homeomorphic to a sphere. The curvature hypothesis is essential for the sphere theorem , and for •Kj(N) = 0, for all 1 < j < \p], and the Sphere Theorem is sharp : E x a m p l e 7.1: Let TVfc = S' (y[&) x S2p~j ( v / ^ ? ) C S2p+1 (1) for any 1 < j < P- Then TV is a minimal fc-submanifold of a unit Euclidean sphere such that for each tangent vector to its first factor S^ \\^~)i RicJV = ^ ( j - l ) < 2 p ( l - i ) = fc--I 3 V P where fc = 2p. But TTJ(N) ^ 0, TT2P-J(N) ^ 0 and TV is apparently not a sphere, for all 1 < j < p. In particular, if j = p, the Clifford embeddings TVfc = Sp ( ^ ) x Sp ( ^ ) C S2p+l (1) with Ric w = (fc - | ) but TTP(TV) ^ 0 for all p. There also exist totally geodesic embeddings Nk = Sp(l) in the unit sphere with Ric^ = (fc - | ) where p = k but np(N) ^ 0. Furthermore,

102

Shihshu Walter

Wei

the Ricci curvature hypothesis is vacuous if p is precisely k by Proposition 3.22 in 57. On the other hand Fengbo Hang and Fanghua Lin prove that the space C°°(M,N) is strongly dense in the Sobolev space L\(M,N) if and only if 7T[p](JV) = 0 and M satisfies ([p] — 1 ) - extension property with respect to N for p < n ( 28 - 29 ). Thus we have Theorem 7.5: (Wei 57 ) Let TV be as in Theorem or a compact kdimensional irreducible homogeneous space with the smallest positive eigenvalue Ai of the Laplacian A on functions. Then if (1) p ScalN p— 1 k is true, then N is p-SSU and the following assertions (2) through (9) hold: (2) Every p-stable map ip : M —> N from a compact manifold M is constant. (3) Every p-stable map ip : N —> K into a complete manifold K is constant. (4) The identity map Id^ is p-unstable. /r\

\

^

2ScalN

(6) ir1(N) = --- = *]p](N) = 0. (7) The homotopy class of any map to or from iV contains a map of arbitrarily small p-energy. (8) CX{M, N) is dense in L\(M, N), where dM is possibly nonempty. (9) The Dirichlet problem for p-harmonic maps into N has a solution. Furthermore, (1) => (2) => (4) <^> (5), and (1) => (3) => (4) <=> (5). In the case p = 2 (, as distinct from the case p > 2), - ^ j ScfN = 2^al_2 i.e. (1) is the same as (5). Hence (1) <=> (2) •£» (3) •*=> (4) and we recover Theorem 6.3 34 ' 41 . As a further application of The Sphere Theorem 7.4., we have a Classification Theorem of compact, irreducible, p-SSU symmetric spaces (cf. 57 6.6). In the case p = 2, the above theorem recaptures a classification theorem of Howard-Wei and Ohnita 41,34 . 8. A Bernstein-type Theorem Being partially motivated by the theory of p-harmonic maps, which include maps / — (/i, • • • , /fc+i) from manifolds M into fc-ellipsoids in Rk+1 satis-

p-Harmonic

Maps with Applications

103

fying div(|d/| p - 2 )V/i) + pi\df\pfi = 0 for some functions pt > 0 55>57, we are concerned with essential positive supersolutions satisfying Q(v) = div(A(x, v, Vv)Vu) + 6(:r, v, Vv)v < 0

(4)

on a Riemannian n-manifold M. In (1), A denotes a smooth cross section in the bundle whose fiber at each point x in M is a nonnegative linear transformation on the tangent space TX(M) into TX(M), b is a smooth

real-valued function, and Vv denotes the gradient of v. ( Thus in terms of normal coordinates, Q(v) =

YH,J=I

sf T (°t,i^r") + ^v

at

^o, where (aij) is

a nonnegative matrix. J The author proves a Bernstein-type Theorem 8.1: (Wei 59 )If M is a complete surface in R3 with its second fundamental form B such that M admits an essential positive supersolution of Q{u) = 0 with coefficients satisfying conditions (1) < AV<j), V(f> > < ci|V(/)|2 V(/> G C$°(M) and (2) b(x,u,Vu) > c2\B\2 in which cx < 3c 2 , then M is a plane. The case A(x,u, Vu) = Identity map, c\ = C2 = l,Q(u) = 0 has a positive (super)solution, and on M the mean curvature H = 0, is due to Fischer Colbrie - Schoen, Do Carmo - Peng, Pogorelov: Theorem 8.2: (Fischer-Colbrie and Schoen; Do Carmo and Peng; Pogorelov 21>14-43) Every complete stable minimal surface in R3 is a plane. The case A(x,u,Vu) = Id., c\ = c^ = l,Q(u) = 0 has a positive (super)solution, and H = constant, is due to Hoffman - Osserman - Schoen: Theorem 8.3: (Hoffman, Osserman and Schoen 31 ) Every complete surface M of constant mean curvature in R3 whose image under the Gauss map lies in some open hemisphere is a plane. The case A(x,u, Vu) = Id., c\ = c2 = l,Q(u) = 0 has a positive (super)solution, | ^ = ff^O, J|^ < 0, and H is of one sign is due to Cheng-Yau: Theorem 8.4: (Cheng and Yau 13 ) Let H be & function defined on R3 with | ^ - = f|| = 0 and §^ < 0. If H is of one sign and M is a graph ^3 = f{xiix2) defined over R2 whose mean curvature is given by H, then M is a plane.

104

Shihshu Walter

Wei

Theorem 8.1 extends from the linear case L(u) = ciAu + bu = 0 to the nonlinear case Q(u) = 0, and generalizes and unifies the above three beautiful theorems 8.2, 8.3, and 8.4( which can also be proved by using (p)harmonic maps in a different approach, cf. 59 3.1, 3.3, and 3.4 ). Furthermore, the hypothesis that the mean curvature H is of one sign in Theorem 8.4 can be dropped, and our result is optimal (cf. 59 2.12). 9. Isoperimetric Inequalities and Sphere Theorems Let M be an n—dimensional smooth, compact, connected Riemannian manifold. A recent elegant inequality of Morgan and Johnson 39 says that, if for example the sectional curvature K of M is less than KQ, then an enclosure of small volume V has at least perimeter P satisfying Pn>(l-CK0Vn)P\

(5)

where P* is the perimeter of the Euclidean ball of volume V. In 67, Meijun Zhu and the author take a global version of (5) and introduce, for any compact connected Riemannian manifold M, an isoperimetric constant T*(M) := inf{C(M) : Pn > nnunVn-1(l

- C(M)V$)

holds }.

(6)

The isoperimetric constant r* := T*(M) : depends deeply on the geometric properties of M, and is closely related to a sharp Sobolev inequality in 68. In turn, r* may even completely determine M: T h e o r e m 9.1: (Wei-Zhu 67 ) Let M be a complete simply-connected Riemannian manifold with Ric(M) > n — 1. Then (I). The isoperimetric constant r* satisfies T.

>

n(n-l)gj 2(n + 2)w^

(7)

For n = 2 or 3, the equality in ((7)) holds if and only if M is isometric to Sn with the standard metric.

(II). For n = 2 or 3, if the isoperimetric constant r* is close to "^"~ \ , then vol(M) is close to

vol(Sn).

2(n+2)o>J

T h e o r e m 9.2: (Wei 67 ) If the sectional curvature Sec(M) > 1 and T*(Mn) is close to n(-n-x\ , then vol(M) is close to vol{Sn). 2(n+2)w™

p-Harmonic Maps with Applications

105

For compact manifold M with positive Ricci curvature, we define the isoperimetric constant 7* of Gromov by 7* : = sup{7 : vol(dfl)

> jvol(dQo)

for

any

domain

f2 C M } ,

(8)

where fi0 C Sn is a spherical cap such t h a t ™|)g°; = J ° i ( M ) . We prove

Theorem 9.3: (Wei 67) Assume that Ric{M) > n - 1. 7* is close to 1 if and only if vol(M)

is close to

vol(Sn).

T h e isoperimetric constants provide a new perspective on manifolds with Ric > n — 1, and a new approach to the equivalence of vol(M) ~ vol(Sn). T h e other equivalent relations in terms of Gromov-Hausdorff distance, radius, and ( n + l ) - s t eigenvalue, are due to Coldings 1 1 ' 1 2 and Petersen 4 5 . Acknowledgments T h e author wishes to t h a n k the organizers of the Conference on Nonlinear Evolution Equations and Dynamical Systems at Huangshan, and the Dep a r t m e n t of Mathematics at University of Science and Technology of China for their kind invitation, support and hospitality. References 1. S.I. Al'ber, The topology of functional manifolds and the calculus of variations in the large, J. Russian Math. Surveys. 25, no 4, 51-117 (1970). 2. M.T. Anderson, The Bernstein problem in complete Riemannian manifolds, Berkeley Ph.D. Thesis, (1981). 3. M. Ara, Instability and nonexistence theorems of F-harmonic maps, Illinois J. Math. 45, no 2, 657-679 (2001). 4. F.E. Burstall, Harmonic maps of finite energy from non-compact manifolds, J. London Math. Soc, 30, 137-147 (1984). 5. E. Bombieri, E. DeGiorgi, and E. Giusti, Minimal Cones and Bernstein Problem, Invent. Math., 7 (1969), 243-268. 6. Bernstein, F., Uber die isoperimetrische Eigenschaft des Kreises auf der Kugeloberflache und in der Ebene, Math. Ann. 60, 117-136 (1905). 7. R.L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145, 1-49 (1969). 8. K. Burns, Convex supporting domains on surfaces, Bull. London Math. Soc, 17, 271-274 (1985). 9. E. Cartan, Lecons sur la Geometrie des Espaces de Riemann, (2nd edition) Gauthier-Villars, Paris (1951). 10. Cheng, S.Y., Eigenvalue comparison theorems and its geometric applications, Math. Z. 143, no. 3, 289-297 (1975).

106

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11. T.H. Colding, Shape of manifolds with positive Ricci curvature, Invent. Math. 124, 175-191 (1996). 12. T.H. Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124, 477-501 (1997). 13. S.Y. Cheng and S.T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28, 333-354 (1975). 14. M. do Carmo and C.K. Peng, Stable complete minimal surfaces in R3 are planes, Bull. Amer. Math. Soc. 1, 903-906 (1979). 15. W. Y. Ding, G. Tian, Energy identity for a class of approximate harmonic maps from surfaces. Comm. Anal. Geom. 3, no. 3-4, 543-554 (1995). 16. F. Duzzar, Regularity of p-harmonic maps into a closed hemisphere, Boll. Un. Mat. Ital. B(7) 5, 157-170 (1991). 17. L. C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rat. Mech. Anal, 7, 827-850 (1983). 18. P. Eberlein, When is geodesic flow of Anasov type II, J. Differential Geom., 8, 565-577 (1973). 19. J. Eells and J.H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86, 109-160 (1964). 20. H. Federer and W. Fleming, Normal and integral currents, Ann. of Math., 72, 458-520 (1960). 21. D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33, 199-211 (1980). 22. Gromov, M., Paul Levy's isoperimetric inequality, preprint IHES/M/80/320. 23. Gromov, M., Structures metriques pour les varietes riemannienes, edited by J. Lafontaine and P.Pansu, CEDIC, Paris, 1981. 24. R.S. Hamilton, Harmonic maps of manifolds with boundary, Lecture Notes In Math, no. 471 (1975), Springer, Berlin. 25. R. Hardt and F.H. Lin, Mapping minimizing the Lp norm of the gradient, XL, Common. Pure and Applied Math., 555-588 (1987). 26. R. Hardt and F.H. Lin, Regularity at infinity for area-minimizing hypersurfaces in hyperbolic space, 88 no.l, Invent. Math., 217-224 (1987). 27. P. Hartman, On homotopy harmonic maps, Canad. J. Math., 19 673-687 (1967). 28. F.B. Hang and F.H. Lin, Topology of Sobolev mappings, Math. Res. Lett. 8, 321-330 (2001). 29. Topology of Sobolev mappings II, preprint. 30. S. Hildebrandt, H. Kaul, and K - 0 Widman, An existence theorem for harmonic mappings of Riemannian manifolds, Acta Math., 138 1-16 (1977). 31. D.A. Hoffman, R. Ossorman and R. Schoen, On the Gauss map of complete surfaces of constant mean curvature in R and Br, Comment. Math. Helv. 57, 519-531 (1982). 32. N. Hungerbhler, m-harmonic flow, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4) 24 (1997), no. 4, 593-631 (1998). 33. M.C. Hong, C.Y. Wang, On the singular set of stable-stationary harmonic

p-Harmonic Maps with Applications 107 maps, Cal. Var. Partial Differential Equations 9, no 2, 141-156 (1999). 34. R. Howard and S.W. Wei, Non-existence of stable harmonic maps to and from certain homogeneous spaces and submanifolds of Euclidean space, Trans. A.M.S., 294 no. 1 319-331 (1986). 35. J. Jost, Two-dimensional geometric variational problems, New York, Wiley, 1991. 36. Jiirgen Jost, A conformally invariant variational problem for mappings between Riemannian manfifolds, preprint of Centre for Math. Analysis, Australian National University (1984). 37. L.Lemaire, Applications harmoniques de surfaces riemanniennes, J. Diff. Geom., 13 51-78 (1978). 38. S. Luckhaus, Partial Holder continuity for minima of certain energies among maps into a Riemannian manifold, Indiana Univ. Math. J., 37 349-367 (1988). 39. Johnson, D. and Morgan, F., Some sharp isoperimetric theorem for Riemannian manifolds, Indiana Univ. Math. J., 49, no. 3, 1017-1041 (2000). 40. L. Mou and P. Yang, Regularity for n-harmonic maps, J. Geom. Anal. 6, no. 1 91-112 (1996). 41. Y. Ohnita, Stability of harmonic maps and standard minimal immersion, Tohoku Math. J. 38 259-267 (1986). 42. Y. L. Ou and S. W. Wei, Classification and construction of p-harmonic morphisms, preprint. 43. R. S. Palais, Critical point theory and the maximum principle, Proc. Sympos. Pure Math., 15, 185-212 (1970). 44. T. Parker, Bubble tree convergence for harmonic maps, J. Diff. Geom. 44, no. 3, 595-633 (1996). 45. Petersen, P., On eigenvalue pinching in positive Ricci curvature, Invent. Math, 138, 1-21 (1999). 46. A. V. Pogorelov, On the stability of minimal surfaces, Dokl. Akad. Nauk SSSR 260, 293-295 (1981, Zbl. 495.53005), English transl.: Soviet Math. Dokl. 24, (1981), 274-276. 47. P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, CIME, Verona, Ediz, Cremonese, 141-195 (1974). 48. J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. 113 1-24 (1981). 49. J. H. Sampson, Some properties and applications of harmonic mappings, Ann. Ec. Norm. Sup., XI (1978). 50. R. Schoen and S.T. Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds of nonnegative Ricci curvature, Comm. Math. Helv., 51, 333-341 (1976). 51. R. Schoen and S.T. Yau, Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature, Ann. of Math., 110 127-142 (1979). 52. J.Schwartz, Nonlinear functional analysis, Gordon and Breach, New York, (1969). 53. S. Smale, Generalized Poincare's Conjecture in dimension greater than four, Ann. of Math., 74, 391-406 (1961).

108 Shihshu Walter Wei 54. H. Takeuchi, Some conformal properties of p-harmonic maps and a regularity for sphere-valued p-harmonic maps, J. Math. Soc. Japan, 46 no. 2, 217-234 (1994). 55. S. W. Wei, The minima of the p-energy functional, Elliptic and Parabolic Methods in Geometry, A. K. Peters 171-203 (1996). 56. S. W. Wei, On p-harmonic maps and their applications to geometry, topology and analysis, Tamkang J. of Math. 28, no. 2, 145-167 (1997). 57. S. W. Wei, Representing homotopy groups and spaces of maps by p-harmonic maps, Indiana Univ. Math. J. 47, no. 2, 625-670 (1998). 58. S. W. Wei, An extrinsic average variational method, Contemp. math. 101, 55-78 (1989). 59. S. W. Wei, Nonlinear partial differential systems on Riemannian manifolds with their geometric applications, Journal of Geometric Analysis 12, no 1, 147-182 (2002). 60. The Structure of Complete Minimal Submanifolds in Complete Manifolds of Nonpositive Curvature (to appear in Houston Journal of Math). 61. p-harmonic maps with image in a geodesically small disc. 62. B. White, Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Math., 160, 1-17 (1988). 63. C. Y. Wang, Bubbling Phenomena of Certain Palais-Smale Sequences from Surfaces to General Targets, Houston J. of Math. 22, no. 3, 559-590 (1996). 64. S.P. Wang, and S.W. Wei, Bernstein conjecture in hyperbolic geometry, Seminar on Minimal Submanifolds, edited by E. Bombieri, Annals of Math Studies no. 103, 339-358 (1983). [W] C. Y. Wang, Bubbling Phenomena of Certain Palais-Smale Sequences from Surfaces to General Targets. Houston J. of Math. 22, no. 3, 559-590 (1996). 65. C.Y. Wang and S.W. Wei, Energy identity for a class of approximated inharmonic maps, Journal of Differential and Integral Equations Vol 15, no 12, 1519-1532 (2002). 66. S.W. Wei and C M . Yau, Regularity of p-energy minimizing maps and psuperstrongly unstable indices, J. Geom. Analysis 4, no. 2 247-272 (1994). 67. S. W. Wei, and M. J. Zhu, Sharp isoperimetric inequalities and sphere theorems, preprint. 68. Zhu, M., Sharp Sobolev and isoperimetric inequalities with mixed boundary conditions, preprint.

T H E INFINITELY M A N Y CONSERVATION LAWS OF T H E LAX I N T E G R A B L E D I F F E R E N T I A L - D I F F E R E N C E SYSTEMS*

DA-JUN ZHANG* AND DENG-YUAN CHEN Department of Mathematics, Shanghai University, Shanghai 200436, People's Republic of China * djzhang @mail. shu. edu. en

A simple and systematic approach of finding conservation laws for the Lax integrable (l+l)-dimensional differential-difference systems is described. The conservation laws of the systems associated with two kinds of (l+l)-dimensional spectral problems are considered. By introducing the generalized Riccati equation related to pseudo-difference operator, we generalize this procedure to the high dimensional differential-difference systems. The conservation laws of the two-dimensional Toda lattice hierarchy are discussed.

