kosmann-schwarzbach y., grammaticos b., tamizhmani k.m. (eds.) integrability of nonlinear systems (springer lnp 638, 2004)(342s)

In the sical theory, the tools of Poisson geometry appear in an essential way, whilefor quantum systems, the representation theory of Lie groups and algebras, and clas-of the infinite-dim

GMPIB, Universit´e Paris VII

Tour 24-14, 5eétage, case 7021

2 place Jussieu

75251 Paris, France

K M TamizhmaniDepartment of MathematicsPondicherry UniversityKalapet

Pondicherry 605 014, India

Y Kosmann-Schwarzbach, B Grammaticos, K M Tamizhmani (eds.), Integrability of

Nonlinear Systems, Lect Notes Phys 638 (Springer-Verlag Berlin Heidelberg 2004), DOI

10.1007/b94605

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This second edition of Integrability of Nonlinear Systems is both streamlined and

revised The eight courses that compose this volume present a comprehensivesurvey of the various aspects of integrable dynamical systems Another exposi-tory article in the first edition dealt with chaos: for this reason, as well as fortechnical reasons, it is not reprinted here Several texts have been revised andothers have been corrected or have had their bibliography brought up to date.The present edition will be a valuable tool for graduate students and researchers.The first edition of this book, which appeared in 1997 as Lecture Notes inPhysics 495, was the development of the lectures delivered at the InternationalSchool on Nonlinear Systems which was held in Pondicherry (India) in January

1996, organized by CIMPA-Centre International de Math´ematiques Pures etAppliqu´ees/International Center for Pure and Applied Mathematics and Pon-dicherry University In February 2003, another International School was held

in Pondicherry, sponsored by CIMPA, UNESCO and the Pondicherry

Gover-nment, dealing with Discrete Integrable Systems The lectures of that school

are now being edited as a volume in the Lecture Notes in Physics series by

B Grammaticos, Y Kosmann-Schwarzbach and Thamizharasi Tamizhmani, andwill constitute a companion volume to the essays presented here

We are very grateful to the scientific editors of Springer-Verlag, Prof WolfBeiglb¨ock and Dr Christian Caron, who invited us to prepare a new edition

We acknowledge with thanks the renewed editorial advice of Dr Bertram E.Schwarzbach, and we thank Miss Sandra Thoms for her expert help in the pro-duction of the book

The Editors 1

1 Analytic Methods 2

2 Painlev´e Analysis 2

3 τ -functions, Bilinear and Trilinear Forms 2

4 Lie-Algebraic and Group-Theoretical Methods 3

5 Bihamiltonian Structures 3

Nonlinear Waves, Solitons, and IST M.J Ablowitz 5

1 Fundamentals of Waves 5

2 IST for Nonlinear Equations in 1+1 Dimensions 9

3 Scattering and the Inverse Scattering Transform 11

4 IST for 2+1 Equations 19

5 Remarks on Related Problems 24

Integrability – and How to Detect It B Grammaticos, A Ramani 31

1 General Introduction: Who Cares about Integrability? 31

2 Historical Presentation: From Newton to Kruskal 33

3 Towards a Working Definition of Integrability 41

3.1 Complete Integrability 43

3.2 Partial and Constrained Integrability 47

4 Integrability and How to Detect It 48

4.1 Fixed and Movable Singularities 49

4.2 The Ablowitz-Ramani-Segur Algorithm 50

5 Implementing Singularity Analysis: From Painlev´e to ARS and Beyond 54

6 Applications to Finite and Infinite Dimensional Systems 66

6.1 Integrable Differential Systems 66

6.2 Integrable Two-Dimensional Hamiltonian Systems 68

6.3 Infinite-Dimensional Systems 71

7 Integrable Discrete Systems Do Exist! 73

8 Singularity Confinement: The Discrete Painlev´e Property 77

9 Applying the Confinement Method:

Discrete Painlev´e Equations and Other Systems 79

9.1 The Discrete Painlev´e Equations 79

9.2 Multidimensional Lattices and Their Similarity Reductions 86

9.3 Linearizable Mappings 87

10 Discrete/Continuous Systems: Blending Confinement with Singularity Analysis 87

10.1 Integrodifferential Equations of the Benjamin-Ono Type 89

10.2 Multidimensional Discrete/Continuous Systems 90

10.3 Delay-Differential Equations 90

11 Conclusion 90

Introduction to the Hirota Bilinear Method J Hietarinta 95

1 Why the Bilinear Form? 95

2 From Nonlinear to Bilinear 95

2.1 Bilinearization of the KdV Equation 96

2.2 Another Example: The Sasa-Satsuma Equation 97

2.3 Comments 98

3 Constructing Multi-soliton Solutions 99

3.1 The Vacuum, and the One-Soliton Solution 99

3.2 The Two-Soliton Solution 100

3.3 Multi-soliton Solutions 101

4 Searching for Integrable Evolution Equations 101

4.1 KdV 102

4.2 mKdV and sG 102

4.3 nlS 103

Lie Bialgebras, Poisson Lie Groups, and Dressing Transformations Y Kosmann-Schwarzbach 107