1. Introduction Conservation laws (CLs) are ubiquitous in applied mathematics. In some case, they express conservation of physical quantities. Even when they do not, they are usually of mathematical interest 1 . The existence of an infinite number of CLs can also act as one of important algebraic characters for the integrable systems 2 . Since MGK's discovery3 of an infinite number of CLs for the KdV equation, many methods have been developed to find them. The purpose of this paper is to consider the CLs for discrete soliton systems. In general, for a (l+l)-dimensional integrable differential-difference equation H(u(t,n)) = 0, there exists the semi-discrete CL, namely ^-tF(u(t,n))

= (E-l)J(u(t,n)),

(1)

where t is a continuous variable and n is discrete, E is a shift operator, F(u(t,n)) is the conserved density and J(u(t,n)) is associated flux. We can list here some known successful methods for finding the infinitely many conserved quantities or CLs in form of (1). For example, using Backlund *This project is supported by the National Natural Science Foundation of China. 109

110 Da-Jun Zhang & Deng-Yuan Chen

transformation 4 , using the formal solutions of eigenfunctions5, using a trace identity for multi-component systems 6 ' 7 , using asymptotic expansion of scattering data 8 , etc. All these methods are based on their continuous versions 9 " 12 . Recently, the authors proposed a simple approach to constructing the CLs for the (l+l)-dimensional Lax integrable differentialdifference systems 13,14 . The essential idea of this approach is a simple equality:

l n ^ y ^ = (£-l)M(n),

(2)

by which the CL can easily be obtained from the concerned Lax pair. Further with the help of the Riccati equation related to the eigenvalue problem and some simple formulas such as oo

1

oo

1

ina-s) = -£-
K

l

~

= £y, q

( 3)

fc=o

the infinitely many CLs are consequently derived. This simple method was also described by Zhu et al 1 5 ' 1 6 ' 1 7 . For the high dimensional systems, the CLs can be obtained through the Sato's theory 18 ' 19,20 which is regarded as a powerful tool to study high dimensional systems and the related properties such as Lax pairs, symmetries, CLs and constraints 21,22,23 . In the Refs.23, 24, the two-dimensional Toda lattice (2DTL) hierarchy has been discussed starting from two pseudodifference operators L = E + Uhn + U2,nE~1 + U3,nE"2 + • • • , M = VQ^E-1

+ wi,n + v2,nE + v3,nE2 + ••• ,

(4) (5)

where Ui n and i>i>n are the functions of the variables n, xi, x2, • • •, yi, yi, • • • (only n is discrete), i.e. Uin = Vi(n) =Vi(n, xi, x2, • • • , yi,y2, • ••). Sato's approach provided two kinds of CLs for the 2DTL hierarchy, the semi-discrete form and continuous form (between two continuous variables). However, this approach only allows us to obtain the infinitely many conserved densities but no explicit associated fluxes.23 In this paper, we first redescribe systematically the method in Refs. 13,14 . The CLs related to two most popular (l+l)-dimensional discrete spectral problems are considered. The one is the Ablowitz-Ladik form8, i.e. /Mn(A,un)Mi2(A,u„)\

E

,

fhn\

^ = Ul(A,Un) M22(X,Un) ) *»' +» = U n )

,Rx

^

Conservation

Laws of Lax Integrable Differential-Difference

Systems

111

where un = u(n,t) denotes the potential field (ui(n,t),• • • ,ui(n,t))T and Mij are all the Laurent polynomials of A; the other is the Blaszak-Marciniak form 34 , i.e. m

tfy>„ = Xip„, * = J2uj(n)Ej,

(s<m£

Z).

(7)

j=s

As the detailed examples, we discuss the CLs of the Wu-Geng lattice hierarchy and the Volterra lattice hierarchy. We also list similar results for some new lattices obtained recently. Second, we generalize the above procedure to the high dimensional differential-difference systems. By introducing the generalized Riccati equation related to the pseudo-difference operator, we obtain the infinitely many CLs of the 2DTL hierarchy. This approach not only presents more forms of the CLs than the Sato's approach but also allows us to obtain the infinitely many explicit associated fluxes. The paper is also organized along the above line. Sec.2 considers the CLs for the (l+l)-dimensional differential-difference systems. Sec.3 considers the CLs for the high dimensional systems. 2. The CLs for the (l-|-l)-Dimensional DifferentialDifference Systems 2.1. For the System

of the Ablowitz-Ladik

Form

For convenience, we write f(n) = fn and Ekf(n) Z). Suppose that the linear problem (6) and A

_ fNu{X,un)

= f(n + k) = fn+k,

(k €

Ni2(X,un)\ (8)

^-{N^X^N^X,^))^

are the Lax pair of the system which we considered, where Nij are also the Laurent polynomials of A. It follows from the above Lax pair that ^

+ 1

01,n

- M

1 1 +

% ± i = M

2 1

M > , 01 ,n

(9)

^ + M 22 ,

(10)

Qni,n)t = N11 + N12p±,
(11)

112

Da-Jun Zhang & Deng-Yuan

Chen

( l n < A 2 , n ) t = i V 2 l,n 1 ^ + W:22'

(12)

»2,n

Then, By virtue of the formula (2), the CL turns out from Eq.(9) and Eq.(ll): {ln(Mu+Ml20n)}t

= (E-l){N11+N126n),

6n = i,n.

(13)

Similarly, another form of the CL can be derived from Eq.(lO) and Eq.(12): [ln(M2iw„ + Mi2)]t = (^-l)(A r 2iu;„ + iVi2),

w„ = ^i,n/
(14)

Next, the quotient of (9) and (10) yields the Riccati equation M210" 1 + M 2 2 Mil + Mi 2 0„ ' i.e.

M n 0 n + i + M129

(15)

= M2I + M220

Also, for ujn we have (16)

M n w n + Mi 2 = M 2 iu; n w„ + i + M22Wn+i-

Generally, expanding 8n or wn in powers of A can solve these Riccati equations. Then, by equating the same powers of A, two sets of infinitely many CLs can be derived from (13) and (14) respectively. All the isospectral equations related to the Lax pair (6) and (8) possess the same conserved densities but different associated fluxes. We note that these two sets of CLs are equivalent because of 6n • ujn = 1. In what follows, we consider a concrete example. In the Ref.35 Wu and Geng have derived a new hierarchy from the linear problems E(j>n

1 -I- Xunv u Xvn n 1n

<j>n,

nt =

JV11

N.12

N2i

-Nn

4>n

(17)

The first equation in the hierarchy is Unt

1 Vn+1

1 V„

1 V„t

Un

(18) ^n—1

with the related t-matrix

N0

( * 1 \ I v"x Wn-l

2/

(19)

Conservation Laws of Lax Integrable Differential-Difference Systems 113

According to the above brief sketch for (6), we have the Riccati equation XunVnU>n = -Un + tJn+i

- U)n + \v„WnVn+l,

(20)

where wn = (f>i2,n- It can be solved by the expansion oo

3=1

with the following recursion formula

=i

For the evolution equation (18), the CL (14) is cast into [ln(l + Xvnun)]t = (E - 1)(

w„).

(22)

«n-l

By rewriting oo

ln(l + Xvnujn) = - In A - \nunvn

+ ln(l + u n t i „ ) ] ^ + 2 ) A " j ) OO

-

OO

- - In A - ln W „r, n + ^ ( - l ) * " 1 - ( « „ < ; „ ^ u # + 2 > A ^ ) f c , fc=l

3= 1

it is easy to work out the first three common conserved densities: pX' — - lnunvn,

PV

pi1' = unvn 2

(— H ), vn vn+i

= — [ - ( - + — ) + — V i - h— ( - + ~ ) ] 2 , unvn

un vn

vn+i

un+iv^+1

2 unvn

vn

vn+i (23)

and the associated fluxes 40) =

— , J(nl) = Un-iVn

2.2. f o r t/ie System

i — j , 42) = Un-iUnv£

L

T ^ ( - +—

Un-iU^y*

of the Blaszak-Marciniak

Vn

)• (24)

Vn+i

Form

The simplest case of (7) is V>„+1 = -V/'n -Pntpn-1,

(25)

which is the spectral problem of the Volterra lattice hierarchy (lnp n ) t - TkK0,

(26)

114

Da-Jun Zhang & Deng-Yuan Chen

where T=(l

+ E)(PnE-E-1Pn)(E-l)-1,

K0 = -(E-E-1)pn.

(27)

We note here that employing a transformation ipn+j = n, we can consider the equivalent form of (25): 4>n+l = n ~ "rPn^n-1,

(28)

which is much easier than the previous one in some case. Now let the time evolution be
+ Bn(\,pn)<j>n-i,

(29)

then, with the help of the formula (2), we can write out the CL of Eq.(26): Qn6n+1)t

= (E- \){An + Bn6~l),

(30)

oo

where 6n = 0„/„_i = 5 j 0 „ A~^ can be derived from the Riccati equaj=i tion #n+l#n = On — TP™'

(31)

which is related to (28). Of course, it is easy to obtain the infinitely many CLs by using the formulas (3). In the following, we list the CLs for some new lattices obtained recently by Hu and his co-workers. The lattice 36 Un.t = Un(«>n-1 ~ Wn+l), (Wn+i +Wn + Wn-i)t + (wn+i

+ j(Un+l ~ Un-l)

+Wn + Wn-i)(wn-l

(32)

- Wn+i) = 0,

with the Lax pair 1pn+3 = -Pni>n+1 + A^„ ~ " J ^ n - l ,

l(>n,t = -Qn'tpn ~ 1>n+2,

(33)

(34)

where pn = Wn+2 + ™n+l + » n ,

9n = Wn+l + Wn,

possesses the CL 14 (lm? n ) t = ( £ 7 - l ) ( - g „ - i ? „ t ? n + i ) ,

(35)

Conservation Laws of Lax Integrable Differential-Difference Systems 115

where the Riccati equation is 1 Al?„ = -Un+l+Pn+l1?n#n+l+$n#n+l#n+2#n+3i

lb

°° 1?n'*

$n = - T

= 2_/ T2j^

The lattice 37 : (un+1-run)ut+2(un+1-un)tt(un+1-un)t

=

eu"+2~2Un+1+Un-eu"+1-2Un+Un-1 (37)

with Lax pair n + Mn + ^n^n^n-l - ?"n-l?"„«-2,

(38)

n,t = - r n 4 > n - l ,

(39)

where p„ = 1in+l,tt+Wn,tt + (u n + l, t -U„,t) , gn =Un-l,t-«n+l,t. ^n = e "+

\

possesses the CL 14 Qn0n)t = (E-l)rn0n-i,

(40)

where the Riccati equation is Atf„ = l + P n l J n - Q n r n ^ n - i ^ n + r n - i r n ^ n - a ^ n - l ^ n ,

$n = X ^

=

z J ~TT'

(41) The lattice 38 ' 39 (Un+1 +Un + U„-l)« - 3un(un+i

+ Un-l)t + 3un+lU„+l,t + 3un_iUn_i,f

- 7(«n+l - 2lt„ + M„_l) + (Un+i - 2un + Un-i)[(un+l ~ Un-{f ~ (Un+1 - Un)(un - U„_l)] = 0

(42) with Lax pair <£n-3 = Vn4>n-2 + Qn^n-1 + A0 n ,

n,t = -rn4>n

~ n-l,

(43)

(44)

possesses the CL 14 (ln0 n ) t = ( £ ; - l ) ( r n + i? n _i),

(45)

116

Da-Jun Zhang & Deng-Yuan

Chen

where the Riccati equation is qn+1tin

= -A -p„+ii?n-iO„ + 0n_20„-i0„,

t?„ = -p- = y > # > A ' .
The CDGKS lattice 40 (lnkn)t

= (E - E-^gn,

(47)

where £ / „ = fn+a, kn = 1 + a2un, 5"- —

63 i n 5*> ^n 10a "

x 81 2 20a 5rk"nl [a ^ "£

1 * EJ ( n'Vn)9 -\i2 * ,

j=-i

^s") + h

J=-I

2

2

+ i s »t a2 E (^«) -1 E (^s«) +\\ j=-2

and s„ = kn+aknkn-a

j=-1

with the Lax pair

"4>n = Afc„Vn+2a ~ 9 A ^ „ + a + kn1pn-a,

(48)

V>nt = V 2 P n V , n - 4 o - T ? n ' 0 n - 2 a + T ' n V ' n + 9 A g n l / ' n + 2 a - 8 1 A 2 p „ + 2 a V ' n + 4 a ,

(49)

where P n — ~„ g"-n*n—2a>

Q

?~n — " ( P n + 2 a

1

1

1

9n = 2Q-5fc„/c„- a [a 2 E t 5 ' " " ) " 9 E a

Pnjj

j=-2

o

( ^ J s " ) - gl>

j=-2

possesses the CL (hx9n+a)t

= (E-

- ll^PnOn^-aKUjn-ia )(^2Pn0n K-Jn-2^

~ ^ n ^ n - a

+

rn

+ 9Aqn^n+2a^n+o ~ 81A Pn+2o^n+4a^n+3a^n+2a^n+o)i (50) where the Riccati equation is i

0 n = \kn6n+2a8n+aQn

~ 9A#„+O0n +

fcn,

0n =

OO

= J^ Vn-a

6

3

X3 .

(51)

j=Q

With the help of the formula (3), we can write out the explicit forms of these CLs, i.e., the infinitely many CLs.

Conservation

Laws of Lax Integrable Differential-Difference

Systems

117

3. The CLs for the High Dimensional DifferentialDifference Systems In this section, we generalize the above method to the high dimensional differential-difference systems obtained through the Sato's approach. 41 Consider the linear problems 23 L(j>n = \<j>n,

(52)

4>nxm = Bm(f>n,

(53)

nvm = Cm
(54)

and Mtpn = ^Ipn,

(55)

1pnxm = Bmlpn,

(56)

VWn = CmV'n,

(57)

where L and M are the pseudo-difference operators (4) and (5), Bm(n) — (L m )+, Cm(n) = (Mm)- where ( ) + means that part of the shift operator containing only non-negative powers and ( )_ means that part containing only negative powers. The related Lax equations LXm = [BmtL],

Lym = [Cm,L],

MXm = [Bm,M],

MVm = [Cm,M] (58)

give an infinite number of nonlinear differential-difference equations for Uj)fl and Vitn, which are called the 2DTL hierarchy, 23 for the 2DTL equation 23 d 2 ln^o,n _ ——5— = 2v0,n - v0,n+i - v 0 ,„-i

,-n. (59)

can be derived from the lowest equations in this hierarchy. Differentiating Eq.(2) directly with respect to xm and noticing (53) yield -Qn6n)Xm = ( E - l ) ^ ,

(60)

where 6n = n/
(61)

118

Da-Jun Zhang & Deng-Yuan Chen

(lnWn+1)Im = (JE7-l)%^L,

(62)

(ln W n + 1 ) U m = ( E - l ) % ^ ,

(63)

where un = ipn/ipn-iTo obtain the explicit conserved densities and associated fluxes, we introduce the generalized Riccati equations associated with the pseudodifference operators L and M. It is easy to find from L, M and their eigenvalue problems (52) and (55) that X6n = 1 + Wl,n$n + U2,n9n-l6n

+ W3, n 0 n _2#n-10n + " " " ,

1 -0Jn

(64)

, = V0,n + Ul,nW„ + »2,nW n W n + i -\

,

x

(65)

which we call the generalized Riccati equations. These two equations can be solved by setting oo

,

)

i

0n = ^0^ X- ,

oo

w„ = f ! 2 - = 5>£>A',

(66)

where ^ W^1} = Uo.n,

= 1, ^ 2 ) = « ! , „ , ^ 2 ) = « ? , „ + «2,n, • • • ; W^2) = Vo,nVltn,

W^3) = Vo.nfal.n + U 2 , n U0,n+l),

Then, it is not difficult to obtain the explicit forms of the CLs (60)-(63). 5 xv. i- u * e d ,dlnn <91n<£n T In the light of - — ( — ) = — (—-r ), it is easy to find the OXg

OXfji

OXJJI

OXs

following continuous CL of the 2DTL hierarchy: £>m(pn

£>s(pn

( — 7 — ) * . = (—7—)*™.


.

icn\

(*#"*)•

.

(67)


Similarly, we have

- 7 — )v. = (-j— »™. (s?m),
(68)


tBm(j)n

(—7

,Ca<j>n^ )». = (—7 )x m ,

(—7—)*. = ( - 7 — )xm,

(s^m),

. . (69) (70)

Conservation Laws of Lax Integrable Differential-Difference Systems 119

{r )

rr *-= {~^r)v-' /&mWn\

,^sWn\

(s#m)i

(71) ('70^

Obviously, the two solvable generalized Riccati equations allow us to write out the explicit forms of these above six continuous CLs. 4.

Conclusions

We have described a simple way of finding the CLs for the differentialdifference systems. This method starts from the Lax pair directly and makes the use of the Riccati equation. It is also generalized to the high dimensional systems related to pseudo-difference operator. Acknowledgments One of the authors (D.J.Zhang) would like to express his sincere thanks to Professor Y.S.Li for his enthusiastic guidance and encouragement. The authors are also grateful to Professor D.Levi for his discussions. References 1. P. E. Hydon, J. Phys. A: Math. Gen. 34, 10347-10355, (2001). 2. A. S. Fokas, Stud. Appl. Math., 77, 253-299, (1987). 3. R. M. Miura, C. S. Gardner and M. D. Kruskal, KdV equation and generalizations, II. Existence of conservation laws and constants of motion, J. Math. Phys., 9, 1204-1209, (1968). 4. M. Wadati, Prog. Theo. Phys. Supp., 59, 36-63, (1976). 5. M. Wadati and M. Watanabe, Prog. Theo. Phys., 57, 808-811, (1977). 6. T. Tsuchida, H. Ujino and M. Wadati, J. Math. Phys., 39, 4785-4813, (1998). 7. T. Tsuchida, H. Ujino and M.Wadati, J. Phys. A: Math, and Gen., 32, 22392262, (1999). 8. M. J. Ablowitz and J. F. Ladik, J. Math. Phys., 17, 1011-1018, (1976). 9. M. Wadati, H. Sanuki and K. Konno, Prog. Theo. Phys., 53, 419-436, (1975). 10. K. Konno, H. Sanuki and Y. H. Ichikawa, Prog. Theo. Phys., 52, 886-889, (1974). 11. T. Tsuchida and M. Wadati, J. Phys. Soc. Japan , 67, 1175-1187, (1998). 12. V. Zakharov and A. Shabat, Sov. Phys. JETP, 34, 62-69 (1972). 13. D. J. Zhang and D. Y. Chen, Chaos, Solitons & Fractals, 14, 573-579 (2002). 14. D. J. Zhang and D. Y. Chen, The conservation laws of some differentialdifference systems, preprint, Shanghai University, 2001.

120 Da-Jun Zhang & Deng-Yuan Chen 15. Z. N. Zhu, X. N. Wu, W. M. Xue and Q. Ding, Phys. Lett. A, 297, 387-395, (2002). 16. Z. N. Zhu, X. N. Wu, W. M. Xue, and Z. M. Zhu, Phys. Lett. A, 296, 280-288, (2002). 17. Z. N. Zhu, W. M. Xue, X. N. Wu and Z. M. Zhu, J. Phys. A: Math. Gen. 35, 5079-5091, (2002). 18. M. Sato, RIMS Kokyuroku Kyoto Univ. 439, 30-46, (1981). 19. M. Sato and Y. Sato, in Nonlinear partial differential equations in applied science, edited by H. Pujita, P. D. Lax and G. Strang, (Kinokuniya/NorthHolland, Tokyo), pp.259-271, (1983). 20. Y. Ohta, J. Satsuma, D. Takahashi and T. Tokihiro, Prog. Theor. Phys. Suppl., 94, 210-241, (1988). 21. K. Kajiwara, J. Matsukidaira and J. Satsuma, Phys. Lett. A, 146, 115-118, (1990). 22. J. Matsukidaira, J. Satsuma and W. Strampp, J. Math. Phys., 31, 1426-1434, (1990). 23. K. Kajiwara and J. Satsuma, J. Math. Phys., 32, 506-514, (1991). 24. B. Konopelchenko and W. Strampp, J. Math. Phys., 33, 3676-3686, (1992). 25. J. Sidorenko and W. Strampp, J. Math. Phys., 34, 1429-1446,(1993). 26. Y. Cheng, J. Math. Phys., 33, 3774-3782, (1992). 27. Y. Cheng and Y. S. Li, Phys. Lett. A , 157, 22-26, (1991). 28. Y. Cheng and Y. S. Li, J. Phys. A: Math. Gen., 25, 419-431, (1992). 29. Y. S. Li, J. Univ. Sci. Tech. China, 23, 1-7, (1993). 30. D. Y. Chen, J. Math. Phys., 43 (2002) 1956-1965. 31. S. Kanaga Vel and K. M. Tamizhmani, Chaos, Soliton and Fractals, 8, 917931, (1997). 32. K. M. Tamizhmani and S. Kanaga Vel, Chaos, Soliton and Fractals, 137-143, (2000). 33. K. M. Tamizhmani, S. Kanaga Vel, B. Grammaticos and A. Ramani, Chaos, Soliton and Fractals, 11, 1423-1431, (2000). 34. M. Blaszak and K. Marciniak, J. Math. Phys., 35 4661-4682, (1994). 35. Y. T. Wu and X. G. Geng, J. Phys. A: Math. Gen., 31 L677-L684, (1998). 36. X. B. Hu and H. W. Tam, Inverse Problems, 17, 319-327, (2001). 37. Y. T. Wu and X. B. Hu, J. Phys. A: Math, and Gen., 32, 1515-1521, (1999). 38. X. B.Hu, Y. T. Wu, Phys. Lett. A, 246, 523-529, (1998). 39. X. B.Hu, D. L. Wang and X. M. Qian, J. Phys. A: Math, and Gen., 32, 7901-7906, (1999). 40. X. B. Hu, Z. N. Zhu and D. L. Wang, J. Phys. Soc. Jpn., 69, 1042-1049 (2000). 41. D. J. Zhang, Conservation laws of the two-dimensional Toda lattice hierarchy, preprint, Shanghai University, 2002.

QUIVERS A N D HOPF ALGEBRAS*

FRED VAN OYSTAEYEN Department Wiskunde en Informatica Universiteit Antwerpen B-2610 Antwerpen (Wilrijk), Belgium voyst@uia. ua. ac. be

PU ZHANG Department of Mathematics, University of Science and Technology of China Hefei 230026, Anhui, P. R. China pzhang© ustc. edu. en

This lecture was given in UIA at Antwerp, and in USTC at Hefei, aiming at outlining a construction of non-commutative, non-cocommutative pointed Hopf algebras via quivers, given by Cibils and Rosso3. We thank Sen Hu for his interest to include it in this proceeding. To save the space, we omit details here.

1. P a t h Algebras and P a t h Coalgebras We first recall some basics from Auslander, Reiten, Smal^ Montgomenry 2 .

1,s

, and Chin,

Definition 1.1: A quiver Q — (Qo,Qi,s,t) is an oriented graph, with Qo the set of vertices, Q\ the set of arrows, s and t two maps from Qi to Qo, where s(a) and t(a) are respectively the starting vertex and the ending vertex of a. We assume that Qo and Qi are countable sets. If the both are finite, then Q is called a finite quiver. A path p of length I in Q is p = on • • • a\ with each a* S Qi, where t(cti) = s(a,+i), 1 < i < I — 1. A vertex is 'Supported in part by the EC AsiaLink project "Algebras and Representations in China and Europe" (ASI/B7-301/98/679-11) and by theNSF of China (10271113) 121

122

Fred van Oystaeyen

& Pu Zhang

regarded as a path of length 0. Denote by s(p) and t(p) the starting vertex s(ai) and the ending vertex t{a{) of p. Definition 1.2: Let A; be a field and Q a quiver. Denote by kQ the kspace with basis the set of all paths in Q. Define KQa to be the algebra with underlying fc-space kQ, and with multiplication given by qp := 0m • • • Piai • • • ai if t(ai) = s(P\), and 0 otherwise, for paths p = a/ • • • a.\ and q = 0m-"Pi- Then kQa is an associative algebra, which is called the path algebra of Q. For each i £ Qo, denote the corresponding element in kQ by e\. If Q is a finite quiver, then kQa has the identity element 1 = X^eQ e*' Note t n a ' ; kQa = 0 „ > o kQn is a No-graded fc-algebra, where kQn is the fc-space with basis the set of all paths of length n; and that kQa is finite-dimensional if and only if Q contains no oriented cycles. Definition 1.3: Let Q be a quiver. The underlying vector space of the path coalgebra kQc is kQ, and the comultiplication A of kQc is defined by A(p) = p (g> s(p) + X]j=l al • • • ai+l ® ai • • • al + tip) ® P

for any path p = on • • • a.\ with each a, £ Q\. In particular, A(ej) = e* ® e, for each vertex i e Qo; and the counit e is defined as e(p) = 0 if I > 1, and 1 if I = 0. Questions, (i) To construct the all possible Hopf structures H on the path algebra kQa (i.e., the underlying algebra structure of H is exactly the path algebra kQa); or, (i') To construct all the possible Hopf structures on the path coalgebra kQc. The two questions are dual. We will work on (1') for some reason. We list some basic properties of path coalgebras for latter use. For a coalgebra C, the set of group-like elements is defined to be G{C) := { c G C ] A(c) = c c, c ^ 0}. It is clear e(c) = 1 for c € G{C). A subcoalgebra D of a coalgebra C is a subspace with the property A(D) C £) ® .D. A non-zero coalgebra C is said to be simple if C has no other subcoalgebras except 0 and C itself. A semisimple coalgebra is a direct sum of simple coalgebras. A coalgebra C is said to be pointed if each simple subcoalgebra of C is of dimension one. Note that a coalgebra C is pointed if and only if each simple subcoalgebra of C is contained in kG(C), see ref. 5, p.59. The coradical filtration {Cn} of a coalgebra C is defined as follows: Co := the sum of all simple subcaolgebra of C; and for

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n =£ 1, Cn := A _ 1 ( C ® C„_i + C 0 <S> C). Then we have Cn C C„+i,n > °;

C

A C

= Un>o Cni

( « ) C E t o Ci ®
Lemma 1.1: (i) Path coalgebra kQc is pointed with G(kQc) = QQ. (ii) The coradical filtration {Cn} of the path coalgebra kQc is exactly

2. Cotensor Product Coalgebras Let C be a coalgebra, M and N be a right C-comodule and a left Ccomodule, with structure maps 5R and 5^, respectively. Define the cotensor product MDN to be the kernel of the map 5fi®id — id®5L'- M ®N —> M ®C ®N. Let M be a C-bicomodule. Define the n-th cotensor product MD™ to be MUn := ker(5R <8> id • • • ® id ® - i d ® 5L • • • ® id) ( | ker(id <8> <5ft <8> id <8> • • • ® id — id ® id ® ^L ® • • • <8> id) O • • • O ker(id ® id ® • • • ® (5^ ® id — id ® id ® • • • ® id <5L). Set cotc(M) := C©MffiM D2 ©- • -©M111™®- • •. Define a coalgebra structure on cotc(M) as follows. Define the counit e by e\Mat = 0 for i > 1, and e\c = £cDefine A | c := A c , A | M := SL + 6R. Then A(M) C C ® M © M < g > C C cotc(Af) <8> cotc(M). For E m i ® m 2 G M D 2 , define A(y^mi m 2 + ^ m i m 2 + ^ m i ® <5fl(m2) G C ® M D 2 © M ® M © M D 2 ® C C cotc(M) c o t c ( M ) . In general, for E

m

i ® • • • <8> m„ € M D n , define

+E

mi

® m 2 ® " ' ® m™ + ' ' ' + E

+ E

m i

® " ' ® m n - l ® <5fi("*n)

mi

®m2 ® " ' ®m"

6 C ig) M D n © M M 0 ' " - 1 ) © M D 2 M D ( n - 2 ' © • • • © M 0 ^ - 1 ' M © M D n C C cote - ^ ®

cotc(M).

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Note that A is well-defined, and that cotc(M) is indeed a coalgebra with structure maps A and e, which is called the cotensor product coagebra of bicomodule M over C. Note that the omitted argument above also shows that MDn is a C-bicomodule with structure maps SL <8> id • • • <8> id and id • • • <8> id SR. By verifying that kQn ~ (kQi)an for n > 0, one gets Lemma 2 . 1 : (i) n > 0 via SL(P)

(ii)

For any quiver Q, kQn is a kQo-bicomodule for any

•= t(p) ®P

and

SR(P)

:=p®

We have the coalgebra isomorphism kQc ~

s(p). cotkQ0(kQi).

We need the universal property of the cotensor product coalgebras. If tp : X —> C ia a coalgebra map, then X has the induced C-bicomodule structure via ijj. lb

Theorem 2.1: Let X —» cotc(V) be a coalgebra map. Set tpn :— pntp : X —> VUn for n > 0, where pn : cotc(V) —> V°n is the projection. Then (i) Vo : X —• C is a coalgebra map. (ii) f/'i : X ——• V is a C-bicomodule map, where X is the induced C-bicomodule via ipo. (iii) For n>2,ipn is exactly the C-bicomodule map given by ipn : X

A

^

X®X®---®X^

V®n,

where A<") := ( A ^ _ 1 ) idx)^x for n > 2 and A ^ := Ax. (iv) -ip = f/iQ © Vi © "'" © ^n © ' ' " • Thus, ip is uniquely determined by ipo and fa. Theorem 2.2: Let V'o : X —> C be a coalgebra map, and tpi : X —> V a C-bicomodule map. Let tpn : X —> V®n be the composition V>„ : X ^ ^

X ® X ® • • • ® X ^ H V®», n > 2.

Then ?/>n is a C-bicomodule map with Im(i[>n) C y D " . If for each x € X there are only finite i such that if>i(x) ^ 0, then ip : X —> cotc(V) is a coalgebra map, where tp = Yli>o ^»-

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3. Hopf Bimodules over Group Algebras We are concerning about the Hopf structures on path coalgebra kQc. We will see below that if kQc admits a graded Hopf algebra with length grading, then kQi admits a /cQo-Hopf bimodule structure. For this reason we start from studying the Hopf bimodules over group algebras. Definition 3.1: Let H be a Hopf algebra. An i/-bimodule M is called an i7-Hopf bimodule provided that M is also an /f-bicomodule such that the structure maps 5L : M —-> H M and SR : M —> M H are both .H-bimodule maps, where the left and right .ff-module structures of H M (resp. M (g> H) are diagonal. Let C be a coalgebra, and x, y £ C. For a M := {m € M \ ^ ( m ) = y m}; for a Mx := {m G M \ 6R(m) = m x) and for V X M := {m G M\5L(m) = y®m, 5R(m) = y

left C-comodule M, denote right C-module M, denote a C-bicomodule M, denote m®x}.

E x a m p l e 3.1: Let Q be a quiver (not necessarily finite). If path coalgebra kQc admits a graded Hopf algebra structure, then kQn is a fcQo-Hopf bimodule, Vn > 0, with y{kQn)x = fc-span of {p G Qn\ t(p) = y, s(p) = x} = y(kQn)x, Vx,j/GQoNow, we investigate the structure of a Hopf bimodule over group algebra kG . Lemma 3.1: (i)

Let M be a left kG-comodule. Then M = 0

V

M.

y€G

(V)

Mx.

Let M be a right kG-comodule. Then M = 0 x€G

(ii)

Let M be a kG-bicomodule. Then M = 0

V

MX.

x,y€G

Lemma 3.2: Let M be a kG-Hopf bimodule. Then g(vMx)h = h 9xh v x 9y M , Vg,h,x,y G G. In particular, dim.k( M ) is constant if yx^1 belongs to the same conjugacy class ofG. Proposition 3.1: Let M be a kG-Hopf bimodule. Then M can be decomposed as a direct sum of kG-Hopf subbimodules of M, i.e., M = 0 C e C c M, where C is the set of conjugacy classes ofG and CM := 0 c 6 C x e G CXMX = ©c€C,xeG XC -^ x - In particular, if G is abelian, then M = 0 „ e G ' 9 ' M and l^M := 0 x e G 9 X M x . We will call CM the C-component of M.

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Denote by B(kG) the category of fcG-Hopf bimodules. The morphism set is the one of fcG-bimodule maps which are simultaneously fcG-bicomodule maps. Lemma 3.3: Let M G B{kG). For each C £ C and a fixed element u(C) in C, U ( C )M 1 is a right kZc-module via conjugate m»g := g~lmg, Vg £ 1 Zc, m e "C-^M , where Zc is the centralizer of u{C) in G. For each conjugacy class C of G, choose an element u(C) £ C, and denote by Zc the centralizer of u(C) in G. We have Lemma 3.4: (Cibils-Rosso3)

Let M = {Mc)cec

& I\cecmod

V(M) := 0 C e C kG ® Mc ®kzc kG = © C e C kG ®

kZ

c-

^et

{Mc)t%.

Then V(M) has a kG-Hopf bimodule structure with C-component kG an Mc®kzcKG d VV(M)X = x®Mc®gi, wheregi is an element inG such that g~1u(C)gi = x~xy. Also, we have a kZc isomorphism u(-c^V(M)1 ~ Mc. Proof.

The left fcG-module structure of V(M) is trivial. The right kG-

module structure of V(M) is diagonal, i.e, if G = Zc gi \J • • • \J Zc gn is a coset decomposition, then (g ® mc ® 9i)»h = gh® mc ® g%h = gh® mch! ® gj where gth = h'gj, h' € Zc- The fcG-bicomodule structure of V(M) denned as 6L(g <8>mc® gi) = gg^lu(C)gi

®{g®mc®

is

gi),

5R{g ® mc ® 9i) = {g <E> mc ® 3i) ® 5In order to prove V(M) is a fcG-Hopf bimodule, it suffices to prove that both 6L and SR are fcG-bimodule map. We omit the details here. The following theorem describes the category B(kG) in the terms of the categories of modules over certain subgroups of G. Theorem 3.1: (Cibils-Rosso3) For each conjugacy class C of G, choose an element u(C) G C, and denote by Zc the centralizer of u(C) in G. Then we have an equivalence of the categories B(kG) c± I\cecTno(^ kZcIn particular, if G is abelian, then B(kG) ~ FLeG mod kG. Proof. Consider the functor W : B(kG) —> Tlcec m°d kZc given by W(M) := ( " ^ M ^ c e c (see Lemma 3.3), and the inverse functor V defined in Lemma 3.4.

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127

4. Hopf Quivers Let G be a group and C the set of conjugacy classes. A class function X : C —> No is called a ramification, and denoted by \ = Y2cec XcC. Definition 4.1: (Cibils-Rosso3) Let x = J2cec XcC be a ramification of G. The corresponding Hopf quiver Q = Q(G, x) has the set of vertices <5o = G, and for each x £ Qo, each c £ C £ C, one has x c arrows from x to ex. Let C be a coalgebra, D and 2? two subspaces of C. The wedge of D and £ is defined to be D /\ E = A~x(D ® C + C ® E). Montgomery 6 has introduced the quiver T(C) of C as follows: the set of vertices of T(C) is the set of isoclasses of simple subcoalgebras of C; and for any two simple subcoalgebras Si and 52, there are exactly dimi<((Si A S2)/(Si + S2)) arrows from Si to 52- It is proved in ref.6 that C is indecomposable coalgebra if and only if T(C) is a connected quiver, see Theorem 2.1 in ref. 6. Let C be a coalgebra with the set G(C) of group-like elements. For x, y £ G(C) denote by Px,y{C) :— {c £ C | A(c) = c ® x + y c}, the set of x, j/-primitive elements in C. It is clear that e(c) = 0 for c £ PXiV(C). Note that k(x — y) C PXiy(C). An x,y-primitive element c is said to be non-trivial if c ^ k(x — ?/). Lemma 4 . 1 : Let C be a coalgebra, x,y £ G(C). Then we have dim k ((ky A kx)/(kx + ky)) = dim k P x , y (C) - 1. That is, the number of arrows from kx to ky in T(C) dim k P x , y (C) - 1.

is exactly

Let C be a pointed Hopf algebra with G(C) = G. By Lemma 4.1 the quiver T(C) in this case can be interpreted as: the set of vertices of T(C) is G, and for x,y £ G, the number of arrows from x to y is dimkP x , y (C) — 1. For example, for each quiver Q, the quiver of path coalgebra kQc is exactly Q, i.e. T(kQc) = Q. Proposition 4.1: Let H be a pointed Hopf algebra with T(H) defined above. Then T(H) is a Hopf quiver. Lemma 4.2: Let Q be a quiver such that kQc admits a graded Hopf structure with length grading. Then Q is a Hopf quiver; and # { p G Qn| s(p) = x, t(p) = y} remains to be a constant if yx~x belongs to the same conjugacy class.

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Lemma 4.3: Let Q be the Hopf quiver of (G, \ = Y^cec XcC). Then 3 a kG-Hopf bimodule M such that dim*; yMx = X[yx-1]5. Hopf Structures on Path Coalgebras Let Q be a quiver such that kQc admits a graded Hopf structure with length grading. Then Q is a Hopf quiver and kQi admits a fcQo-Hopf bimodule structure with SL and SR defined as SL(P) = p<S> s(p) and 6R(P) = t(p) <8>p. So, we start from a quiver Q with Qo a group such that kQ\ admits a kQoHopf bimodule structure with SR and 6L given above. We will construct a graded, associative multiplication on kQc. For this, let Vo : X := kQc ® kQc —• kQo be the composition of maps: X

Po

-^,° kQ0 ® kQ0 —• kQ0

using the multiplication in kQo, and ipi be the composition of maps: X «>®"i®5»®«> kQQ ® kQx e kQt ® kQo —> fcQi using the fcQo-bimodule structure on kQ\. Then it is clear that Vo is a coalgebra map, and that V>i is a fcQo-bicomdule map. For n > 2, set ipn to be the composition of maps : X

i-»

X®X®---®X^-f(A;Qi)®n,

where A 2 = ( i d ® r ® i d ) ( A ® A) : X —• X<S>X, A is the comultiplication of kQc and r is the standard twist. Note that for any x € X, there are only finitely many i such that ipi(x) ^ 0. Then by Theorem 2.2, we have a coalgebra map V> = $ 3 ^ = kQ° ® fc(5C —»

fc< C

3 -

CoTkQ^kQ^).

i>0

We claim that ^ is a graded, associative multiplication on /c<3°. In order to prove this, we fix some notations. Recall that A 2 is defined inductively as A 2 : = (A 2 n _ ® «0A2 for n > 2 and A 2 := A2. An n-split of a path a is a sequence ( a n , • • • ,c*i) of paths such that a = a „ • • • oc\. An n-thin split of a path a is a sequence (an, • • • , a i ) of vertices and arrows such that a = an • • • a\. Note that for any path a, A(a) = ]C(Q2 a,) a 2 ® a i > where the sum is over all 2-splits (02, a i ) of a. By induction on n, we have Lemma 5.1: For n>2,

and two paths a and 0, we have

Quivers and Hopf Algebras 129

A<"

1)

(a®i9) = E ( a n , . . , a 1 ) , w n , . . , f t ) a n ® i 8 n ( » - - - ® o i ® i 9 i

where the sum is over all n-splits (an, • • • , a i ) and (/3n, • • • ,f3\) of a and (3, respectively. Set Xm := © i + j = m kQi ® kQj. Then X = © m > 0 X m . The following fact shows that ip is graded. L e m m a 5.2: For any n and m, we have ipn(^m)

= 0

ifm^n.