Introduction 107

1 Lie Bialgebras 110

1.1 An Example: sl(2,C) 110

1.2 Lie-Algebra Cohomology 111

1.3 Definition of Lie Bialgebras 113

1.4 The Coadjoint Representation 114

1.5 The Dual of a Lie Bialgebra 114

1.6 The Double of a Lie Bialgebra Manin Triples 115

1.7 Examples 116

1.8 Bibliographical Note 119

2 Classical Yang-Baxter Equation and r-Matrices 119

2.1 When Does δr Define a Lie-Bialgebra Structure on g? 119

2.2 The Classical Yang-Baxter Equation 122

2.3 Tensor Notation 126

2.4 R-Matrices and Double Lie Algebras 128

2.5 The Double of a Lie Bialgebra Is a Factorizable Lie Bialgebra 131

2.6 Bibliographical Note 132

3 Poisson Manifolds The Dual of a Lie Algebra Lax Equations 133

3.1 Poisson Manifolds 133

3.2 The Dual of a Lie Algebra 135

3.3 The First Russian Formula 136

3.4 The Traces of Powers of Lax Matrices Are in Involution 138

3.5 Symplectic Leaves and Coadjoint Orbits 139

3.6 Double Lie Algebras and Lax Equations 142

3.7 Solution by Factorization 145

3.8 Bibliographical Note 146

4 Poisson Lie Groups 146

4.1 Multiplicative Tensor Fields on Lie Groups 147

4.2 Poisson Lie Groups and Lie Bialgebras 149

4.3 The Second Russian Formula (Quadratic Brackets) 151

4.4 Examples 151

4.5 The Dual of a Poisson Lie Group 153

4.6 The Double of a Poisson Lie Group 155

4.7 Poisson Actions 155

4.8 Momentum Mapping 158

4.9 Dressing Transformations 158

4.10 Bibliographical Note 162

Appendix 1 The ‘Big Bracket’ and Its Applications 162

Appendix 2 The Poisson Calculus and Its Applications 165

Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: A Review and Extensions of Tests for the Painlev´ e Property M.D Kruskal, N Joshi, R Halburd 175

1 Introduction 175

2 Nonlinear-Regular-Singular Analysis 180

2.1 The Painlev´e Property 182

2.2 The α-Method 185

2.3 The Painlev´e Test 187

2.4 Necessary versus Sufficient Conditions for the Painlev´e Property 191 2.5 A Direct Proof of the Painlev´e Property for ODEs 192

2.6 Rigorous Results for PDEs 194

3 Nonlinear-Irregular-Singular Point Analysis 195

3.1 The Chazy Equation 196

3.2 The Bureau Equation 199

4 Coalescence Limints 201

Eight Lectures on Integrable Systems

F Magri, P Casati, G Falqui, M Pedroni 209

1st Lecture: Bihamiltonian Manifolds 210

2nd Lecture: Marsden–Ratiu Reduction 215

3rd Lecture: Generalized Casimir Functions 220

4th Lecture: Gel’fand–Dickey Manifolds 225

5th Lecture: Gel’fand–Dickey Equations 229

6th Lecture: KP Equations 235

7th Lecture: Poisson–Nijenhuis Manifolds 241

8th Lecture: The Calogero System 247

Bilinear Formalism in Soliton Theory J Satsuma 251

1 Introduction 251

2 Hirota’s Method 252

3 Algebraic Inentities 255

4 Extensions 259

4.1 q-Discrete Toda Equation 259

4.2 Trilinear Formalism 260

4.3 Ultra-discrete Systems 263

5 Concluding Remarks 266

Quantum and Classical Integrable Systems M.A Semenov-Tian-Shansky 269

1 Introduction 269

2 Generalities 271

2.1 Basic Theorem: Linear Case 273

2.2 Two Examples 283

3 Quadratic Case 291

3.1 Abstract Case: Poisson Lie Groups and Factorizable Lie Bialgebras 293

3.2 Duality Theory for Poisson Lie Groups and Twisted Spectral Invariants 296

3.3 Sklyanin Bracket on G (z) 304

4 Quantization 305

4.1 Linear Case 305

4.2 Quadratic Case Quasitriangular Hopf Algebras 319

Mark J Ablowitz

Department of Applied Mathematics,

Campus Box 526,

University of Colorado at Boulder,

Boulder Colorado 80309-0526, USA

Universit´e Paris VII,

Tour 24-14, 5e´etage, case 7021,

20014 Turku, Finlandhietarin@utu.fi

Nalini Joshi

School of Mathematicsand Statistics F07,University of Sydney,NSW 2006, Australianalini@maths.usyd.edu.au

Yvette Kosmann-Schwarzbach

Centre de Math´ematiques,

´Ecole Polytechnique,

91128 Palaiseau, Franceyks@math.polytechnique.fr

Martin D Kruskal

Department of Mathematics,Rutgers University,

New Brunswick NJ 08903, USAkruskal@math.rutgers.edu

Franco Magri

Dipartimento di Matematica

e Applicazioni,Universit`a di Milano-Bicocca,Via degli Arcimboldi 8,

20126 Milano, Italymagri@matapp.unimib.it

The Editors

Nonlinear systems model all but the simplest physical phenomena In the sical theory, the tools of Poisson geometry appear in an essential way, whilefor quantum systems, the representation theory of Lie groups and algebras, and

clas-of the infinite-dimensional loop and Kac-Moody algebras are basic There is aclass of nonlinear systems which are integrable, and the methods of solution forthese systems draw on many fields of mathematics They are the subject of thelectures in this book

There is both a continuous and a discrete version of the theory of integrablesystems In the continuous case, one has to study either systems of ordinarydifferential equations, in which case the tools are those of finite-dimensional dif-ferential geometry, Lie algebras and the Painlev´e test – the prototypical example

is that of the Toda system –, or partial differential equations, in which case thetools are those of infinite-dimensional differential geometry, loop algebras and thegeneralized Painlev´e test – the prototypical examples are the Korteweg-de Vriesequation (KdV), the Kadomtsev-Petviashvilii equation (KP) and the nonlinearSchr¨odinger equation (NLS) In the discrete case there appear discretized ope-rators, which are either differential-difference operators, or difference operators,

and the tools for studying them are those of q-analysis.

At the center of the theory of integrable systems lies the notion of a Lax pair,describing the isospectral deformation of a linear operator, a matrix differentialoperator, usually depending on a parameter, so that the Lax operator takes

values in a loop algebra or a loop group A Lax pair (L, M ) is such that the time

evolution of the Lax operator, ˙L = [L, M ], is equivalent to the given nonlinear system The study of the associated linear problem Lψ = λψ can be carried out

by various methods

In another approach to integrable equations, a given nonlinear system is ten as a Hamiltonian dynamical system with respect to some Hamiltonian struc-ture on the underlying phase-space (For finite-dimensional manifolds, the term

writ-“Poisson structure” is usually preferred, that of “Hamiltonian structure” beingmore frequently applied to the infinite-dimensional case.) For finite-dimensionalHamiltonian systems on a symplectic manifold (a Poisson manifold with a non-

degenerate Poisson tensor) of dimension 2n, integrability in the sense of Liouville (1855) and Arnold (1974) is defined by the requirement that there exist n con-

served quantities that are functionnally independent on a dense open set and

in involution, i e., whose pairwise Poisson brackets vanish Geometric methods

are then applied in various ways

The Editors, Introduction, Lect Notes Phys.638, 1–4 (2004)

http://www.springerlink.com/
Springer-Verlag Berlin Heidelberg 2004c

1 Analytic Methods

The inverse scattering method (ISM), using the inverse scattering transform(IST), is closely related to the Riemann-Hilbert factorization problem and tothe ¯∂ method This is the subject of M.J Ablowitz’s survey, “Nonlinear waves,

solitons and IST”, which treats IST for equations both in one space variable, (1+1)-dimensional problems, and in 2 space variables, (2 + 1)-dimensional problems,and whose last section contains a review of recent work on the self-dual Yang-Mills equations (SDYM) and their reductions to integrable systems

2 Painlev´ e Analysis

In the Painlev´e test for an ordinary differential equation, the time variable iscomplexified If all movable critical points of the solutions are poles, the equationpasses the test It contributes to the determination of the integrability or non-integrability of nonlinear equations, defined in terms of their solvability by means

of an associated linear problem In the Ablowitz-Ramani-Segur method for thedetection of integrability, the various ordinary differential equations that arise

as reductions of a given nonlinear partial differential equation are tested for thePainlev´e property

In their survey, “Analytic and asymptotic methods for nonlinear singularityanalysis”, M.D Kruskal, N Joshi and R Halburd review the Painlev´e propertyand its generalizations, the various methods of singularity analysis, and recentdevelopments concerning irregular singularities and the preservation of the Pain-lev´e property under asymptotic limits

The review by B Grammaticos and A Ramani, “Integrability”, describesthe various definitions of integrability, their comparison and implementationfor both finite- and infinite-dimensional systems, and for both continuous anddiscrete systems, including some recent results obtained in collaboration withK.M Tamizhmani The method of singularity confinement, a discrete equivalent

of the Painlev´e method, is explained and applied to the discrete analogues of thePainlev´e equations

Hirota’s method is the most efficient known for the determination of solitonand multi-soliton solutions of integrable equations Once the equation is writ-

ten in bilinear form in terms of a new dependent variable, the τ -function, and

of Hirota’s bilinear differential operators, multi-soliton solutions of the originalnonlinear equation are obtained by combining soliton solutions J Hietarinta’s

“Introduction to the Hirota bilinear method” is an outline of the method withexamples, while J Satsuma’s “Bilinear formalism in soliton theory” develops

the theory further, treats the bilinear identities satisfied by the τ -functions,

and shows how the method can be generalized to a trilinear formalism valid

for multi-dimensional extensions of the soliton equations, to the q-discrete and

ultra-discrete cases and how it can be applied to the study of cellular automata

When the Poisson brackets of the matrix elements of the Lax matrix, viewed

as linear functions on a Lie algebra of matrices, can be expressed in terms of a

so-called “r-matrix”, the traces of powers of the Lax matrix are in involution,

and in many cases the integrability of the original nonlinear system follows It

turns out that a Lie algebra equipped with an “r-matrix” defining a Poisson

bracket, e.g., satisfying the classical or modified Yang-Baxter equation (CYBE,MYBE) is a special case of a Lie bialgebra, the infinitesimal object associatedwith a Lie group equipped with a Poisson structure compatible with the groupmultiplication, called a Poisson Lie group Poisson Lie groups play a role in thesolution of equations on a 1-dimensional lattice, and they are the ingredients ofthe geometric theory of the dressing transformations for wave functions satisfying

a zero-curvature equation under elements of the “hidden symmetry group” The

quantum version of these objects, quantum R-matrices satisfying the quantum

Yang-Baxter equation (QYBE), and quantum groups are the ingredients of thequantum inverse scattering method (QISM), while the Bethe Ansatz, construc-ting eigenvectors for a quantum Hamiltonian by applying creation operators tothe vacuum, can be interpreted in terms of the representation theory of quantumgroups associated with Kac-Moody algebras