For p > n, denote by Z>£ the set of p-sequence of 0 and 1 such that the number of l's is n. For d = (di, • • • ,d p ) € Pg, set d 6 Pp_„ to be the complement sequence of d, i.e., d + d = (1,1, • • • ,1). Let a be a path of length n and d £T>%. Define da to be the p-thin split of a given by da := ((da)p, ••• , (da)i), where (da)i is a vertex if di = 0 and is an arrow of di = 1. Note that da is a uniquely determined p-thin split of a. Let a and /? be paths of length n and m, respectively, and d G Z>™+n. Define (a•/?),* to be the element in kQn+m: {a»/3)d := [(da)m+n.(d~P)m+n\ • • • i(da)i.(d(3)i], where [(da)i.(df3)i\ denotes the action of A;Qo-bimodule of kQ\. With these notations we have L e m m a 5.3: Let a and [3 be paths of length n and m, respectively. Then we have

J2 ia*P)d-

ipn+m(a®P)=

dexc+n Proof i,n+m{a®P)

^n+m)A2n+m-1)(a®f3)

= =

^®(»+m)(

^

an+m

® 0„+m

® • • • ® <*1 ®

0l)

(<*n+m.-" , < * l ) , ( / 3 n + m , - " ,0i)

where the sum is over all (n + m)-splits of a and /?, respectively. Hence, •4>„+m(a®P)=

J2

= E

i/>f ( n + m ) (an+m®/Wm«>---®ai®/3i)

(«'&*•

a

dgX)n+m

Note that | P £ + m | = ("+ m ). For n,m,l, set P " ^ m ( + ' to be the set {(di,d2,d3)\di

€V%+m+l, d2eV2+m+t,

d3€T>?+m+l,d1+d2

+ d3 = (i,ir--

,1)}.

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For paths a,(3,7 of lengths n,m,l,

respectively, and d = (di,d2,d3)

G

^ T ' . define ( a . / 3 » 7 ) d := [(rfia)n+m+i-(^2/3)n+m+i-(rf37)n+m+/] ' " * [(dia)1.(d2(3)l

.(d 3 7)i] G

where d i a = ((dia) n +m+/,-•• ,(dia)i),
be paths of lengths n,m,l,

V'n+m+/(V'n+m(a®/3) ® 7 ) =

respectively. Then we

]P (a./? .7)4 d=(di,d2,*)e^+ m m < + '

= 1pn+m+l(a ® Vm+/(/3 ® 7 ) ) .

Note that the associativity of multiplication ip can be also deduced by using Theorem 2.1. Lemma 5.5: Let Q be a quiver with Qo a group. Assume that kQi admits a kQo-Hopf bimodule structure with SL and SR defined before. Then the path coalgebra kQc admits a graded Hopf structure with length grading. Proof. We already have a graded, associative multiplication map ip such that ip : kQc ® kQc —• kQc is a coalgebra map. This makes kQc a pointed bialgebra. Since pointed bialgebra whose set of group-like elements is a group is always a Hopf algebra (Takeuchi 10 ), it follows that (kQ, ip, A) is a graded Hopf algebra. Example 5.1: In the graded Hopf structure on kQc given in Lemma 5.5, we have p.a = [t(0).a]\p.s(a)] + [p.t(a)][s(p).a], for a,(3 G QX. Theorem 5.1: (Cibils-Rosso3) Let Q be a quiver. Then the following are equivalent: (i) Q is a Hopf quiver of some (G, x); (ii) Qo is a group and kQi is a fcQo-Hopf bimodule (SL and 5n denned as before); (iii) kQc admits a graded Hopf structure (with the length grading).

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131

Proof, (i) =>• (ii) : Assume that Q is a Hopf quiver of a group G with ramification \. By Lemma 4.3 we have a fcQo-Hopf bimodule M such that dimkyMx = Xlyx-1]- Since kQi = ®XiyeGy(kQi)x, dimky(kQ1)x = #{a; —• y} = Xlyx-1]: by the definition of a Hopf quiver, it follows that one can identify kQi with M by identifying each y(kQ\)x with VMX, and then kQi has an induced kQo-Hopt bimodule structure, with 5L and SR exactly being the ones defined as 5L(&) = t(a) ® a, Sji(a) = a <8> s(a), Va £ Qi. (ii) => (Hi) follows from Lemma 5.5 and (Hi) =>• (i) follows from Lemma 4.2. Theorem 5.2: (Cibils-Rosso3) Let Q = Q(G,x) be a Hopf quiver. Then there exists a one to one correspondence between the following two sets: { the isoclasses of graded structures on kQc with length grading } and the set {(Mc)cec

G Y[ mod kZc cec

\ dimfc Mc = Xc}-

Proof. Let H be an graded Hopf structure on kQc, with length grading. Then kQ\ is a /cQo-Hopf bimodule by Theorem 5.8, and dimfc "'^(fcQi) 1 = Xc- By Theorem 3.8 we obtain (Mc)c G I l c e c Tno^ ^ c w ^ ^ dimfc Mc = XcIf we have two graded Hopf structures Hi and Hi on kQc, both with length grading such that fcQo-Hopf bimodule kQ\ induced by H\ and the fcQo-Hopf bimodule kQi induced by H2 are isomorphic, then by Theorem 2.1 we know that the multiplications of H\ and H2 are uniquely determined by the corresponding kQo-Hopi bimodule structures on kQi, hence H\ and if2 are isomorphic as Hopf algebras. The theorem above shows that any graded Hopf structure on kQc, with length grading, is given as we constructed. Thus, this construction reduces graded Hopf structures to module structures of subgroup Zc of QoExample 5.2: Let G = (g) be the cyclic group of order n, with ramification X = \g]- Then the corresponding Hopf quiver Q is the basic cycle of length n. Denote this quiver by Z„, and the arrow from gl to g%+l b y a j , 0 < i < n — 1. We will determine the all graded Hopf structures on kU^. First, construct fcG-Hopf bimodules V(M). Let M = (Mx)xeG G YlxeG m°d ^ > w i t n dimM x = X[x] = 1 if x = g, and 0 if x ^ g. Thus M is just a one-dimensional /cG-module and hence M = kv with vg = qv, q € fc, qn = 1. It follows that

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V{M) = 0 x £ G kG Mx ®kG kG = kG®kv

= span{g ! ® v|0 < i < n - 1}.

Then, make kQi a &G-Hopf bimodule by identifying gi+1{kQi)gi with V(M)s' = fc(g'®u). We identify a* with p*®u. Then 3 ^ = 5(5'®u) = g ^ 1 u = a i + i and a ^ = (gi ® w)s = # i + 1 ®vg = qgi+1 ® u = g a i + i . Finally, using the &G-bimodule structure on fcQi to write out the multiplication of kZn explicitly. Let p\ be the path of length I and with s (Pi) = i- Denote by (l+lm) be the Gaussian binomial coemcent for l,m e N 0 , which is defined by (l+Tn) := ^ g ^ 1 , where lq\ := 1,---J, and /, := 1 + q + • • • + ql~1(= £ ^ ) . Then gl+1

otiQij = [t(ai).aj}[oti.s(aj)] +

[ai.t(aj)][s(ai).aj]

= Q:'(Xi+j+ioti+j + qJ+1ai+j+1ai+j

= q3 f j

ai+j+iai+j.

By induction we obtain (*)

P\PT = 1jl (* + j ^

*£!?.

*> m > 0'

0 < i, j < n - 1.

When 9 runs over all n-th roots in k of unit, the formula (*) gives the all pairwise non-isomorphic graded Hopf algebra structures on &Z£ (with length grading). In particular, if n = 1, then one obtains the all Hopf structures on k[x], the polynomials in one variable: l + m

\xn+m,

and A ( i ) = l ® a : + a ; ® l .

As we see from the lines , if G is an infinite cyclic group, then (*) also gives the all graded Hopf structures on /cZ£, when q runs over all non-zero elements in k. E x a m p l e 5.3: Ree's Shuffle algebra. Let G be the unit group. Clearly any vector space V is a A;G-Hopf bimodule. The corresponding Hopf quiver consists of one single vertex and a family of loops indexed by a linear basis of V. In this case, CoT kG (V) = k © V © V®2 © • • • © V®n © • • • , which is denoted by Sh(V) (see ref.9) and is called Ree's Shuffle algebra

Quivers and Hopf Algebras 133 T h e comultiplication of Sh(V)

is given by

n-l A(t>i • • • vn) = 1 ® (vi ® • • • wn) + ^2 (v\ ® • • • <S> Vi) ® («i+i ® • • • <8> Vn) i=l + (ui ® • • • ® u n ) ® 1, «i € V, 1 < i < n, n € N. Recall t h a t a (n, m)-shufne tp is just an (n + m ) - p e r m u t a t i o n such t h a t ip(l) < -0(2) < • • • < i>{n) and ip(n + 1) < tp(n + 2) < • • • < ip{n + m ) . Denote the set of all (n, m)-shuffles by ( n | m ) . Clearly there is a one to one correspondence between (n\m) and X>" + m , by identifying ip with d^ = ( d l , d 2 , • • • , d „ + m ) such t h a t of* = 1 whenever i is contained in W(l)^(2),."^(n)}. Let a = v\ • • • ® Dn and /? = u n + i ® • • • v n + m , then in the Hopf algebra Sh{V) = C o T k G ( V ) (,°fP)di,

= ^ - i ( l ) ®1ty-i ( 2 ) ® • • • ® t t y - i ( „ + m ) ,

by the definition of (a*f3)d- By Lemma 5.3 we have the multiplication of

Sh(V) (ill ® • • • ® l»n) • («n+l <8> • • • ® Un+m) =

and the antipode 5 is given by S(v\ • • • u n ) = (—l) n u n ® • • • u i . References 1. Auslander, M., Reiten, I., Small/), S.O., Representation Theory of Artin Algebras, Cambridge Studies in Adv. Math. 36, Cambridge Univ. Press, 1995. 2. Chin, W., Montgomery, S., Basic coalgebras, AMS/IP Studies in Adv. Math., vol 4, (1997), 41-47. 3. Cibils, C , Rosso, M., Hopf quivers, J.Algebra 254 (2002), 241-251. 4. Kassel, C , Quantum Group, Graduate Texts in Math. 155, Springer-Verlag, New York, 1995. 5. Montgomery, S., Hopf Algebras and Their Actions on Rings, CBMS Regional Conf. Series in Math. 82, Amer. Math. Soc, Providence, RI, 1993. 6. Montgomery, S., Indecomposable coalgebras, simple comodules and pointed Hopf algebras, Proc. of the Amer. Math. Sco. 123(1995), 2343-2351. 7. Ree, R., Lie elements and an algebra associated with shuffles, Ann. Math. 68 (1958), 210-220. 8. Ringel, C. M., Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984.

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Fred van Oystaeyen & Pu Zhang

9. Sweedler, M.E., Hopf Algebras, New York: Benjamin, 1969. 10. Takeuchi, M., Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan 23(1971). 561-582.

THEORY OF BIDIRECTIONAL SOLITONS O N WATER*

JIN E. ZHANG Department of FIN A, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

YISHEN LI Department of Mathematics and Center of Nonlinear Science, University of Science and Technology of China, Hefei 230026, P. R. China

A theory of bidirectional solitons on water has been developed by using the classical Boussinesq equation. A solid mathematical foundation of the bidirectional water wave interaction has been well-established with our theory.

1. I n t r o d u c t i o n T h e K d V equation

Ct + Cz + 2CC* + gC*x* = o,

(i)

where C, is wave elevation, x and t are space and time coordinates, is often used to model weakly nonlinear and weakly dispersive waves traveling from the left to the right on a uniform layer of water with a depth scaled to be 1. T h e K d V equation is integrable, has a bi-Hamiltonian structure and allows an exact iV-soliton solution, but its solutions are only physically meaningful for unidirectional water waves. T h e K d V equation cannot be used to model bidirectional water wave interaction due to its intrinsic n a t u r e . T h e Boussinesq equation (singular) 3

1

*JEZ has been supported by the Research Grants Council of Hong Kong. YSL has been supported by the National Basic Research Project: Nonlinear Science, and by the Ministry of Education of China. 135

136

Jin E. Zhang & Yishen Li

is also used to model weakly nonlinear and weakly dispersive water waves traveling in a direction either from the left to the right or from the right to the left. It is integrable, has a bi-Hamiltonian structure and allows an exact bidirectional AT-soliton solution including overtaking and head-on collisions, but few people realize that its solutions of head-on collision are not physically meaningful for water waves. In fact, the maximum run-up of a soliton with amplitude a < 1 on a vertical wave predicted by equation (2) is la — 3a 2 /2 + 0 ( a 3 ) , which is much smaller than the physically correct result 2a + a 2 /2 + 0(a3). Its prediction of the maximum run-up is not physical because the contribution from the potential energy of the soliton (—3a2/2) is negative, see Ref. 6 for a complete discussion on the details of the problem. The problem can be very serious if the Boussinesq equation (2) is used to model the interaction of a solitary wave with a vertical wall. The classical Boussinesq equation f Ct + [(1 + C)"]i = \ut+

-\uxxx,

uux + Cx = 0,

where u is the projection of surface velocity of water particles on x axis, is the only serious candidate that can be used to model bidirectional solitons on water, see Ref. 4 for a derivation of this equation and a few other variations (including higher order ones) of the generalized Boussinesq equations. It is known to be integrable and equivalent to Broer-Kaup (BK) system that has a tri-Hamiltonian structure, but its exact bidirectional JV-soliton solution has not been found until our recent papers 1'2<3>6. As shown by us in Ref. 6, its solution of head-on collision is physically meaningful for water waves. In fact, the maximum run-up of a soliton with amplitude a -C 1 on a vertical wave predicted by equation (3) is equal to the physically correct result 2a + o 2 /2 (see equation (56) of this paper for details). Therefore the equation can be used to model the run-up of ocean waves on dykes and dams 5, 8, 9. This paper summarizes our recent results on the theory of bidirectional solitons on water, developed by using the integrable Boussinesq surfacevariable equation (3). In Ref. 1, 2, we present the three Darboux transformations (DTs) of the equivalent BK system and use the DTs to construct a multisoliton soliton solution in a recursive formula. In Ref. 3, 6, we prove that the equation is equivalent to a member of Ablowitz-Kaup-Newell-Segur (AKNS) system, and construct a general formula of the bidirectional multisoliton solution by using DT on the AKNS system. We also discuss the mechanics of soliton interaction in Ref. 6. In Ref. 7, we present the con-

Theory of Bidirectional Solitons on Water

137

servation laws, tri-Hamiltonian structure and associated hierarchy of the equation (3), and discuss some related problems that are of interest for further research. 2. The Algebraic Properties of the Model Equation (3) and its Related Equivalent Equations With scaling transformation x —> x,

(4)

o

equation (3) becomes

fCt + [(i + 0«]* + J W J J X

— ^)

(5)

\ Ut + UUX + Cx = 0.

The Lax pair of the system (5) is f fax = (A2 + Xu + \u2 - C - 1)0, \ + (A - \u)<j>x.

(6)

To study solitons on water, we require following regularity conditions at infinity: dxi

fry, = 0, 'dx1 X—»±00

= 0, x—>±oo

i = 0, 1, 2,

(7)

The system (5) has infinite number of conservation laws dmn dt

dfn , dx 1

mn+i

1 /

^2 \ k + I = n, \ k € [0, n)

/o = —2(C + 1 + 2 u 2 ) '

/"

,

m t m i l - ^ l , 9x

= m

(8)

n = 0, 1, 2, Idu

n=l,2,•

"+! ~ 2Um"'

n = 1

' 2 ' '•'•

(9)

(10)

The system (5) has a tri-Hamiltonian expression Vt = I>i JW3 = ©2<JW2 = V3SHi,

(11)

138

Jin E. Zhang & Yishen Li

(12) 3

+ \{^B-C^+\^A^)),

(13)

B = (2/i-u a .)a + a(2ft-« a .) + a(u + a) 2 ,

(14)

~\C-\^A-D

A = - ( u a + au), C={2h£,

ux)d + d(2h - ux) + (u - d)2<9,

(15)

= - J ( " - ^ ) [ ( 2 / i - U x ) 5 + a ( 2 / i - u x ) ] - i [ ( 2 / i - u x ) 5 + a(2/i-w : c )]( u + 5), (16) Wi = - X!^o 5 M a ; '

W2 = - / ! ^

\hudx,

«3 = " Too (§*«' + 1^ - X ) «**•

(l7)

where

"-(»),-(2)- « - ( « / » ) •

(I8)

JW denotes the vector of variational differentials and d = d/dx. The system (5) is associated with a hierarchy Vt = V1Vn(~Jh\=VlVn5Hi,

n = l , 2, • • • ,

(19)

where

^ / r ^ a - ^ +h+^ j

(20)

By using the transformation u=-v,

C = -l + w-

-vx,

(21)

we can convert the system (5) to BK system j vt = \{v2

+2w-vx)x,

\wt

\wx)x.

— (vw+

(22)

Introducing the following transformation q

= e!udx,

r=-(l

+ <;-^ux)e-Iudx

(23)

Theory of Bidirectional Solitons on Water

139

or u=—, ( = -l-qr+-ux, q 2 we have an equivalent system for q and r, j qt + \qxx -qr2

(24)

-q = 0,

(25)

2

\ r t - \rxx + qr + r = 0. which is a member of the AKNS system. 3. The Darboux Transformation of the BK System The BK system (22) has a lax pair 3>X = M $ ,

$ = (^1,02)',

M=(

_

2 w

A

_

i

)

(26)

!

2

* t = JV$,

AT=

4V

2

\ - A u > - ^(u>x ~wv) X* + j ( v x - v J ) /

.

(27)

We consider three basic Darboux transformations

$* = r *

(i = 1,2,3),

T1=a1(Al

+ ai

\ ci ^2

/ «2 &2

\

*1Y a,! J

rp3 _ fot(X + a) ab

which map (26) and (27) to $ ; = M^\

$* = N^\

(29)

l

where M* and N are the same as M and iV but with w, v replaced by w%, vl. We have proved in Refs.l, 2 the following proposition: Proposition 3.1: If we take

01 =1(x,Ai), C2 = - " 2 ,

02 = <^2(a:,Ai), «2 = e 2 J ( v - , ;

o2 = - - - p - , 2 Vl )dx

,

d2 =-w~-^ V2

Ipl =l(x,X2),

X2,

b2 = -, ^

lp2 = 4>2{x,X2),

140

Jin E. Zhang & Yishen Li

a-

_ ( A ^ l 0 2 ~ Ai2) ,

_ (A2 - Ai)02<02 c, A a = e^^v3-^dx,

A = 01^2-^2^-1,

, _ (Al - A2)0lV>i o,

, _ (Al02<0i - A20iV>2) d^ , j = el/("-»>

u,*, «', i = 1, 2, 3,

then we have 1

,2

=

x — _ d\ _l£. di '

_ ^ _ ^ + 2 rpi 4>2 v3 = v-

A

iI _

('4>2\ / v^2 \ \ij

W2 = ( 2 A 2 W

„Ai0 „^W22 _ s02 01 ~0i'

_VW_

Wx)tl_w2(p.Y

'^2

\^2j

y q ^ , ^ 3 = (1 + 26)(2c + w),

and w%, vl, i = 1, 2, 3 are new solutions of equation (22). When C = 0, u = 0, we have v — 0, w = 1. If we take u = 0, w = 1 as our "seed", and take |Ai| > 1, 01 = cosh[fi(a:,*)],

0 2 = cisinh[£i(x,£)] + Ai cosh[£i(x,£)],

fiOM) = ci(x + Aii),

(30)

ci = yjA2 - 1,

then both (v1, w1) and (i;2, w2) give a single bell-type soliton solution for C- The soliton goes to the left when Ai > 1, and to the right when Ai < —1. If we take v — 0, w = 1 as our "seed", and take —2 < A2 < Ai < — 1, (01,02) the same as equation (30), ipi = sinh[£ 2 (z,*)],

i>2 = c2cosh[£2(:r,*)] + A2sinh[£2(a;,*)],

&{x,t) = e2(:r + A2i),

c2 = \J\\ - 1,

then (v3, w3) gives a solution with two right-going bell-type solitons overtaking collision. If we take v = 0, w = 1 as our "seed", and take Ai < —1, A2 > 1, (0i,02) the same as equation (30), ipi = cosh[£2(a;, t)},

ip2 = c2 sinh[£2(a:, t)} + A2 cosh[^2(a;, t)],

then (v3, w3) gives a solution with two bell-type solitons head-on collision.