The lectures by Y Kosmann-Schwarzbach, “Lie bialgebras, Poisson Liegroups and dressing transformations”, are an exposition, including the proofs

of all the main results, of the theory of Lie bialgebras, classical r-matrices,

Pois-son Lie groups and PoisPois-son actions

The survey by M.A Semenov-Tian-Shansky, “Quantum and classical grable systems”, treats the relation between the Hamiltonians of a quantumsystem solvable by the quantum inverse scattering method and the Casimir ele-ments of the underlying hidden symmetry algebra, itself the universal enveloping

inte-algebra of a Kac-Moody inte-algebra or a q-deformation of such an inte-algebra, leading

to deep results on the spectrum and the eigenfunctions of the quantum system.This study is preceded by that of the analogous classical situation which serves

as a guide to the quantum case and utilizes the full machinery of classical

r-matrices and Poisson Lie groups, and the comparison between the classical andthe quantum cases is explicitly carried out

brackets This fundamental idea, due to F Magri, is the basis of “Eight lectures

on integrable systems”, by F Magri, P Casati, G Falqui, and M Pedroni, wherethey develop the geometry of bihamiltonian manifolds and various reductiontheorems in Poisson geometry, before applying the results to the theory of bothinfinite- and finite-dimensional soliton equations They show which reductionsyield the Gelfand-Dickey and the Kadomtsev-Petviashvilii equations, and theyderive the bihamiltonian structure of the Calogero system

The surveys included in this volume treat many aspects of the theory of linear systems, they are different in spirit but not unrelated For example, there

non-is a parallel, which deserves further explanation, between the role of q-analysnon-is

in the theory of discrete integrable sytems and that of q-deformations of algebras

of functions on Lie groups and of universal enveloping algebras of Lie algebras in

the theory of quantum integrable systems, while the r-matrix method for

clas-sical integrable systems on loop algebras, which seems to be purely algebraic, is

in fact an infinitesimal version of the Riemann-Hilbert factorization problem.The theory of nonlinear systems, and in particular of integrable systems, isrelated to several very active fields of theoretical physics For instance, the roleplayed in the theory of integrable systems by infinite Grassmannians (on which

the τ -function “lives”), the boson-fermion correspondence, the representation theory of W -algebras, the Virasoro algebra in particular, all show links with

conformal field theory

We hope that this book will permit the reader to study some of the manyfacets of the theory of nonlinear systems and their integrability, and to followtheir future developments, both in mathematics and in theoretical physics

M.J Ablowitz

Department of Applied Mathematics, Campus Box 526, University of Colorado atBoulder, Boulder Colorado 80309-0526, USA

markjab@newton.colorado.edu

Abstract These lectures are written for a wide audience with diverse backgrounds.

The subject is approached from a general perspective and overly detailed discussionsare avoided Many of the topics require only a standard background in applied mathe-matics

The lectures deal with the following topics: fundamentals of linear and nonlinearwave motion; isospectral flows with associated compatible linear systems includingPDE’s in 1+1 and 2+1 dimensions, with remarks on differential-difference and partialdifference equations; the Inverse Scattering Transform (IST) for decaying initial data

on the infinite line for problems in 1+1 dimensions; IST for 2+1 dimensional problems;remarks on self-dual Yang-Mills equations and their reductions The first topic is ex-tremely broad, but a brief review provides motivation for the other subjects covered inthese lectures

Water waves are an interesting physical model and a natural way for us to beginour discussion Consequently let us consider the equations of water waves for anirrotational, incompressible, inviscid fluid:

where η denotes the free surface, and, since the fluid is ideal, the velocity is

derivable from a potential, ¯u =
φ For simplicity, we shall assume waves in one dimension, η = η(x, t), ¯ u = (u, w) =

M.J Ablowitz, Nonlinear Waves, Solitons, and IST, Lect Notes Phys.638, 5–29 (2004)

http://www.springerlink.com/
Springer-Verlag Berlin Heidelberg 2004c

The linear dispersion relation is obtained by looking for a special solution of the

form φ(x, z, t) = Re (Φ(z)e i(kx −ωt) ), η = Re (N e i(kx −ωt)) Equations (1.1) and

two-dimen-T ρ

(η xx (1 + η2) + η yy (1 + η2

x)− 2η xy η x η y)

(1 + η2

x + η2)3/2 , where now η = η(x, y, t) and T is the coefficient of surface tension The di- spersion relation is obtained by looking for wave-like solutions such as η =

Re (N e i(kx+ly −ωt)) and one finds

ω2= (gκ + T κ3) tanh κh,

which the reader can verify

Rather than proceeding with two-dimensional waves, we will first discuss theone-dimensional situation In the case of long waves (shallow water),|kh| << 1,

(1.5) yields,

ω2= ghk2(1−1

3(kh)

The first approximation is ω2= c2k2, c2= gh (c0is the long wave speed) which

is the dispersion relation of the linear wave equation,

where α = c0h

2

6 In fact, a more careful asymptotic analysis allows one to derivethe Korteweg-de Vries (KdV) and Kadomtsev-Petviashvili (KP) equations evenwith surface tension included The asymptotic description is valid when thefollowing conditions hold, considering two-dimensional waves, i.e., this is relevant

to the derivation of the KP equation:

i) wave amplitudes are small: ε = |η|max/h << 1,

ii) shallow water–long waves: (κh)2<< 1,

iii) slow transverse variations: (m/k)2<< 1,

iv) maximal balance: 0((m/k)2) = 0((κh)2) = 0(ε).