Theory of Bidirectional

Solitons on Water

141

4. The Darboux Transformation of the A K N S System The Lax pair of the system (25) reads VX = MV,

* = (1>i,ih)T,

-Xq r X

M =

(31)

+i

••="•• "-[^r T.t > 2'x

"

V*-'

2.

The Darboux transformation on the AKNS system, available in a textbook, is given as follows. Let

s-n

*i-(£,'£)

T-X-I+±T,»->.

(32)

where / is a 2 x 2 identity matrix, 0 is a solution of equations (31), then 0' is a solution of equation <j>'x = M ' 0 ' ,

4>'t = N'tf,

(33)

where M' and N' are the same as M and AT in equations (31), but with q, r, qx and rx replaced by q', r', q'x and r'x. We assume Aj ^ Aj for i ^ j , i = 1, 2, • • •, 2n, and denote l,j = l{x,Xj),

02 j =
We define a 2n x 2n matrix H to be the following ( A™-101,1 A" - 02,1 A" _ 2 0i,l A™_202,1 ••• 01,1 02,1 \ A2~ 01,2 A j " 02,2 A^ - 01,2 A Q - 02,2 • • • 01,2 02,2 H--

(35) \^2n


0 i , 2 n X^~

02,2n

01,2n 02,2n/

Solving the equations / Oi \

/ -A?0M \

(bx\

tf

A, \a2nJ

/ -A?02,l —

ai

H

\w

\-A2 n 01,2n/

\

^2^2,2

= B

\_^2n^2,2n/

(36)

gives us aj and 6», i = 1, 2, 3, • ••, 2n. Then
r' = r - 2 6 i ,

(37)

detff2 a 2 = detff '

6i

detifi detff'

(38)

where

142

Jin E. Zhang & Yishen Li

and H2 is a 2n x 2n matrix of H with the second column replaced by A, H\ is a 2n x In matrix of H with the first column replaced by B. 5. The Mechanics of Soliton Interaction For a layer of quiescent water without any waves, wave elevation is C, = 0 and velocity is u = 0, and corresponding q = 1 and r = — 1. Therefore we take (<7>r) = (1) — 1) a s o u r initial seed to implement Darboux transformation. With this initial seed, we have the following two sets of basic solutions for the spectral problem (31): 4>i j = cosh£j,

(p2j = Cj sinh^- + Xj cosh£j,

4>i j = sinh£j,

4>2,j = Cj cosh^- + Xj sinh^-,

j is an odd number, (39)

j is an even number, (40) 2 where £,• = Cj(x + Xjt) and Cj = JX — 1. The eigenvalue Xj is the wave speed of a soliton. The soliton is right-going if Aj < — 1, left-going if Xj > 1. For a single right-going soliton solution, we can take m = 0 and I = 1 with the following eigenvalues and eigenfunctions: XI = - 1 ,

<£i,i = l,

A| = — A < — 1,

02,1 = - ! .

c =

f = - g - c f c ~ •**).

0ii2=sinh£,

VX2 - 1,

02,2 = ccosh£ — Asinh£,

where x and t have been converted to the original coordinates before the scaling transformation (4). We obtain a single-soliton solution of equation (3) uB(x-Xt;X)

CB (X

= ;

y ~^ , A + cosh^/3(A2-l)(x-At)

2(A2 - 1) (1 + A cosh y/3(X2 - l)(x - At; A) = ^ — = 72 fA + cosh y/3{X2 - l){x - At) J

(41) V

\tj) "•

;

( 42 )

The wave speed A and the wave amplitude a satisfy A = 1 + \a.

(43)

Integrating the wave elevation (42) over the whole space domain gives us the mass under the soliton mB(X) = y ° ° CB(S; X)ds = ^ X

2

- 1 = -j=y/(l

+ o/4)a.

(44)

Theory of Bidirectional

Solitons on Water

143

Differentiating (42) twice and evaluating at the origin gives us C£(0;A) = - 6 ( 2 - A ) ( A - l ) 2 .

(45)

Therefore the soliton has a single peak when A < 2 and double peaks when A > 2. The soliton appears to have some remarkable features. It is singlepeaked when the wave amplitude is not larger than 2, and double-peaked when the wave amplitude is larger than 2. As is well-known that the Boussinesq model is only valid for the water waves with small amplitude, i.e., the wave amplitude smaller than water depth (scaled to be 1 here). Therefore the new feature of double-peaked soliton is not physically meaningful for the water wave. We now construct a multisoliton solution with 2m left-going and 21 right-going solitons, the power of the eigenvalue in the Darboux transformation is taken to be n = m + I. First we rank the solitons by their amplitudes (or speeds). For the 2m left-going solitons, we assume A2m > A2 m -i > • • • > Ai > 1. For the 21 right-going solitons, we assume X21 < ^21-1 < • • • < AJ < —1. With the eigenfunctions defined in (39, 40) for both Xj and AJ, we can obtain the soliton solution as follows:

«=4>

( = -l-q'r'

q'

+ lux,

(46)

2

where q' = l + 2a 2 ,

r' = - l - 2 6 i ,

a 2 and b\ are defined by (38). This is the solution for the interaction of an even number of solitons in both directions. To obtain an odd number of solitons, we can simply set the first eigenvalue to be 1 for a left-going soliton and —1 for a right-going soliton. In other words, an odd number of soliton solution can be treated as an even number of solitons in which one of the solitons has zero amplitude. For a solution with two-soliton overtaking collision, we take m = 0 and / = 1 with the following eigenvalues and eigenfunctions: AJ = —Ai < —1, A2 = - A 2 < —Ai,

0i,i = cosh£i, 01,2 = sinh£ 2 ,

02,i = c\ sinh£i — Ai cosh£i, 02,2 = C2Cosh£2 - A 2 sinh^ 2 ,

where Ai and A2 are two positive numbers. The solution to system (3),

144

Jin E. Zhang & Yishen Li

given by (46), can be written in a closed form as 2(A2 - Ai)[c| - c\ tanh 2 fr tanh 2 & - (A| - A2) tanh 2 (C2 - c\ tanh^i tanh£ 2 ) 2 — (A2 - A1)2 tanh 2 £2 _

fr]

(4g)

(A2 - Ai) (ci tanh £1 - Ai) (c2 - A2 tanh £2) + -E;UX, c2 - ci tanh^i tanh£ 2 — (A2 - Ai) tanh£2 J \/3

(49)

c2 - ci tanh gi tanh £2 + (A2 - Ai) tanh g2 C2 — c\ tanh £1 tanh £2 — (A2 — Ai) tanh £2

v^ £,i = —Ci(x-\it),

Ci = yJ\f-l,

i = l,2,

A2 > Ax > 1,

where x and t have been converted to the original coordinates before the scaling transformation (4), Ai and A2 are the speeds of the two solitons, with A2 larger than Ai. The soliton with the speed A2 is taking over the soliton with the speed Ai. The process of overtaking interaction can be easily seen with the asymptotic limit of the solution (48, 49): as t —» —00, COM) - •


- Aii - Ai; Ai) +

CB{X

- A2t + A 2 ; A2),

u(x,t) —> UB{X - A i t - Ai; Ai) + UB{X - A2t + A 2 ; A2), and as t —• +00, C(x, t) ^(B(X-

u{x,t) —>

UB(X

Ait + Ai; Ai) + Cs(z - A2t - A 2 ;

- Ait + Ai; Ai)

+UB(X

A2),

- A2t - A 2 ; A2),

where CB(S; A) and u s ( s ; A) are the wave elevation and surface velocity of the single-soliton solution given by (42, 41), and the total phase shift of the two solitons are given by the following 2Ai =

= arccosh; —, ^3(A'f - 1) A2 - Ai '

(50)

2A2 =

= arccosh—. A \/3(A 2 - 1) A2-A1

(51)

Since the mass has been obtained previously in (44) as i = l,2, the conservation of momentum can be easily verified by n

A

o

A

8

, A i A

2

- l

2miAi = 2m2A 2 = -3 arccosh- A 2 - A

Theory of Bidirectional

Solitons on Water

145

For a solution with two-soliton head-on collision, we take m = 0 and / = 1 with the following eigenvalues and eigenfunctions: X{ = —Ai < —1,

i,i = cosh£i,

A2 = A2 > 1,

0i,2 = cosh^2,

2,i = ci sinh£i — Ai cosh£i, 02,2 = C2sinh£2 + A2cosh^2-

The solution of system (3) given by (46) can be written in a closed form as follows 2(Ai + X2)(X2 -\\~cl tanh 2 £2 + c\ tanh 2 £1) u = (C2 tanh^ 2 - ci tanh£i) 2 - (Ai + A2)2

(54)

C2 tanh £2 — c\ tanh £1 — Ai — A2 Ai -|- A2

< = - ! + ci tanh £2 — c\ tanh £1 + 1+2

(Ai + A2)(cj tanh£i - Ai)(c2 tanh £2 + A2) + 7-^Wx. (55) C2 tanh £2 — c\ tanh £1 + Ai + A2 \/3 , — c 2 { x + A2t),

£1 = — c i ( a : - A i t ) ,

>A?-i.

Ai>l,

1,2,

where x and t are the original coordinates before the scaling transformation. The soliton with speed Ai is moving from the left to the right. The soliton with speed A2 is moving from the right to the left. At t = 0, the two solitons merge into a single peak. One may verify that £x(0,0) = 0, i.e., the solution is symmetric about the origin. Therefore the maximum amplitude appears at the origin, i.e., Cmax = C(0,0) = ai + a2 +

-a\a2.

(56)

For the head-on collision of two solitons with the same amplitude, a\ a2 — a, the wave elevation at t = 0 can be simplified and given by

C(z,0)

1 2a + ^a2) sech2 -y/3a(4 + a)x

and the velocity at t = 0 is zero for all x. After the head-on collision, each soliton experiences a backward phase shift. The asymptotic analysis of the solution (54, 55) leads to the following limits: as t —> —00, C(i, t) -»

CB(X

u(x, t) ->uB(x-

- Ait - Ai; Ai) +

CB(X

+ A2t + A 2 ; A2),

Ait - Ai; Ai) - uB(x + X2t + A 2 ; A2),

(58) (59)

146

Jin E. Zhang & Yishen Li

and as t —• +00, C(ar, t) - •

CB(Z

- Ait + Ai; Ai) +

u(x,t) -*uB(x-\it

CB(X

+ A1;Xi)-uB(x

+ X2t - A 2 ; A2),

(60)

+ X2t-

(61)

A 2 ;A 2 ),

s

where Cs( ! A) and us(s; A) are the wave elevation and surface velocity of the single-soliton solution given by (42, 41), and the total phase shift of the two solitons are given by 2Ai =

. 2 arccosh* 1 * 2 + 1 , Al+A2 ^3(A? - 1)

2A2 =

2 ;

^/3(A2 - 1)

arccosh^l + i A i + A2

The conservation of momentum can be easily verified by A

r.

A

"

,AlA2 +

l

2miAi = zm 2 A 2 = - arccosh— —. 3 Ai + A2 For the asymptotic behavior of the iV-soliton solution for large t, the phase shift of each soliton after the interaction can be derived from our solution. We have following conjecture: Conjecture. For the N right-going overtaking soliton solution given by (46), the asymptotic behavior of the solution is N

N

lim £{x,t) = ^T/(B(x-Xjt+Aj),

lim C(x,t) =

j=i

Y^(B(x-\jt-Aj), j=i

(63) where the phase shift of the jth soliton is given by N

A,- = Y^ sign(A, — Aj)— i = 1, ^ ^

1

AjAj

= arccosh ~ 1)

(64)

*i

The phase shift for N head-on colliding soliton solution has a similar result. 6. Conclusion A theory of bidirectional solitons on water has been developed by using the integrable Boussinesq surface-variable equation (3). With the physical significance well-recognized, we may now study the existence, uniqueness and stability of the initial value problem of the equation. We can also study the solution of the equation with some small disturbances or external moving forces. These problems are interesting for further research.

Theory of Bidirectional Solitons on Water 147 Acknowledgments We wish to t h a n k Theodore Y. Wu for continuous encouragement. References

1. Li, Y.-S., Ma, W.-X., and Zhang, J. E. Darboux transformations of classical Boussinesq system and its new solutions. Phys. Lett. A 275(1-2), 60-66 (2000). 2. Li, Y.-S., and Zhang, J. E. Darboux transformations of classical Boussinesq system and its multi-soliton solutions. Phys. Lett. A 284(6), 253-258 (2001). 3. Li, Y.-S., and Zhang, J. E. Bidirectional soliton solutions of the classical Boussinesq system and AKNS system. Chaos, Solitons and Fractals 16(2), 271-277 (2003). 4. Wu, T. Y., and Zhang, J. E. On modeling nonlinear long waves. In Mathematics is for solving problems, (ed. L. P. Cook, V. Roytburd & M. Tulin), SIAM, 233-241 (1996). 5. Zhang, J. E. /. Run-up of ocean waves on beaches; II. Nonlinear waves in a fluid-filled elastic tube, Ph.D. thesis, California Institute of Technology (1996). 6. Zhang, J. E., and Li, Y.-S. Bidirectional solitons on water. Phys. Rev. E 67, 016306, 1-8 (2003). 7. Zhang, J. E., and Li, Y.-S. Bidirectional solitons on water and its related problems, preprint (2003). 8. Zhang, J. E., and Wu, T. Y. Oblique long waves on beach and induced longshore current. ASCE J. Eng. Mech. 125(7), 812-826 (1999). 9. Zhang, J. E., Wu, T. Y., and Hou, T. Y. Coastal hydrodynamics of ocean waves on beach. Adv. App. Mech. 37, 89-165 (2000).

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A FINITE DIMENSIONAL INTEGRABLE SYSTEM ASSOCIATED W I T H B K K SOLITON EQUATION*

JINSHUN ZHANG Department of Mathematics, Zhengzhou University, Zhengzhou Henan 450052, People's Republic of China

A BKK soliton hierarchy is proposed by means of a 3 x 3 Lenard operator pairs. Through a natural nonlinearization of the BKK eigenvalue problems, a finite-dimensional Hamiltonian system is obtained. Some interesting solutions of the system are also given.

1. Introduction The theory of integrable systems has been an interesting and important problem. The finite dimensional integrable system is used to describe the problems in mathematical physics, mechanics ,..., such as Kovalevskia top, geodesic flows on the ellipsoid harmonic oscillator equation on sphere, Calogero-Moser system.etc. 1-5 The soliton equation as an infinite dimensional integrable system is one of the most prominent subjects in the field of nonlinear science. A fairly satisfactory understanding has been obtained for the soliton equations over recent decades 6 . Several systematic methods has been employed to obtain explicit solutions of soliton equations, such as the inverse scattering method(IST), Backlund-Darboux transformation, the Hiroda's bilinear method 6 ' 7 , the Lie symmetry method 8 , the algebrageometric method 9 . The nonlinearization approach (or constrained flows) of eigenvalue problems or Lax Pairs has been used to seek the relations between the infinite and finite integrable. There are series of finite integrable system obtained by this approach 10 ~ 14 . In this paper, we consider Broer-Kaup-Kupershmidt (BKK) soliton

"This work was supported by National Natural Science Foundation of China (Project 10071075). 149

150

Jinshun

equation

Zhang

15 16 17

- -

:

Uy —

^UXx

ZWUX ~T~ ^Vxi

Vy = \vxx -

2{uv)x.

We start from the Lax Pairs of BKK soliton equation. By using a map a\ : C i-» s/(2,C), a 3 x 3 matrix differential Lenard operator pair and soliton hierarchy with their Lax pairs are deduced easily. In section 3, the Bargmann constraint between the eigenvalues and potential of the Lax Pair is obtained in a natural way by using another map T\ : C 2 •—» C 3 , through which the Lax pair of the soliton equation is nonlinearized into a N-dimensional Hamiltonian system. The conserved integrals {Fm} are obtained by resorting to the generating function .F(A). It is shown that the N-dimensional Hamiltonian system is integrable in Liouville sense. In section 4, we obtained some interesting solutions of the system. 2. B K K Hierarchy Consider BKK spectral problem

17

:

Define a linear map a\ : C 3 — i > sl(2, C): a\(a) =

),

a£C

.

(2)

Here U =
= <JX{(K-XJ)G},

(3)

Where K, J are Lenard operator pairs: / d K=\2vd-2u V-2

1 0

-v \ 0 , d + 2uj

J =

/ - 2 0 2u \ 0 2 -2v , \ 0 0 0 /

d=—.

(4)

dX

oo

Let G = Yl ^ _:? 5j-i> 9j = (5J)5j>5f) T G C3> it' s

easv to

prove the fol-

3=0

lowing proposition: Proposition 2.1: The matrix V = cr\(G) satisfies Lax equation Vx — [U, V] = 0 if and only if Kgj = Jgj+i, Jff-i = 0, j = - 1 , 0 , 1 , . . . .

A Finite Dimensional

Integrable System Associated with BKK Soliton Equation

151

The Lenard gradients {gj} and BKK vector fields {Xj} can been defined recursively by Kgj = Jgj+i, x

j

Jg-i=0,

= J9j = Kgj-u

3 = -1> 0 , 1 , 2 , . . . .

The explicit recursive formula is : 'Sj+i^-sL-flj+Vflj. g1j+1 = \(d + 2u)g*+1, I

3 = 1,2,....

(6)

2

g]+1 = vg*+1 + vg) + \{d-

2u)g .

The first few members are: g-i = (u,v,l)T,g0

= {-\ux

- u2, \vx 3

g\ = (\uxx

+ ^uux - \uv + u -

-uv,-u)T,

\vx,

22

%v ^uvXx -- \{uv) \{uv)x x -- \v\v + u2v, \ux + u2 "'xxxx -- \UV X

I„, 32 = {~\u xxx I„ g t'XXX

\v)T,

2 3„,2 Q„,2„,2 , 3/„,„,\ , 3„,2„ -_ „,„, uuxx _- \u x - 2>u ux + ^(uv)x + 2 « » - w

_ |I„,„. ra

+ \u2Vx

M

^ -_ fuua; 3„„ + j . 3 \uvu -- f3,„,„, (uwx)x x

+ §MV2 - U3V, ~\uxx

- §UUx - W3 +

r;Uv)T.

+ \vx,\vxx

2(uv)x,0)T,

The corresponding vector fields are X0 = {ux,vx,0)T, X2 = {{\uxx

Xi = {-\uxx

+ \uUx

-2uux

+ V? - TiUv)x, (\vxx

-

- § m ; x - jV2

+

3u2v)x,0)T.

(7) Let

AT

GN = (XNG)+ = J2 *N-J9j-i,

VN =

ax(GN).

Then the BKK hierarchy is obtained from the zero-curvature form: —

\v

I = XN «—» UtN - VNtX + [U, VN) = 0.