With these conditions we consider unidirectional waves; namely, since thesolution of the wave equation (1.8) above has both right- and left-going waves:

η = f (x − c0t, y) + g(x + c0t, y), (1.11)

we consider initial values which select, say right-going waves, i.e., g = 0 In this

case the following KP equation is found,

2η yy = 0, (1.12)

where γ = h2(1− ˆ T )/6; ˆ T = T /3pgh2 The KdV equation,

η t + c0η x+3c0

2h ηη x + γη xxx = 0, (1.13) results if η is independent of y and η → 0 as x → ∞ The normalized KP equations result by rescaling η, t, x, y (we leave this to the reader to verify),

(u t + 6uu x + u xxx)x + 3σ2u yy = 0, (1.14)

σ2 =±1 (σ2 = sgn (1− ˆ T )) We see that there are two physically interesting

choices of sign depending on sgn (1− ˆ T ) In the usual situation, the surface tension is taken to be negligible, hence σ2 = +1; in the literature this is oftencalled the KPII equation When surface tension is large enough for sgn (1− ˆ T ) =

−1 = σ2, (1.14) is called the KPI equation

The normalized form of the KdV equation which follows from (1.14) is

In fact, the KdV equation is comprised of two parts, both of which are tally important in the study of wave phenomena, namely first-order quasilinearhyperbolic waves,

and a linear dispersive wave equation,

The quasilinear equation (1.16) has solutions which become multi-valued in finitetime This is due to the fact that the characteristics of the equation satisfy

where x = ξ denotes the particular characteristic at t = 0, and at t = 0, u(ξ, 0) =

F (ξ) Thus the solution of (1.16) is given by

where ξ = ξ(x, t) is given by solving the implicit equation (1.19) It follows from (1.19)–(1.20) that any decaying lump-like initial data will lead to crossing of characteristics, i.e., from (1.18), larger positive values of u travel faster than

smaller values, and multi-valuedness of the solution

The usual mechanism to arrest multi-valuedness, or crossing of stics, is to supplement (1.16) with a small term with higher-order derivatives,e.g., the Burgers equation (cf Whitham, [1]); in this reference a general review

characteri-of linear and nonlinear waves is given],

which can be solved by transform methods

From the exact solution it is found that (1.16) is the main solution for (1.21)

for 0 < ε << 1 until, asymptotically speaking, the characteristics almost cross.

Then an asymptotically thin shock wave is formed, which, for decaying initial

data, vanishes as t → ∞.

However, as shown by Zabusky and Kruskal [2], the KdV equation ves quite differently from Burgers equation In [2] it was shown that the KdV

beha-equation (1.15), with a small coefficient in front of the u xxx term, i.e., replace

u xxx by ε2u xxx , 0 < ε << 1, which can be obtained from (1.15) by rescaling

x, t, develops numerous special hump-like travelling waves, referred to as

soli-tons (solisoli-tons will be discussed later in more detail), in the asymptotic regionnear the time of multi-valuedness The solitons move to the right, away from the

front, and, as t → ∞, a dispersive tail is left behind (cf [3]) The tail vanishes

as t → ∞ Years later, researchers studied the asymptotic problem of KdV with

a small dispersive term in more detail (cf [4])

Finally, it is worth remarking upon the solution of the linear equation (1.17)

with u(x, 0) = f (x) given, and decaying sufficiently just as |x| → ∞ The solution

is obtained by Fourier transforms and is found to be

f (x)e −ikx dx = ˆ u(k, 0), and where ˆ u(k, t) denotes the

Fou-rier transform at any time t While the general solution establishes existence,

qualitative information can be obtained by further study Asymptotic analysis as

t → ∞ (stationary phase-steepest descent methods) establishes that the solution

(cf [3] for further details)

The method of Fourier transforms generalizes readily, e.g., it is applicable

to any evolutionary PDE with constant coefficients The scheme for solving a

linear problem with a dispersion relation ω(k), e.g., (1.17) where ω(k) = −k3,

by Fourier transforms is as follows:

The KdV equation (1.15) was the first equation solved (on the infinite linewith appropriately decaying data) by inverse scattering methods [5] Subse-

quently Zakharov and Shabat [24] showed that the nonlinear Schr¨odinger tion (NLS), which arises as a centrally important equation in fluid dynamics,nonlinear optics and plasma physics,

where v is an n-dimensional vector, and X, T are n × n matrices Compatibility

of (2.4a,b) implies that v xt = v tx , hence X, T satisfy

X x − T t + [X, T ] = 0, where [X, T ] = XT − T X It is easiest to consider a concrete situation where X

is specified Thus, consider the 2× 2 linear equation for (2.4a),

where A, B, C, D are scalar functions of q, r, and k; k is a parameter which is

essential in the method of direct and inverse scattering Compatibility of (2.5)–

(2.6) implies D = −A, and

A x = qC − rB

B x + 2ikB = q t − 2Aq

C − 2ikC = r + 2Ar.

(2.7)

In [3,6] it is shown that finite power series solutions for A, B, C, A =

N



j=0

A j k j

etc., lead to nonlinear equations Important special cases are listed below When

N = 2, q = ∓r ∗ (= u) results in the NLS equation (2.1) (with either choice of

sign s = ±1) When N = 3 and if r = −1, q ≡ u, we find the KdV equation (1.15); and when q = ∓r ≡ u, the mKdV equation (2.2) is obtained If N = 1, the sine-Gordon equation (2.3) results if q = −r = u x /2 and the sinh-Gordon



where A0(k) = lim

|x|→∞ A(x, t, k) (A0(k) may be a ratio of entire functions), and

L is an integro-differential operator given by

L = 12i

where r = exp(i(kx −ω r t)) and q = exp(i(kx −ω q t)) are linearized wave solutions.