(8)

Which Lax pairs is
(9)

The first two members are(ti = y, ti = t) BKK

WIS

TT

BKK

11 :

I .

U

Utl

~ ~2Uxx ~ 2uux + ^vx, vti = \vxx - 2(uv)x;

t2=(\uXX

v

+ luUx+V?

t2 = iivxx ~ %uvx - \v* +

~IUVX)X,

(11)

3u'v)x.

152

Jinshun

Zhang

3. The Finite Dimensional Hamiltonian System Let ip — ( C 3 T\i
(12)

It's easy to test the following formula Krxfr)

= XJTX(
(13)

Consider N copies of the linear BKK equation (1):

N


^ H ^ r t - X : ; ) ' >-^ - » with distinct eigenvalues A = a\,...,

ajv> where pj =
Tk = {pkqk-Oikql,-pl,ql)T,

k = l,...,N,

Gx = g_x + ^

. A

fc=i

k

• ak

Then we have the Lax matrix:

V(\) = *X(GX) = AX+Y:

^

fc_i

(15)

\ + Qx(p,q) -2 < p,q > -Qx(p,p)\ l + Qx(q,q) -X-Qx(p,q) J N

Where Qx&v) = £ & j= l

J

A

= £ ^n^>

(Vn \V21

V12 -Vn

= dio f l (o 1 > ...,a A r), p =

8=0

(pi,...,Piv)T, and

V9fc

A -2 V1 A

-PkQkJ

Proposition 3.1: The Lax matrix satisfies the following relation: N

VX(X) - [U, V(X)} = ax(J(go

-$>)). fc=i

Proof. By using (3) and (13), we have VX(X) - [U, V(X)} = ax((K =

XJ)Gx)

ax{Kg^+Y,^^) fe=i

= °x(Jgo + E (*-°*>^+g*-*>jT») fe=i AT

=
£ Tfc)). fe=l

(16)

A Finite Dimensional Integrable System Associated with BKK Soliton Equation

153

Prom here we obtain the Bargmann constraint in a natural way: N

.

(17)

The explicit formula can be written as following form by means of (14) K

\v=-2.

'

Then the Spectral problem (1) is nonlinearized into a N-dimensional Hamiltonian system (px = Ap-2q-

p

[qx=p-Aq+q

= - ^ , =

^ ,

where H = ^ < p,p > — < p,Aq > + < p,q >< q,q > . Lemma 3.1: Let A,Be Then

sl(2,C),

A satisfies Lax equation Ax = [A,B].

^-{detA) ax

= 0.

We noticed that V(A) = a\(G\) is a solution of the Lax equation Vx — [U,V] = 0 in the Bargmann constraint, so .F(A) = \detV{\) is invariant along the x flow. Therefore we have the generating function of integrals of Eq. (18): A2

1

°°

F

(2°)

HX) = -detV{X) = --+YJ^km=0

Where F0 = \ < p,p > - < p, Kq > + < p, q >< q,q >= H, Fm = \
Amp >-

771—1

fc=0


> +
Amq >

(21)

< p, Akp > < p,A - - q > < q,Am-k~1q > m k l

Proposition 3.2: {Fm} are integrals of BKK system (18), i.e., {Fm, H}

154

Jinshun

Zhang

Remark 3.1: The 'time' parts of Lax pairs for BKK hierarchy are also nonlinearized into N-dimensional Hamiltonian systems by using (19): Ptrn —

~d^

(22) _

n

Htm

dFm

—

dp

•

It can be proved by means of the generating function that the {Fm} are involutive and independent. Therefore the N-dimensional Hamiltonian system (18) is a complete integrable systems in the Liouville sense. The N involutive in pairs and functionally independent integrals are Fo,F\,...,F^-i1'54. Some Solutions of the System For the Hamiltonian system: (px = Ap-2q\qx=p-hq+ H = -

p

= -^-, = %••

q

~

+

.

we consider a special case. When we take N = l , the system is became: (Px = \p-3pq2, {QX =pXq + q3By using a transformative method, we got three kinds solution. (1) . 9=(

Ae

.i

. 1

j2

P=[

2coshAa; '

2^>

-,1/2/'

A

\3/2.

^coshAa/

;

(2) 1 = \/2fcitanh 1 ; 1 -2A 1 [( 2 A ~ P = \J 2k1ta.nhtT=2Xl^>'i

Al

+ &i tanh £i) cosh £ - / c sinh £ 2 + A coshff],

- fcf tanh £i) coshf + (fci t a n h £ i - Ai)(fcsinh£ + A cosh 0 1 ;

(3) 1

=

V / 2fc 1 tanhg 1 -2Ai[( 2 A ~

Al

P = V / 2fc 1 tanhg 1 ^2AT^ A i - fc? 2

2

+

kl

tanh

tanh

^)

sinh

£ ~ (fccosh£ + Asinh£)],

6 ) sinh£ + (fci tanh £i - Ai)(fccosh£ + Asinh?)],

where k = A — 1,£ = Xx, and Ai is a constant.

A Finite Dimensional Integrable System Associated with BKK Soliton Equation

155

References 1. V. I. Arnold, Mathematical Methods of Classical Mechanics, Spring-Verlag, New York, 1978. 2. L. Haine, Geodesic flow on S0(4) and Abelian surfaces, Math. Ann. 263, 435 (1983). 3. F. Calogero, Lett. Nuovo Cimento 13, 411 (1975). 4. J. Morse, Various aspects of integrable Hamiltonian systems, in: J. Guckenheimer, J. Moser, Sh. Newhouse: Dynamical systems, C. I. M. E. Lecture 1978, Progress in Mathematics 8, Boston 1980, pp. 233-289. 5. J. E. Marsden and T. S. Ratiu, 1994 Introduction to Mechanics and Symmetry (New York: Springer-Verlag). 6. M. J. Ablowitz and P. A. Clarkson, 1991 Solitons, Nonlinear Evolution Equations and the Inverse Scattering (Cambridge: Cambridge University Press). 7. Li Y. S., W. X. Ma, J. E. Zhang, Phys. Lett. A. 275, 60 (2000). 8. P. J. Olver, Applications of Lie Groups to Differential Equations, Second Edition, Graduate Texts in Mathematics, vol. 107, New York: Springer-Verlag, 1993. 9. E. D. Belokolos, A. I. Bobenco, V. Z. Enol'skii, A. R. Its and V. B. Matveev, Algebra-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin, 1994. 10. C. W. Cao and X. G. Geng, Classical Integrable Systems Generated through Nonlinearization of Eigenvalue Problems, Proc. Conf. on Nonlinear Physics Shanghai, 1989, Research Reports in Physics (Berlin: Spring) pp. 66-78, 1989. 11. C. W. Cao, Science in China, A33, 528 (1990). 12. Antonowics M. and Rauch-Wojciechowski S., J. Phys. A: Math. Gen. 24, 5043 (1991). 13. X. G. Geng, C. W. Cao and H. H. Dai, J. Phys. A: Math. Gen. 34, 989 (2000). 14. Y. T. Wu and J. S. Zhang, J. Phys. A: Nath. Gen. 34, 193-210 (2001). 15. D. J. Kaup, Prog. Theor. 54, 396 (1975). 16. Kupershmidt, Commun. Math. Phys. 99, 51 (1985). 17. J. Satsuma, K. Kajiwara, J. Matsukidara and J. Hietarinta, J. Phys. Soc. Jan. 61, 3096 (1992). 18. Y. Cheng and Y. S. Li, Phys. Lett. A 157, 22 (1991).

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T H E NOVEL N-SOLITON SOLUTIONS OF EQUATION FOR SHALLOW WATER WAVES* YI ZHANG Department of Mathematics, Shanghai University, Shanghai 200436, P.R. China Department of Mathematics, Zhejiang Normal University, Jinhua 321004, P.R. China

SHU-FANG DENG AND DENG-YUAN CHEN Department

of Mathematics, Shanghai 200436,

Shanghai University, P.R. China

In this paper, two model equations for shallow water waves are investigated. By using Hirota's bilinear method, the novel N-soliton solutions are obtained. Finally, we analysis the singularities of the novel solution.

1. Introduction In the studies of the soliton equation, the bilinear derivative method first proposed by Hirota provides us with a very powerful tool 1 ' 3 . The key of this method is that a given nonlinear evolution equation must be become of bilinear form by the dependent variable transformation, then the Nsoliton solutions which describes the multiple collisions of solitons can be constructed. Hirota bilinear method has been successfully used to a number of nonlinear evolution equations as nonlinear Schrodinger equation,modified KdV equation, and sine-Gordon equation and so on. Recently, Chen and his collaborators 4,5 ' 6 generalize the Hirota's standard form to get novel N-soliton solutions. In our present letter we try to construct novel multisoliton solutions for two kinds shallow water wave equations following the method of Ref.2. We consider two model equations for shallow water waves as follows: POO

ut - uxxt - 4uut + 2ux \

utdx + ux = 0,

(1)

Jx

and /•OO

ut ~ Uxxt - 3uu t + 3u x / Jx 157

utdx + ux = 0 .

(2)

158

Yi Zhang, Shu-Fang Deng & Deng-Yuan Chen

Equation (1) is referred to as the AKNS equation in Ref.7; while equation (2) is called the Hirota-Satsuma equation 2 . Concerning equation (1) and (2), much research has been conducted 8 ' 9 . In Ref.2, Hirota et al. obtained multi-soliton solutions by bilinear method. Both equations have the same linear dispersion relations and similar nonlinearities, but the solutions of them shows the difference in characteristics. Additional, any scale transformation never reduce equation (2) to equation (1). Furthermore, equation (1) belongs to Lax integrally systems, but equation (2) can not be solved by Inverse Scattering Transformation^.S.T) 2 . This paper is organized as follows. In section 2 we state the results of the explicit construction of new multi-soliton solutions of (1) by the in introduction of additional bilinear equation. In section 3, we give the novel exact N-soliton solutions of equation (2) by our generalized method; Finally in section 4 we briefly analysis the singularities of the solution for equation (1) and (2). 2. Exact JV-soliton Solutions of Equation (1) By the dependent variable transformation u ( M ) = 2(ln/) X I >

(3)

we can write (1) in the bilinear forms Dx[Dx(Dt+Dx-DtD2x)f

. / ] . f2-±Dt(D*f

. / ) . f2+±Dx(DtD3xf

. / ) . f2 = 0, (4)

where D is the well-known operator defined by D?D?f.g={dx-dx.)m(dt-dt>)nf(x,t)gtf,lf)\x=x.,ust,,

(5)

which is not in the bilinear form of / and, as it stands, we can not directly use the standard bilinear method. To overcome this difficulty, we introduce an independent variable y and impose a subsidiary condition Dx(Dy + Dsx)f.f

= 0.

(6)

Using (6), then (4) may be rewritten as \Dx(Dt + DX-

DtD2x) + \Dt{Dy

+ D3x)}f.f

= 0,

where the boundary condition / —> 0 at |a;| —» oo was used.

(7)

The Novel N-Soliton Solutions of Equation for Shallow Water Waves 159

In what follows, we only consider the simultaneous bilinear derivative equation (6) and (7). Expanding the function / as a formal series / =

l

+ e

/(i)+ea/(2)+e3/(3) + -" ,

(8)

and comparing coefficient of same powers of e gives following recursion relations for the /(") Jxy

ftx

' Jxxxx

~ Jtxxx ^ fxx

"'

+ ~^\fty

v"/

+ Jtxxx)

~ 0>

f$ + fxlL = -\Dx(Dy + DDfW.fW, r(2) _ f (2) Jtx Jtxxx*

,(2) l,f(2) Jxx ' 3\Jty

= -\[Dx{Dt

+ DX-

,(2) x < Jtxxx)

,

= -[Dx(Dt

f (3) Jxx

i/,(3) • 3\Jty

+ DX-

'

(11) (12)

DtDl) + \Dt(Dy

+

D*)]fM.fV,

m + f£L = -Dx(Dy + DDfV.fW, /(3) _ f (3) Jtx Jtxxx'

(1")

,(3) ^ Jtxxx)

(13) (14)

DtDl) + \Dt{Dy

+

Dl)]fW.f™,

and so on. Assuming that Z'1^ has the form

fW = Yl'nJe^'

& = Ujt+kjX+pjy+Zj^,

Vj=Oijt+/3jx+jjy+Sj,

(15)

j=i

and the coefficients of above are all real constants, the novel multi-soliton solutions for the equation (1) can be obtained with the assistance of MATHEMATICA. For n = 1, we take fW=me*\

(16)

and by solving (9)-(14), we have

f{2) = -§/'\

(17)

where = - * i , a i = - "'a ' " 0i, 7i = -3fc]f)Si. £n> Pi»-* «—£%?* '- ""

"^Tp-r*

1

(18)

160

Yi Zhang, Shu-Fang Deng & Deng-Yuan

Chen

by the way, we correct the misprinted error of dispersion relation in 2. We may deduce /<"> = 0 ,

(n > 3),

(19)

thus the series (8) truncated, the novel one-soliton solution is given by u(x,t) = 2[ln(l + r?1e«1 - ~^e2^)]xx.

(20)

In the case of n — 2, taking fW=me^+me^,

(21)

then it follows from (9)-(14) that 7

4kf

+[

4kf

fC3 ) = 7

(fc1+/c2)27?l7?2

lf3Uki-k2)\ [ 4fc|(fc 1 +/c 2 ) 4 ' 1

_ 2/3 1 /3|(fc 1 -fc 2 ) 3 Mfci+fe)6 J

r/3i 2 (fci-fe) 4 2/?2/32(fc1-fc2) l Tft+ 4fc?(ifc1+ifc2)4 fc^i+fe)5 f(4) J

^P\kl+k2fm

/3i/3 2 (fci — fe2) 16ife?]fe|(fci + A ; 2 ) 8

3

£l+2f2

f,+2£l Je

'

2£l+2£2

(23)

(24)

/<"> = 0, (n > 5),

(25)

here the relation between /32 with Q 2 is similar to (18). This process can be continued to N-soliton though the computation becomes somewhat tedious. Frequently the novel N-soliton solutions of the equation (1) is given with

/= E {n(^) w(w_1) ^Ai+^r (2 - w) ex P (x:^+i: ww^)}' M=0,l,2 j=l

i

j =l

1<3<1

(26)

where B P

jl Bn

-= (fcJ

(kj + ktf

fc

')

(27)

The Novel N-Soliton

Solutions of Equation for Shallow Water Waves

161

3. Novel N-soliton of Equation (2) The same variable transformation (3) make equation (2) into Dx(Dt + Dx-DtD2x)f.f

= 0,

(28)

The novel multi-soliton solutions for the (2) are found by assuming fW has the form n

/(D = J2»&efc, ij = u>jt + kjX + ^°\Vj

= ctjt + PjX + 1},

(29)

We briefly give the main results. The novel one-soliton solution of the equation (2) is obtained from

u(,,t)

S^ll-V'1

= 2Ni + ^ ' ~

(30)

where Wl =

fcTTi'

ai =

jkj^WPu

(31)

A similar calculation shows that the novel two-soliton solution of the equation (2) has the form u = 2[ln(l + fW + fM + /O)

+

/(4) ) ] x x )

(32)

here /(1)=7/1e«1+7?2e^, f(2) _ •> ~~

g?(fc?-3) -2£, _ 12fc 2 (fc 2 -l) e

(33)

gg(fcg-3) c 2 £ , 12fc^(fc^-l) e

r(fci-fc 2 ) 2 (-3+fc?+fcl-fc 1 fc 2 ) "•" 4fcl + fe2)2(-3+fcJ+^+feifc2)'/1''2

_ 6ff2fci(fc1-fc2)(6-3fc| +

fcj-5fc?+fcj)

3

(fci+fe2) (-3+fef+fc1fe2+fci)

*•

, 6/31fc2(fc1-fc2)(6-3fc^ + fcf-5fc|+fc^) ^ (fc 2 +fc 2 ) 3 (-3+^ +fc1fc2+ fc|) '' 2 , 6p1p2(.-3+kl-kik2+kl)(,6-5k\+k'i-3kl+ki)-i "• (fei + fc2)4(-3+fcf+fcifc2 + fc|)3 W3) _ r ~~ I

J

1

"~

(OA\

/x

Je

£,+£ 2

^(fci-3)(fc 1 -fc 2 ) 4 (-3+fc?-fc 1 fc 2 +fc^) 2 12fc 2 (fc|-l)(fc 1 +fe 2 ) 4 (-3+fc 2 +fcifc2+ fc|)2 /3 2 /3 1 (fc 2 -3)(fc 2 -fci) 3 (-3+fc 2 -fc 1 fc 2 +fc 2 )(6-5A ; 2 +

2

2

23

fc2(fc -i)(fc1+fc2)5(_3+fe +fcifc2+fc )

fc4-3fc2+fc4)

xi.^+2^

'/I;JC

, r /3 2 (fc 2 -3)(fci-fc 2 ) 4 (-3+fc 2 -fc 1 fc 2 +fc 2 ) 2 ~*~ I 12fc 2 (fc 2 -l)(fci+fe 2 ) 4 (-3+fc 2 + fc1fc2+fc2)2 /? 2 /3 2 (fc 2 -3)(fc 1 -fc 2 ) 3 (-3+fc 2 -fc 1 fc 2 +fc 2 )(6-5fc 2 +fc 4 -3fc 2 +fc 4 ) 1^+25! "^ fci(fe2-l)(fci+fc2)6(-3+fc2+fe1fc2+fc2)3 ^'/2je

(35)

;

162

,(4)

7

Yi Zhang, Shu-Fang Deng & Deng-Yuan

ff/£(fci

=

Chen

- 3)(fc22 - 3)(fci - fc2)8(-3 + k\ - kjhj + fc22)4

lUkfmkf-l)(kl-l)(k1

2?1+2?2

+ k2)s(-3 + kj + k1k2 + k^e

(36) / ( n ) = 0, (for n > 5).

(37)

Generally, the novel iV-soliton solutions of the equation (2) have the structure

/ = E,=o,i,2{n;=1(^-)^^-1)(f^)"il^(/3,^.+c,)«c-«> ex

P(E"=i M J + E I V K J

(3g)

HViAji)},

where . ^ _ (fej ~ fci)2(~3 + A;? + fc? - fcjfci) ~~ (kj + k)H-3 + k) + kf + kjki)'

j

_ _ki_ ~~ k) - 1 '

K

™]

and the first sum is taken over all possible combination of fi = 0,1,2. 4. T h e Singularities of t h e Solution for E q u a t i o n s In what follows, we will discuss the singularity of the novel soliton solution for the shallow water wave equation. When j3j ^ 0, the special singular wave pattern exist in the equation (1) and (2), whose analytic singular solution are derived on the basis of (15). It is obvious that novel one-soliton solution for equation (1) and (2) are different, let alone their novel multi-soliton solutions from (26),(27) and (38), but according to 2 the equations have the same one-soliton solution. We note that the single soliton for the equation (1) and (2) are respectively expressed by (20) and (30). Prom (20), it is easy to find the zero points of / are the points of intersection of the following two movable lines m=c,

^i = l n ( c + W c 2 + | | j - l n / 3 2 + ln(2fc2).

(40)

where c is some constant. This means the singular points must exist due to the different slopes of the two lines in (40) and then the singularity locus of u can be described. Similarly, the singularity of equation (1) may be discussed too. Above-mentioned facts are just consistent with the description in Ref.10, i.e., the poles of the reflection coefficients are at most of first order for a rapidly decreasing solution. It is helpful to study singular solutions of partial differential equations that model nonlinear physical systems 11 .

The Novel N-Soliton Solutions of Equation for Shallow Water Waves 163

References 1. R. Hirota, Phys. Rev. Lett., 27, (1192) (1971). 2. R. Hirota and J. Satsuma, J. Phys. Soc. Jpn., 40, 611 (1976). 3. M. J. Ablowitz and H. Segur, Soliton and the Inverse Scattering Transform SI AM, Philadlphia, 1981. 4. D. Y. Chen, S. F. Deng and D. J. Zhang, preprint, Shanghai University, 2000. 5. D. Y. Chen, D. J. Zhang, S. F. Deng, J. Phys. Soc. Jan., 7 1 , 658 (2002). 6. S. F. Deng and D. Y. Chen, J. Phys. Soc. Jan., 70, 3174 (2001). 7. M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Studies in Appl. Math, 53, 249 (1974). 8. X. B. Hu, Y. Li, J. Phys. A: Math. Gen., 24, 1979 (1991). 9. P. A. Clarkson and E. L. Mansfield, Nonlinearity, 7, 975 (1994). 10. M. Wadati and K. Ohkuma, J. Phys. Soc. Jpn., 51,2029 (1982). 11. M. Kovalyov, Appl. Math. Lett., 9, 89(1996).

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A NEW INTEGRABLE HIERARCHY AND ITS EXPANSIVE INTEGRABLE MODEL*

YUFENG ZHANG Institute of Computational Mathematics, Academy of Mathematics and Systems Sciences, Academia Sinica, Beijing 100080, P.R. China School of Information Science and Eng., Shandong University of Science and Technology, Taian 271019, P.R. China

QINGYOU YAN Department of Economics and Statistics, Shandong Finance Institute, Jinan 250014, P.R. China Center of Advanced Design Technology, Dalian University, Dalian 116622, P.R. China

A new isospectral problem, which is an expanding form of the KN spectral problem, is designed. Using Tu-model leads to a new integrable hierarchy with bi-Hamiltonian structure. Then, a new loop algebra G, which is a subalgebra of loop algebra A2, is constructed. It follows that a new isospectral problem with multi-potentials is designed from G. Again using Tu-model obtains expanding integrable system of the derived system above.

1. I n t r o d u c t i o n As we know, searching for integrable systems has been an important topic in soliton theory. At present, current method for obtaining integrable sys1

tems is Tu-model . Its key step lies in choosing a properly spectral problem and solving stationary zero-curvature equation Vx = [U, V] for V. For which, the powers of A in U are not too much higher and too lower are demanded so t h a t the solutions to the stationary zero-curvature equation 2

can be expressed by cyclic operators . Therefore, a dealing approach reads t h a t seeking subalgebras of loop algebra A\ assures the power derivations of basis elements more t h a n 1. For the sake, a subalgebra of t h e loop algebra "This work supported by the NSF of China (50275013, G60174037). 165

166 Yufeng Zhang & Qingyou Yan Ax with power deviations being 2 is constructed first. Then a new isospectral problem is designed, from which a new integrable hierarchy is derived, which possesses bi-Hamiltonian structure. Furthermore, a new loop algebra G, which is a subalgebra of loop algebra JL 2 , is constructed. It follows that an expanding integrable model is derived from G by using Tu-model. 2. A N e w Integrable Hierarchy Choosing a loop algebra A\ with the basis as follows hi \

i (X2n

°

\

/ ^

i (

°

X2n

~'\

[h(m),e±(n)] = e^(m + n), [e_(m),e+(n)] = h(m + n — 1), , degh(n) — 2n, dege±(n) = 2n — 1.

(1)

Consider the isospectral problem tfx = U9,U = h(l) + \{r2 - q2)h(0) + qe+(0) + re_(0).

(2)

Let V = J2 {amh(—m) + bme+(—m) +c m e_(—m)), solving the stationary m>0

zero-curvature equation VX = [U,V\

(3)

yields the recursive relations ' bm+i = Cmx - \{r2 - q2)bm + qam, c m +i = bmx - \(r2 - q2)cm + ram, Oimx

=

—qCm +

T0m,

CLQ = a = const., bo = CQ = 0, ai = 0, hi = ag, c\ = ar, a2 = f (r 2 -
- q2)aq, (4)

Let n K
then (3) may be written as

- ^

+ [^, ^| n) ] = V™ - [U, v[n)j.

(5)

It is easy to find that the terms in the left-hand side of (5) are of degree > 0, while the terms of the right-hand aide of (5) are of degree < 1, therefore,

A New Integrable Hierarchy and its Expansive Integrable Model

167

both sides of (5) are of degree 0,1. Thus, we have -Vg> + [U,vin)] = - 6 n + i e _ ( 0 ) - c„ + 1 e + (0). Taking V ' n ' = V+ , from the zero-curvature equation Ut ~ V£n) + [U, V(n)] = 0

(6)

gives rise to the Lax integrable hierarchy

-ft)-Co 1 ) (-t.)-* (-:;)•

p

>

where J\ is Hamiltonian operator. Prom the recursive relation (4), we find that b

n

+

* \

= L l

( b n \

K-Cn+lJ

where L\ =

(g)

\-C,

l I'af)qd~lrr - iif; ( r 2 - g2)

-fl + gfl- 1 ? i(r2-g2)+r5-1g>/

a + rfl-V

Therefore, (7) can be written as

i

In order to apply the trace identity , we need to introduce new variables

\Gn

Again let V — i I , ., . * \A(b — c)

/

-a

y-Cn

+

ran^J

1, a direct calculation leads to J

f (V, %) = t r ( V f )i(A 2 6 - qa), (V, f ) = i ( - A 2 c + ro), \(V,%)

= ±\{2a +

qb-rc).

Substitution of (9) into the trace identity 6 ,„dU, ._„ d,„{(V,%)\ : M u r > = * ~ 7 U T (vA ^ yM / _

Su" ' a \ '

aA

yields that £X(2a + qb - re) = A ^ ( _

w >—>

A

^

+

^

7

).

168

Yufeng Zhang & Qingyou

Yan

Comparing the coefficients of A (-2n + 2 + 7)G„.

2n+1

obtains that j^{2an

+ qbn — rcn) =

Substituting the initial values in (4) gives 7 = 0. Therefore,

Gn = - £ , where Hn =

(12)

2an+

£%~TCn is Hamiltonian operator.

Since Gn = N ( bn ), where N = ( * ~' *d, \—cnJ \ rd V

bn ^ —c„ /

N*Gn =

V

, _ 9 ^ ?l ^ , thus we have 1 + ro qJ

N*5Hn du

Therefore, the hierarchy of nonlinear equations (9) possesses the following bi-Hamiltonian structure

"--ft)-^-*^

(13)

where J = JxL^N^K = J^N'Li. As reduction cases of (9) or (13), taking n = l,a = 2, we have trivial equations j Qu = 2qx, \ rtl = 2rx. When n = 2, a = 2, a new generalized Schrodinger equations is presented as follows f Qt2 = 2rxx - (q(r2 - q2))x - r(r2 - q2), \ rt2 = 2qxx - {r(r2 - q2))x - q(r2 - q2). Following the method of Ref.3, we may prove the hierarchy (9) is integ r a t e Hamiltonian system in the Liouville sense.

A New Integrable Hierarchy and its Expansive Integrable Model

169

3. A N e w Loop Algebra and Expanding Integrable Model of the Integrable System (9) or (13) In terms of loop algebra (1), we construct a subalgebra of loop algebra A2 0 A 2 "" 1 A2" 0 0 \ 2 2n 0 - A 0 ,e 2 (n) = ± A "" 1 0 ei(n) = 0 0 0 0 0/ 0 A 2 "" 1 0N < e3(n) = | -A 2 ™ -1 0 0 0 0 0, ei(m),e 2 (n)] = e 3 (m + n), [ei(m),e 3 (n)] = e2(m +

0N 0 0, (14) n),

. [e 3 (m),e 2 (n)] = ei(m + n - 1 ) . We find that ei(n), e 2 (n), e 3 (n) have the common communicative relations. Therefore, the integrable hierarchy (9) is also obtained by the above loop algebra and Tu-model. Now we expand (14) into a new loop algebra G as follows

(15) [ex{m),e2{n)

= e 3 (m + n),[ei(m),e 3 (n)] = e2(m + n),

[e 3 (m),e 2 (n) = ei(m + n — 1), [ei(ra), e 4 (n)] = \e±{m + n), [e 1 (m),e 5 (n) = -\es(rn

+ n), [e 2 (m), e 4 (n)] = \e<s(m + n),

[e 2 (ro),e 5 (n) = \e±(m-\-n-

1), [e 3 (m),e 4 (n)] = -^e5(m

+ n),

[e 3 (m),e 5 (n) = i e 4 ( m + n - 1), [e 4 (m), e 5 (n)] = 0, degei(n) = dege4(n) = 2n,degej(n) = 2n— l , i ^ 1,4. Let Gi = span{e!(n),e 2 (n),e 3 (n)},G 2 = span{e 4 (n),e 5 (n)} be subalgebras of G, then it is easy to find that Gx+G2 = G,G1^A1, where = denotes isomorphic relation.

[GUG2] c G 2 ,

(16)

170

Yufeng Zhang & Qingyou Yan

R e m a r k 3 . 1 : Here the loop algebra G is different from the loop algebra in the Refs.[4, 5, 6, 7]. Obviously, in terms of (15) and Tu-model, new expanding integrable models m a y be obtained. It is worth illustration t h a t here expanding integrable models implies integrable couplings of hierarchies of evolution equations. Taking an isospectral problem 0

+ cme3(-m)

+ dme4(-m) +

fme5(-m)),

solving t h e auxiliary equation VX = [U,V]

(18)

gives rise t o amx = -u\cm

+ U2bm, cm+i

= bmx - | ( u | - u\)cm

+

u2am,

"m+l — cmx ~ 2\U2 ~ ul)"m + ulami dm+i = 2dmx - \{v% - u\)dm - uifm - u2fm + u3am + Ui(bm + c m ) , fm+1 = -Ifmx - 3(^2 _ ul)fm + Uldm - U2dm - U3bm + U3Cm + U±am, bo = Co = d0 = /o = 0, a 0 = 0, ci = fiu2, h = ftm, a i = 0, (19) k /1 = 0u4, di = P(u3 + U1U4). Let V+

n ~ 5 Z ( a ™ e i ( n ~m)

+ bme2(n

- m ) + cme3(n

-

m)

m=0 + d m e 4 ( n - m) + fme5(n y{n)

=

X2ny

- m)),

_y(n)^

then (18) can be written as -V™

+ [U, Vln)] = V<£ - [U, Vin)}.

(20)

T h e terms in the left-hand side of (20) are of degree > 0, the terms of the right-hand side are of degree < 1; thus b o t h sides of (20) are of degree 0 , 1 . Therefore, -V™

+ [U, Vin)] = - [ c n + 1 e 2 ( 0 ) + 6 „ + 1 e 3 ( 0 ) + ( ^

|/„+ieB(0)

+ f / » + i - y ( 6 n + i + c„+1))e4(0)].

A New Integrable Hierarchy and its Expansive Integrable Model 171

Taking V = V^\ then zero-curvature equation Ut - K ( n ) + [U, V] = 0 determines the Lax integrable hierarchy as follows /ui\ Ut

(

cn+i

U2

=

V /

\

bn+i dn + l I " a / n + 1 _ (6n + l+CTi+l)"4 1 "r 2 2 /n +l 2

0 - 1 0 0 \ / &n+l \ 1 0 0 0 —Cn+1 2

2

4

J

4

V 2/n+l /

V o o o-W

/

/ &n+l \ -Cn+1

(21)

2d n +i V 2/„+i /

In terms of (19), we have / bn \ -c, 2d,

\2/„y where a

—9 4- uid

\

3 + U23

_1

2(u4 + «3a

_1

«2 M2)

2( — U3 + U 4 9 - 1 « 2 )

ui

0

0

^(«2 - u ? ) + u2d'1ui

0

2(-u4 + u 3 a _1 ui)

2a-|(u2

2( —U3 + U4C?- Ul)

«1

0 •«?)

— U\ — U2

-28-

A(u\-u\)j

Thus, (21) can be rewritten 0ui ut =

U2

n

JL

2/3(tt3 + mui)

U3

\ujt

\

(22)

2f3uA

Comparing the constructions of J i , L i in (9) with those of J, L in (22), in terms of definition of integrable couplings , concludes that (22) is the integrable coupling of the system (9), i.e. (22) is a kind of integrable expansive model of the hierarchy (9). Remark 3.2: In this paper, we again present integrable coupling of a new hierarchy of evolution equations. But this method is different from the approaches in Refs. 5, 6, 7. Specially, they are different loop algebras. Therefore, using various loop algebras maybe obtain different integrable couplings. One open problem remains. How do we construct multi-Hamiltonian

172

Yufeng Zhang & Qingyou

Yan

structures, infinite conserved laws of integrable couplings? It is worth while studying in the future. Acknowledgement The first author is very thankful to professor Hu Xingbiao, professor Guo Fukui and Dr. Fan Engui for their enthusiastic guidance and help. References 1. 2. 3. 4. 5. 6.

Guizhang Tu, J. Math. Phys. 1989, 30(2): 330-338. Guo Fukui, Acta Math. Phys., 1999, 19(5): 507-512. Engui Fan, J. Phys. A: Math. Gen. 34, 2001: 513-519. Yufeng Zhang and Hongqing Zhang, J. Math. Phys. 2002, 43(1): 466-472. Yufeng Zhang etc., Phys. Lett. A 2002, 299: 453-548. Yufeng Zhang and Hongqing Zhang, J. Math. Research and exposition, 2002, 21(2): 289-294. 7. Guo Fukui and Zhang Yufeng, Acta Physica Sinica, 2002, 51(5): 951-954. 8. M. Lakshmanan and K. M. Tamizhani, J. Math. Phys. 1985, 26: 1189-1200. 9. K. M. Tamizhmani and M. Lakshmanan, J. Phys. A: Math. Gen., 1983, 16: 3773-3762.

E X T R E M A L F U N C T I O N S OF S O B O L E V - P O I N C A R E INEQUALITY MEIJUN ZHU Department of Mathematics University of Oklahoma Norman, Oklahoma 73019, USA

1. Introduction Let (Mn,g) be a n-dimensional compact Riemannian manifold without boundary. For any p £ [ l , n ) , w e consider the Sobolev quotient ( J M . \u\v>dvg)P/V'' where p* = np/(n — p) is the Sobolev conjugate of p. Under certain constraint u € C, we ask the following Basic question: Is

inf Fp(u)

achieved

by

some

uo€C?

The constraints shall naturally arise from some geometric or physical problems. For instance, typical constraints are the follows: 1). Constraint on the average. C := {u £ W1'P(M2) : JMiu = 0}; Or more general, there is a function /(s) such that C := {u G WX'P(M2) : 2). Constraint on the weighted average. For some function k(x) € C{Mn), C:= { « £ W1>P{M2) : JM2 k{x)u = 0}; Or more general, there is a function f(s) such that C : = { M € W^{M2) : JM2 k(x)f(u) = 0}. The motivation to study this type of problems is twofold. First of all, from analytic point of view, this is a interesting and challenging question. Since p« is the critical Sobolev exponent, the compactness of a minimizing sequence is always a difficult issue to be considered. At the same time, the functions of a minimizing sequence usually change sign. Secondly, such a question is closely related to some geometric problems, for example, the Poincare's isoperimetric inequality problem and the estimates of the length of shortest closed geodesic. We shall describe more about this in the following. 173

174

Meijun

Zhu

In his famous paper 14 published in 1905, Poincare suggested that the shortest simple closed geodesic curve on a two dimensional convex surface (with nonnegative Gaussian curvature, so it is not necessary an ovaloid which often refers to a convex surface with strict positive Gaussian curvature) could be found by minimizing the arclength functional among all simple closed curves that bisect the total curvature. Let us consider a two dimensional convex surface M2. We define £ge = { all simple closed geodesic curves on M 2 } , and C = { all simple closed curves on M

which bisect total

curvature}.

2

Denote the geodesic curvature of any curve I on M by kg(x). If / S £ge, then ft kg — 0. Thus, from Gauss-Bonnet theorem we know that Cge C C. To find the shortest simple closed geodesic curves on M2, it is natural to find a curve Zo which achieves the following infimum: \lo\ = inf |Z|. iecge However, it was unclear back in 1905 how to show that Cge is not empty (notice that the fundamental group 7ri(M 2 ) is trivial). The ingenious and bold conjecture of Poincare actually indicates that one can find the same minimizer but in a much larger set:

N = inf|/| ! This in turn would imply that Cge is not empty. The conjecture was studied, for example, by Berger and Bombieri 6 (see, also, Berger 5 ) , and was proved by Croke 8 in 1982 for any smooth two dimensional ovaloids by fairly elementary arguments using space of piecewise geodesic curves. Roughly speaking, Berger and Bombieri's argument in 6 is based on the study of the extremal function of inf Fi(u) for u G C, where C consists of all function v € BV(M2) such that v = \Mi — XM2 f° r t w o C 1 smooth subsets Mi, M 2 = M 2 \ M i C M 2 , and JM2 k(x)v = 0, where k(x) is the Gaussian curvature of M2. Poincare's idea prompts us to ask whether the extremal function of inf F\(u) still exists if we take the infimum in a large set. For instance, we ask: is inf u€BV(M2),

JM2

FAu)

(2)

k(x)u=0

achieved? If the answer is yes, whether the extremal function is a simple function whose nodal line is a geodesic curve.

Extremal Functions of Sobolev-Poincari Inequality 175

The advantage to study (2) is that it may relate Poincare's isoperimetric inequality to the problem of Gromov on the estimates of the length of the shortest simple closed geodesic. Recall Gromov question: For a compact n-dimensional Riemannian manifold M n , let L(Mn) be the shortest length of simple closed geodesic curves and V(Mn) be its volume. Gromov 12 asked whether there is a universal constant C(n) depending only on n, such that V{Mn)

~

v

v ;

'

Gromov's question is related to some early results of Loewner 13 (see, for example, Berger 4 ) and Pu 15 , and later was studied by Treibergs 17 , Croke 9 , Rotman 16 and many others. In particular, Croke 9 proved (3) for any two spheres with C(n) = 31 2 ; He also guessed that the optimal constant (the lowest upper bound) might be 2»3 1 / ' 4 . It is now standard to verify that inf u S B V ( M 2 ) , fM2k(.x)u=0

Fi(u) < lim 0 _ >1+ P

inf

FJu) = 2y/n.

u € B V ( M 2 ) , JM2

k(x)u=0

Thus, if we can show that the nodal line of the extremal function of inf F\ (u) is a geodesic curve, we can conclude that

Notice that T/TT < 2»3 1 / 4 , and y/n is achieved by the standard sphere S2. General speaking, it is not easy to prove the existence of extremal functions, as one can recall the resolution of the Yamabe problem. Here I shall report some of our recent results obtained in 18. When the underline manifold is the standard sphere 5 " , we define the Sobolev-Poincare quotient by

/„(„) := where uA = ^ ^

(f

Ss.W

(5)

Js„ u. We have

Theorem 1.1: If 1 < p < (1 + y/l + 8n)/4), inf{/ p (u) : u £ C 1 ( S " ) \ { 0 } } is achieved.

then Pi{Sn,p,n)

:=

The main idea in the proof is to show that there is a minimizing sequence which strongly converges (in Lp" sense) to a nonzero function. If one can

176

Meijun

Zhu

prove that f IVulp ,/»,,"'

1 n

Pl(S ,p,n)<—i—=

inf

,

(6)

the convergence of a minimizing sequence will follow from some standard arguments. However, for general p such a strict inequality may not be true. For example, from Bernstein inequality on S2 one can show that Is2 | V M | inf = -^— = 2 ^ n/{n 1) in 1)/n ueBV(s*)\{o}(fs2\u-uA\ - ) k{2,1) Nevertheless, by choosing a suitable (but standard) test function we can show the following

Theorem 1.2: Let (Mn,g) be a n-dimensional compact Riemannian manifold without boundary embedded in Rn+1. If p £ [2n/(n + 1), (1 + y/1 + 8n)/4)) for n > 2 or p e (n/(n - 1), (1 + VI + 8n)/4)) for n > 4, then

P 7 (^ n ,P,n):= n

^

^ " |V^| P

< —1—,

;

«ec~\{o}(/Mn|u-Wa|P.)p/p* and the infimum is achieved.