There are numerous generalizations and extensions of these ideas The reader

is encouraged to consult the many papers and monographs related to this subject(cf [7] for an extensive bibliography)

In this section we quote the main results in the scattering theory of the linearproblem associated with the KdV equation (1.15) In fact, as discussed in Sect 3,

the KdV equation arises from (2.5)–(2.6) with r = −1, q ≡ u, and taking a finite expansion in powers of k, A =

3



j=0

A j k j, etc In fact, in this case, (2.5)–(2.6) takes

a simpler form when written as a scalar system,

v xx + (u(x, t) + k2)v = 0 (3.1)

v t = (u x + γ)v + (4k2− 2u)v x , (3.2) where γ is an arbitrary constant The reader can verify that (3.1)-(3.2) are consistent with the KdV equation (1.15), assuming ∂k/∂t = 0, i.e., it is an

isospectral flow

Scattering theory is developed on the spatial part of the compatible system –

in this case (3.1) In fact, (3.1) is the well-known time-independent Schr¨odingerequation Scattering and inverse scattering associated with (3.1) have a longhistory which we shall not review here The results described below hold for

functions u(x, t) decaying sufficiently fast at infinity, e.g., satisfying

is invariant under k → −k, the boundary conditions (3.4) imply that

The fact that (3.1) is a second order equation means that φ, ψ, ¯ ψ are linearly

related,

φ(x, k) = a(k) ¯ ψ(x, k) + b(k)ψ(x, k) (3.6) Actually a(k) and b(k) are readily related to the reflection coefficient r(k) and the transmission coefficient t(k) in quantum mechanics, k ∈ IR,

2ikx N (x,¯ −k) (3.11)

Indeed, (3.11) is a fundamental equation in this approach It is a generalizedRiemann-Hilbert boundary value problem (RHBVP), which is a consequence ofthe following facts:

i) M (x, k) and a(k) can be analytically extended to the upper half k-plane

(UHP) and tend to unity as|k| → ∞ (Im k > 0).

ii) ¯N (x, k) can be analytically extended to the lower half k-plane (LHP) and

tends to unity as|k| → ∞ (Im k < 0).

These analytic properties can be proven by a careful analysis of the integral

equations which represent M and ¯ N , namely,

establishes the analytic behavior of a(k) (recall that M (x; k) is analytic for

Im k > 0) In fact, the zeroes of a(k) are relevant in what follows It can be established that for u(x) real there are a finite number of simple zeroes, all of

which lie on the imaginary k axis: a(k j ) = 0, k j = ik j , k j > 0, j = 1, N.

Sometimes the RHBVP (3.11) is written as

(µ+(x, k) − µ − (x, k)) = r(k)e 2ikx µ − (x, −k), (3.17) where µ+(x, k) = M (x, k)/a(k), µ − (x, k) = ¯ N (x, k) We also note the relations-

u(x)M (x, k)e −2ikx dx. (3.18)

From (3.16), (3.18) we can compute r(k), t(k) (see (3.7)) in terms of initial data The above relationships follow from the direct problem Namely, given u(x) satisfying (3.3), then a(k) and b(k) are given by (3.16), (3.18), hence r(k) from

(3.7) and the analytic properties from the integral equations (3.12)–(3.13) andthe RHBVP from (3.17)

The inverse problem requires giving appropriate scattering data to uniquelydetermine a solution to (3.17) Equation (3.17) is a RHBVP with a shift In fact,there is no closed form solution to (3.17) The best one can do is to transform(3.17) to an integral equation or system of integral equations determined by thescattering data

The simplest case occurs where a(k) = 0 for Im k > 0 Taking a minus tion of (3.17) (after subtracting unity from both µ ± (x, k)) where the projection

projec-operators are defined by

and upon comparing (3.23)–(3.24) we see that

So far we have not allowed for the possibility that a(k) can vanish for Im k >

0 If a(k) vanishes at k j = iκ j , κ j > 0, j = 1, N , the final result is that the GLM equation is only modified by changing the function F (x):

scattering of the 2×2 problem (2.5) and many other scattering problems related

to integrable equations in 1 + 1 dimensions is similar We refer the reader to[3,6,7] for more details.

The inverse scattering transform is completed when one determines the dependence of the scattering data For this purpose, we consider the time-evolution equation (3.2), or, for a fixed eigenfunction, the equation satisfied

time-by M = φe ikx , where φ is defined by (3.4),

M t = (γ − 4ik3+ u x + 2iku)M + (4k2− 2u)M x (3.31)

with the asymptotic behaviors,

M = a(k, t) + b(k, t)e 2ikx as x → +∞ (3.32)

Note that we now denote all functions with explicit time-dependence The latterequation of (3.32) follows from (3.4), (3.6), (3.9) These asymptotic relationsimply

This kernel fixes the integral equation (3.27), and hence the solution u(x, t) to

the KdV equation follows from (3.28) in terms of initial data The scheme ofsolution is similar to that of Fourier transforms:

Inverse Scattering

There are a number of results which follow from the above developments.a) For scattering data that correspond to potentials satisfying (3.3), the solution

to the GLM equation exists With suitable conditions on u and its derivatives,

global solutions to the KdV equation can be established b) Long-time tic analysis of the KdV equation can be ascertained The solution is comprised

asympto-of a discrete part consisting asympto-of N soliton (see below) waves moving to the right, and a dispersive tail which decays algebraically as t → ∞ c) The discrete part

of the spectrum can be solved in terms of a linear algebraic system In the GLMequation, the discrete spectrum corresponds to a degenerate kernel From theRHBVP, the following linear system results,

i(κ p + κ l)exp(−2κ p x + 8κ3t)N p (x, t) = exp( −2κ l x) (3.36)

where N l (x, t) ≡ N(x, k = iκ l , t), and from the solution of (3.36) we reconstruct the solution of KdV, u(x, t), via

i t − x i , C i (0) = 2κ i exp(2κ i x i ), i = 1, 2 The two-soliton

solution shows that the sum of two solitary waves of the form given by (3.38) isthe asymptotic state of (3.39), but there is a phase shift due to the interaction

We also note that knowledge that the function a(k, t) is a constant of the

motion can be related to the infinite number of conservation laws of KdV (cf [3,7])

It should also be noted that discretizations of (2.5), (3.1) lead to interestingdiscrete nonlinear evolution equations which can be solved by IST The bestknown of these equations is the Toda lattice,