(7)

/c p (n,p)'

The case of p € [2n/(n + 1), (1 + v T + 8 n ) / 4 ) ) in Theorem 1.1 is obviously covered by the above theorem. It is unclear whether (7) is still true for p > (1 + y/1 + 8n)/4) or not. On the other hand, one might be able to check that if n > 2 inequality (7) holds for p = 1, thus it is true for p e [1,1 + <$o) for some positive number SQ. Unfortunately, we have no information about this 6Q. We may guess that the strict inequality (7) holds for all p € [l,n) and n > 2 except the case of p = 1 and n = 2. Even though we do not know whether (7) holds for general p or not, however, if we return to the special manifold Sn and change the constraint on u slightly, we obtain a nice result for all p £ ( l , n ) . Theorem 1.3: If 1 < p < n, then inf

,/g"

|V

"|P/

:=

Pn(Sn,p,n),

is achieved. Based on the symmetrization result on Sn (see, for example, Baernstein 2 ), we can assume that there is a minimizing sequence depending only on one variable. Amazingly, the case of p < 2n/(n + 1) in Theorem

Extremal Functions of Sobolev-Poincard

Inequality

177

1.1 can be handled in the same spirit in the proof of Theorem 1.3 (note that this upper bound of p matches the lower bound of p in Theorem 1.2 perfectly). 2. Outline of the Proofs Proof of Theorem 1.3. For a small positive parameter 0 < e < < 1, we define qe = p» — e, and S£ := inf u ec 1 (S n )\{o},/ 5 „ |u|«=-2«=o Je(u) - inf S„n IV"I" —

m I

«eCl(S")\{0},/

s n

|„|9«-2u=0

(8)

(/sn|u|*e)P/««-

Let (ai,...,a n _2,0) be the spherical coordinates of 5™, where 0 < a, < 27r, — § < # < § • Standard variational method shows that inf J e (u) is attained. Further, the symmetrization argument (see, e.g. Baerstein 2 ) yields that the extremal function ue(x) only depends on 9 and is a monotonically non-decreasing function of 9. We can normalize uc such that /

|«e|*=l.

(9)

Thus, ue satisfies the following equation: rV(|VUer2Vw£) + S e K | ^ - V = 0 i n

5"

\ du£(9)/d6 > 0. If ||U £ ||L°° < C, then from the elliptic estimates ( see, for example, 10 ) we know that ||u e || c i,a < C for some a G (0,1), and conclude that up to a subsequence of e, we —* uo in C 1 , Q as e —> 0, where uo is the minimizer of Jo. Hence theorem 1.3 is proved. So we shall focus on ruling out the case ll^ellL00 —> oo

as

e —* 0.

(11)

We denote 0e as the zero point of ue, and assume that 0e —» QQ (up to a subsequence of e). Further, without loss of generality we can assume that

J u£(0) < 0 for - f < 6 < 6C \ue(6) and ue(n/2) = ma.xue(9) =

>0

for 6e <6<

f,

||U£||L~-

Proposition 2.1: Given S > 0, for any 9 < n/2 — 5, ue{9) —> 0

uniformly

as

e —» 0.

178

Meijun Zhu

Proof. The proof can be carried out using energy dependent blow up analysis. We refer readers to, for example, n for more details. Now, for any r € (—7r/2,7r/2), we define /x£ = max ee [_ w / 2 , T ] |u e (0)|. Prom Proposition 2.1 we know that fi£ —* 0 as e —> 0. Let ve{9) = —ue(9)/fie. Then v€ satisfies: ( V(\Vve\P-2Vvc)

+ S^-Py^'1

\ 0 < |u£| < 1 for -

TT/2

=0in

<6<

5" n

< 9 < r}

{-TT/2

T.

It follows from the standard elliptic theory that ve —+ i>o in any compact set of {a; = (ai,...,an-i,6) £ Sn : —n/2 < 9 < r } , where VQ satisfies: fV(|Vuo| p_ " 2 Vuo) = 0

\ max \v0(9)\ = 1,

for

-TT/2<6>
0 < \v0\ < 1 for - TT/2 < 9 < r.

In both cases (r > #o or r < ^o) we obtain a contradiction due to the maximum principle! Proof of Theorem 1.1. Let us first establish Theorem 1.1 in the case of p < 2n/(n + 1 ) . We follow the main stream in the proof of Theorem 1.3. For a small positive parameter 0 < e < < 1, we define qe = p» — e, and Ze:=

inf u€Cg°\{0},

Ie(u):= / s „ «=0

inf

, rJsn.\

« S C 0 - \ { 0 } , / s „ U = 0 (JSn

'

. (13)

|u|9«)P/9.

Standard variational method and the symmetrization argument show that inf Ie (u) is attained by ue(x) which depends only on 9 and is a monotonically non-decreasing function of 9. We normalize ut such that ue\q< = 1. (14) / .S" If | | U £ | | L ~ < oo, we are done. Otherwise, we assume, up to some subsequence of e, that ||ue||z,°= —> oo

as e —+ 0.

Let 6e be the zero point of ue and denote ce = Js„ \u£\q'~2ue. Up to a further subsequence of e, we can assume that 9e —> #o- It follows easily from Holder inequality and (14) that |Ce| < C. Without loss of generality, we can assume that f uA9) < 0 for - 2f _< 9 < 9t { W \ue{9) > 0 for 9e < 9< f,

(15

Extremal Functions of Sobolev-Poincari

and U£(TT/2) — maxue(9) = given by:

||U£||L°°.

Inequality

179

The Euler-Lagrange equation of ue is

f V(|Vu e |P- 2 Vu £ ) + Z £ |u £ |««- 2 u £ - Z£c£ = 0 in

Sn

\ due(6)/d6 > 0. Easy to see that \Z€ct\ < C. We define \ie = — uc(—f). As in the proof of Proposition 2.1, one can show that /i £ —> 0. If 0o > —7r/2, we let v£ = -u€/fie. Then v£ satisfies f V(|Vve[p~2V«e) + Z e /4*" p w?' _1 - ^ c e / 4 ~ p = 0 in \due(6>)/d6>>0, 0 < v e < l f o r -7r/2<6><6» £ .

Sn n

{-TT/2

< 0 < *«} (17)

For any fixed p < 2n/(n + 1 ) , we choose positive constants s and £ such that (fc-l)t + s = g £ - l ,

and

fe

~ 1)* +

g = L

^ lg ^

Then from Holder inequality, we obtain:

<(/ s ju £ n^.(/ s t > £ |r = C. ( / ' ; / 2 |u £ |) s

(using

/ g n « = 0)

0. Thus it follows from the standard elliptic theory that v£ —• VQ in C 1,Q (ii') for any compact set K of {x = («i, ...,a n _i,#) € 5 " : —7r/2 < 8 < 8o}, where vo satisfies: J V(|Vuo| p_2 Vw 0 ) = 0 \ WO(-TT/2) = max_ 7r/2 < 0 <e o M # ) | = 1.

for - TT/2 < 0 < 0O 0 < u0 < 1 for -n/2<9< 0O.

Contradiction to the maximum principle! If #o = — §, then for sufficiently small e, 0£ < 0, thus uc(6) > 0 for 0 > 0. Therefore JSnnr_n/2<e<w4} u e ^ 0- ^ follows that |u £ (-7r/2)|.(9 e +~) >C [ •<

JS"n{0 e <0<7r/4}

ue>C

[ JSnn{0<9
ue> Cu e (0).

This yields that |ue(—7r/2)| > u £ (0) for sufficiently small e. We then consider ve = —ue//j,e in the lower hemisphere and obtain ve —> VQ in C 1 ' Q (A') for

180

Meijun Zhu

any compact set K of the lower hemisphere, where VQ satisfies: j V(|Vi;o| p ~ 2 Vvo) = 0

for - n/2 < 6 < 0

[t>o(-7r/2) = max_ 7r/2 <0 2n(n+ 1), we only need to prove Theorem 1.2. Proof of Theorem 1.2. (7) can be checked out by choosing some suitable test functions and tedious computations. We refer readers to 18 for more details. Here we prove Theorem 1.2 under the assumption that (7) holds. We quote the following lemma from Aubin's book 1. Lemma 2 . 1 : Given S > 0, there is a constant C(S) such that \u\p'dvg)p'p'

(/

< (kp(n,p) +6) [

\Vgu\pdvg + C(S) f

\u\pdvg

£Wl>p{Mn).

holds for all u

Let {u^"1'} be a minimizing sequence with JMn u^dvg = 0 and ||w ||p.,Mn = (/A*- |«(TB)l"*dwfl)1/p* = I- Clearly, \\u^\\m,P{Mn) < C. m After passing to a subsequence, u^ ' converges weakly to some u e Wl'p{Mn), and u ' m ' converges strongly to u in Lq for any q < p». Thus J M „ udvg = 0. Due to Brezis-Lieb's lemma 7 , it is not difficult to see that (m)

f

( | U ( " 0 | P . _ | u (»0 _ u\P-)dvg = f

\u\p'dvg + o(l),

and consequently f

(|U(">)|P- _ |u(™) _ u\p')dvg

< 1 + o(l),

/

\u\p'dvg < 1.

where o(l) denotes some quantity tending to zero as m tends to co. Choose 6 such that kP^P)+5 ~ Pi{MniP->n) ^ Therefore, by the Sobolev embedding theorem and Lemma 2.1, we have, for P / (M»,p,n) = JMn |V9u<m>|* + o(l) = JM» |V 9 (« (m) - u)\p + JMn |V9u|" + 0(1) =

/M-

I v (« ( m ) - ")IP + X 5 & 7 / « " I u(m) - "l P + / M m

—

n

( / M , \u< > -a\*-Y>'*>' +Pi(M ,p,n)(fMn fcP(n,p)+«WM

|V9«IP

+ o(l)

&>•)'/*' + o ( l )

> kHn,P)+6 fMn W(m) ~ "l p > + P/(M",p,n) J M „fi»>-+ o(l) = ( t f ( „ ; f H r f i ( M n , f , " ) ) / M . I« ( m ) - « I P - + P / ( M » , p , n ) + o(l)

Extremal Functions of Sobolev-Poincari Inequality 181 This yields t h a t fM„ \u^m^ — u| p * = o ( l ) . It follows easily t h a t u is a minimizer. Theorem 1.2 is proved. References 1. Aubin, T., Nonlinear Analysis on Manifolds. Monge-Ampere Equations. Berlin, Springer-Verlag, 1982, Grundlehern math. Wiss 252. 2. Baernstein, A., A unified approach to symmetrization. Partial Differential Equations of Elliptic type. Sympos. MAth. XXXV, edited by A. Alvino, etc. Cambridge Univ. Press, Cambridge, 1994, 47-91. 3. Bernstein, F., Uber die isoperimetrische Eigenschaft des Kreises auf der Kugeloberflache und in der Ebene. Math. Ann. 60, 117-136 (1905). 4. Berger, M., A l'ombre de Loewner. Ann. Sci. Ecole Norm. Sup. 4, No. 2, 241-260 (1972). 5. Berger, M.S., Simple closed geodesies on ovaloid and calculus of variations, Seminar on minimal submanifolds, 261-270, Ann. of Math. Stud., 103, Princeton Univ. Press, Princeton, NJ 1983. 6. Berger, M.S. and Bombieri, E., On the Poincare's isoperimetric problem for simple closed geodesies. Journal of Functional Analysis 42, 274-298 (1981). 7. Brezis, H. and Lieb, E., A relation between pointwise convergence of functions and convergence of functionals. Pro. Amer. Math. Soc. 88, no. 3, 486-490 (1983). 8. Croke, C , Poincare's problem and the length of the shortest closed geodesic on a convex hypersurface. J. Diff. Geom. 17, 595-634 (1982). 9. Croke, C , Area and the length of the shortest closed geodesic. J. Diff. Geom. 27, 1-21 (1988). 10. DiBenedetto, E., C +a local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal, no.8, 827-850 (1983). 11. Ghoussoub,N and Gu,C, and Zhu, M., On a singularly perturbed Neumann problem with the critical exponent. Comm. P.D.Es. 26, no. 11-12, 1929-1946 (2001). 12. Gromov, M., Filling Riemannian Manifolds. J. Diff. Geom. 18, 1-147 (1983). 13. Loewner, C , unpublished manuscript. 14. Poincare, H., Sur les lignes geodesiques des surfaces convexes. Trans. Amer. Math. Soc. 6, 237-274 (1905). 15. Pu, P., Some inequalities in certain non-orientable Riemannian manifolds. Pacific J. Math. 2, 55-71 (1952). 16. Rotman, R., Upper bounds on the length of the shortest closed geodesic on simply connected manifolds. Math. Z. 233, no. 2, 365-398 (2000). 17. Treibergs, A., Estimates of volume by the length of shortest closed geodesies on a convex hypersurface, Invent. Math. 80, 481-488 (1985). 18. Zhu, M., On the extremal functions of Sobolev-Poincare inequality, Submitted.

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LIST OF T H E PARTICIPANTS

Nonlinear Evolution Equations and Dynamical System ICM 2002 Satellite conference at Yellow Mountains, August 15-18, 2002, China Chen Chunli, Department of Physics, Shanghai Jiaotong University, Shanghai, P.R. China, 200030, [email protected] Chen Zuchi, USTC, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, China 230026, [email protected] Chen Guiqiang, Department of Mathematics, Northwestern University, Evanston, IL 60208, USA, [email protected] Cheng Yi, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, China 230026, [email protected] Deng Shufeng, Department of Mathematics, Shanghai University, Shanghai, P.R. China, 200436, [email protected] B. Dubrovin, SISSA-ISAS, Via Beirut 2/4, 34100 Trieste, ITALY, dubrovin@sissa. it L.D. Faddeev, Sankt-Petersburg Steklov Mathematical Institute, Laboratory of Mathematical Problems of Physics 191011, Sankt-Peterburg, Fontanka, 27, Russian, [email protected] Gao Yun, Department of Mathematics and Statistics, York University, Toronto, Canada, M3J 1P3 and USTC, [email protected] Gui Changfeng, Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, USA, [email protected] He Jingsong, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, China 230026, [email protected] Hu Hengchun, Department of Physics, Shanghai Jiaotong University, Shanghai, P.R. China, 200030, [email protected] Hu Sen, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, China 230026, [email protected] Hu Xing-Biao, Academia Sinica, Academy of Mathematical Sciences and Systems Sciences, Institute of Computational Mathematics and Scientific Engineering Computations, State Key Lab Sci and Engn Comp, Beijing, P.R. China, 100080, [email protected] Huang Yi-Zhi, Department of Mathematics, Rutgers University, 110 183

184

List of the

Participants

Frelinghuysen Road, Piscataway, NJ 08854-8019 USA, yzhuang@math. rutgers.edu Huang Zhenghong, Department of Computer Science, Chong Qing Technology and Business University, Chong Qing, P.R. China, [email protected] Ji Xiaoda, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, China 230026, [email protected] V. Kac, Department of Mathematics, MIT, Cambridge, MA 02139 USA, [email protected] K. Khanin, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK, kk262@newton cam.ac.uk Jungseob Lee, Department of Mathematics, Ajou Univ., Wondcheondong, Paldal-gu, Suwon 442-749, South Korea, [email protected] D. Levi, University Roma Tre, Dipartimento Fis E Amaldi, Via Vasca Navale 84,1-00146 Rome, Italy Univ Roma Tre, Dipartimento Fis E Amaldi, 1-00146 Rome, Italy, [email protected] Li Mengru, Department of Mathematics, Zhengzhou University, Henan, P.R. China, 450052, [email protected] Li Ji, Department of Physics, Shanghai Jiaotong University, Shanghai, P.R. China, 200030, [email protected] Li Zhen, Department of Mathematics, Zhengzhou University, Henan, P.R. China, 450052 Li Yinghua, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, China 230026, [email protected] Li Yishen, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, China 230026, [email protected] Liu Jin, Department of Mathematics, Shanghai University, Shanghai, P.R. China, 200436, [email protected] Lu Yunguang, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, China 230026, [email protected] Ma Hongcai, Department of Physics, Shanghai Jiaotong University, Shanghai, P.R. China, 200030, [email protected] Peng Jiaqui, Graduate School of the Chinese Academy of Sciences, Chinese Academy of Sciences, Yuquan Road, Beijing Tang Xiaoyan, Department of Physics, Shanghai Jiaotong University, Shanghai, P.R. China, 200030, [email protected] Wang Hongye, Department of Mathematics, Zhengzhou University, Henan, P.R. China, 450052, [email protected] Wei Shishu, Department of Mathematics Physical Sciences Center,

List of the Participants

185

601 Elm Avenue, University of Oklahoma Norman, Oklahoma 73019 [email protected] Xie Weiqing, Department of Mathematics and Statistics, College of Science, California State Polytechnic University, 3801 West Temple Avenue, Pomona, CA 91768,USA, [email protected] Yan Qingyou, Center of Advanced Design Technology, Dalian University, Dalian, P.R. China, 116622, [email protected]. Ye Xiangdong, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, China 230026, [email protected] Zhang Da-Jun, Department of Mathematics, Shanghai University, Shanghai, P.R. China, 200436, [email protected] Zhang Jingshun, Department of Mathematics, Zhengzhou University, Henan, P.R. China, 450052, [email protected] Zhang Pu, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, China 230026, [email protected] Zhang Shunli, Department of Physics, Shanghai Jiaotong University, Shanghai, P.R. China, 200030, [email protected], [email protected] Zhang Yi, Department of Mathematics, Shanghai University, Shanghai, P.R. China, 200436, [email protected] Zhang Yufeng, School of Information Science and Engineering, Shangdong University of Science and Technology Tian, P.R. China, 271019, zhangy [email protected] Zhou Zi-Xiang, Institute of Mathematics, Fudan University, Shanghai, P.R. China, 200433, [email protected] Zuo Dafeng, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, China 230026, [email protected] Zheng Zhong, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, China 230026, [email protected] Zhu Meijun, Department of Mathematics Physical Sciences Center, 601 Elm Avenue University of Oklahoma Norman, Oklahoma 73019, [email protected]

onlinodr Evolution tqudtionsand

This book contains the papers presented at the ICM2002 Satellite Conference

on Nonlinear Evolution Equations and Dynamical Systems. About 5 0 mathematicians and scientists attended •ting - including Dubrovin (ICTP), Faddeev (Steklov), Kac (MIT) and Witten (IAS). The book covers several fields, such as nonlinear evolution equations and integrable

systems, infinite-

dimensional algebra, conformal field theory and geometry.

World Scientific www.worldscientific.com 5200hc