∂2u n

∂t2 = exp(−(u n − u n −1))− exp(−(u n+1 − u n )), (3.40)

which is related to the linear discrete Schr¨odinger scattering problem,

α n v n+1 + α n −1 v n −1 + β n v n = kv n , (3.41)

and u ∗ is the complex conjugate of u There are also double discretizations of the

NLS equation which can be solved by IST These discretizations are obtained

by discretizing the temporal equation (2.6) One example of a doubly discreteNLS equation is given by

i∆ m u m n

It should also be mentioned that there is another class of nonlinear evolutionequations in 1+1 dimensions that are solvable by IST This is the class of singularintegro-differential equations The paradigm equation is

u t+1

δ u x + 2uu x + Tu xx = 0, (3.48)

where δ is a constant and Tu is the singular integral operator,

The BO equation was derived in the context of long internal gravity waves in

a stratified fluid [14–16], whereas the ILW equation was derived in a similarcontext in [17,18] In [7] the IST analysis associated with the ILW equation and

BO equation is reviewed The unusual aspect of the IST scheme is the fact thatthe scattering operator is a differential RHBVP Related generalizations are alsodiscussed in [7]

In Sect 2 we discussed the relevance of the KP equation in two-dimensional waterwaves The normalized KP equation is given by (1.14) In this section, a broadoutline of the main results of IST for the KP equation will be outlined Just asthe KdV equation was the first 1+1 equation linearized by IST methods, the KPequation was the first nontrivial equation linearized by 2+1 IST methods Afterthe methods were established for the KP equation, they were quickly generalized

to other equations such as the Davey-Stewartson equation and the 2+1 N waveequation (cf [7])

The compatible linear system for KP is given by

v t + 4v xxx + 6uv x + 3u x v − 3σ(∂ −1

x u y )v + γv = 0, (4.1b) where ∂ −1

x =

 x

−∞

dx  , and γ is an arbitrary constant In the case when σ2=−1,

i.e., KPI, then (4.1a) is the nonstationary Schr¨odinger equation When σ2= +1,

i.e., KPII, then (4.1a) is a “reverse” heat equation which is well-known to beill-posed as an initial value problem However, as a scattering problem, the case

when σ2= +1 can be analyzed effectively However the situation when σ2= +1

is very different from σ2=−1.

A) KPI: σ2=−1

In this case it is convenient to make the transformation,

v(x, y, k) = m(x, y, k) exp(i(kx − k2y)), (4.2) whereupon (4.1a) is transformed to (σ = i),

We want to find an eigenfunction which is bounded for all x, y, k Such an

eigen-function satisfies the following integral equation,

It is clear that if k = k R + ik I , then G(x, y, k) is not well defined for k I = 0 But

there are natural analytic functions for Im k > 0 In fact, by contour integration,

G ± (x, y, k) = 2π i

 ∞

−∞

e ipx −ip(p+2k)y {θ(y)θ(∓p) − θ(−y)θ(±p)}dp, (4.8)

where G ± stands for the limit k → k R ± i0 A study of the properties of the integral equation (cf [8,9]) shows that there are solutions m ± (x, y, k) to (4.4) It

is natural to ask how these functions are related In [10; see also 7], the followingnonlocal generalized RHBVP is derived,

N (x, y, k, l) = exp(β(x, y, k, l) + ˜ G − (uN ). (4.9d)

An essential difference between (4.9) and the result for KdV is that m ± (x, y, k)

may have poles since, in the derivation of (4.9), we employed the fact that m ±

satisfies a Fredholm integral equation (4.4) In KdV, the underlying integralequations were of Volterra type, and consequently their solutions have no poles

However, in the form (3.17), we see that µ+(x, k) = M (x, k)/a(k), and poles are introduced through the zeroes of the scattering data a(k) It turns out that the poles of the eigenfunction, m ± (x, y, k), suitable normalizing coefficients and the

“reflection” coefficients f (k, l) complete the IST picture We shall assume that the eigenfunctions m ± (x, y, k) have only simple poles – indeed, in recent work

it is demonstrated that this is not necessary; the poles can be of any order [11]

The functions m ± are assumed to have the representations

where µ ± are analytic functions for Im k > 0 and µ ± → 0 as |k| → ∞ The

following important relation holds,

u(x, y)φ(x, y)dx dy (4.14)

In recent work [11], it has been shown that values of the integral Q(k, φ) can

be an integer and that Q is actually an underlying index of the problem The

time-dependence of the data is obtained from (4.1b) and it is found that

The time dependence (4.15) with (4.9), (4.10) and (4.11) complete the IST

fra-mework, assuming only simple poles for m as expressed in (4.10) The class of lump solutions which are real, nonsingular and decay as 0(1/r2), r2= x2+ y2,

as r → ∞, is obtained from (4.11)–(4.12) where k j − = ¯k j+ , γ j − = ¯γ j+ (bar

stands for the complex conjugate) A one-lump solution (N = 1) is found to be

In (4.20) and hereafter all double integrals are taken from −∞ to ∞ in both

variables of integration Thus, unlike (4.7), Green’s function expressed by (4.20)

has no regions of analyticity In fact, G = G(x, y, k R , k I ) where k = k R + ik I;

i.e., G depends on the real and imaginary parts of k Note that by use of contour

θ(y(p2+ 2k R p)) exp(ipx −p(p+2k)y)dp (4.21)

This implies that the function m via (4.18) is also a function of k R , k I and

therefore is analytic nowhere in the k-plane By taking the anti-holomorphic

derivative of (4.18), i.e operating on (4.18) by ∂

∂k =

12

using, as we sometimes do, the notation G(x, y, k) to denote G(x, y, k R , k I) etc.,

nity (cf [12]), the direct problem establishes relations (4.23)–(4.24) The inverse

problem is fixed by giving R(k R , k I) (there are no discrete state solutions knownwhich lead to real, nonsingular, decaying states for KPII) in order to deter-

mine m(x, y, k) and then u(x, y) The inverse problem is developed by using the

generalized Cauchy integral formula (cf [13]),

z = z R + iz I , and dz ∧ d¯z = 2idz R dz I As k → ∞, we can establish that m ∼ 1,

hence the second term on the right-hand side of (4.26) is unity Using (4.23) wefind that

u(x, y) = − ∂

∂x

2i π

5 Remarks on Related Problems

In the previous sections we have discussed the integrability of a class of nonlinearequations in 1+1and in 2+1 dimensions It is natural to ask whether there are any4-dimensional integrable systems Indeed, there is at least one such importantsystem, the self-dual Yang-Mills (SDYM) equations They are the result of thecompatibility of the following linear pair,

where γ α , γ α¯, γ β , γ β¯ are four dependent variables, often called gauge potentials,

and α, ¯ α, β, ¯ β are four independent variables which can be written in terms of

the usual Cartesian coordinates as

There has been significant interest in the SDYM equations as a “master” grable system Ward [19] has conjectured that perhaps all “integrable” equations,e.g soliton equations, may be obtained as a reduction of the SDYM equations.The reduction process has three aspects (see [7]).

inte-i) Employ the gauge freedom (5.6) of the equations Frequently the choice ofgauge can simplify the analysis and make the search for integrable reductionsconsiderably easier

ii) Reduction of independent variables, i.e., γ a (α, ¯ α, β, ¯ β) can be functions of

α, or α, β, etc.

iii) Choice of the underlying gauge group (algebra) in which one carries out the

analysis Sometimes it is a matrix algebra, e.g., su(n), gl(n); but in many interesting cases the gauge algebra is infinite dimensional, e.g., sdiff(S3)

It is often easiest to make identifications via the linear pair of SDYM For

example, suppose γ a ∈ gl(N), γ a = γ a (α, β), γ β¯ = iJ = diagonal matrix,

γ α¯ = iA0= diagonal matrix Then, calling γ α = Q, γ β = A1(5.1a–5.1b) reduceto

∂Ψ

∂Ψ

In fact, (5.7) is the linear pair associated with the N wave system (when N = 3

it is the 3 wave system) We need go no further in writing the equations (cf [7])except to point out that once the spatial part of the linear system is known, thenactually the entire hierarchy can be ascertained Other special cases include KdV,NLS, sine Gordon etc

It is also worth remarking that the well-known 2+1 dimensional soliton stem can be obtained from SDYM if we assume that the gauge potentials areelements of the infinite dimensional gauge algebra of differential polynomials

sy-For example, suppose γ α¯ = γ β¯ = 0, Q = Q(α, y, β), A1 = A1(α, y, β), J, A0

are diagonal matrices and

Compatibility of (5.9) yields the N wave equations in 2+1 dimensions (here

the independent variables are α, y, β; i.e., β plays the role of time) Again the

hierarchy is generated from the spatial part of the linear system Other specialcases include the KP and Davey-Stewartson systems A detailed discussion of2+1 reductions can be found in [20].

It should also be mentioned that SDYM reduces to the classical 0+1 sional Painlev´e equations In [21] it is shown that all of the six Painlev´e equationscan be obtained from SDYM with finite-dimensional Lie Groups (matrix gaugealgebra)

dimen-In [22] it was shown that using an infinite-dimensional gauge algebra, sdiff

(S3), SDYM could be reduced to the system

∆(t) = ∆(γt)

(ct + d)12 ∆ → 0 , Im t → ∞, (5.14)

a, b, c, d integers Such a function ∆(t) is called the discriminant modular form;

explicit formulae representing the function are

where E2(t) is the Eisenstein series, and σ1(n) are particular number-theoretic

coefficients From (5.15) or (5.16) the general solution is obtained from (5.13)

where a, b, c, d are taken to be arbitrary coefficients with ad − bc = 1 If we use

(5.12), then a convenient “B¨acklund” type transformation is

y II (t) = y I (γt)

(ct + d)2 − 6c

Equations (5.12)-(5.17) demonstrate that the solutions to Chazy’s equation (5.11)

and the solution to the Darboux-Halphen system (5.10) (by finding w1, w2, w3

in terms of y, ˙ y, ¨ y) are expressible in terms of automorphic functions.

An interesting question to ask is whether the solution of (5.10–5.11) can beobtained via the inverse method In fact, in a recent paper [23] it has been shownthat the linear compatible system of SDYM can be reduced to a monodromyproblem The novelty is that in this case the monodromy problem has evolvingmonodromy data+– unlike those associated with the Painlev´e equation wherethe monodromy is fixed (isomonodromy) Then the linear problem can be used

to find the solutions of (5.10) which are automorphic functions, and via (5.10), tosolve Chazy’s equation (5.11) Generalizations of the Darboux-Halphen systemare also examined and solved in [23] It is outside the scope of this article to gointo those details

Acknowledgments

This work was partially supported by the Air Force Office of Scientific Research,Air Force Materials Command, USAF under grant F49620-97-1-0017, and by theNSF under grant DMS-9404265 The US Government is authorized to reproduceand distribute reprints for governmental purposes notwithstanding any copyrightnotation thereon The views and conclusions contained herein are those of theauthor and should not be interpreted as necessarily representing the officialpolicies or endorsements, either expressed or implied, of the Air Force Office

of Scientific Research or the US Government

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