Soliton Equations and Hamiltonian Systems (Advanced Series in Mathematical Physics, V. 26)

The theory of soliton equations and integrable systems has developed rapidly during the last 30 years with numerous applications in mechanics and physics. For a long time, books in this field have not been written but the flood of papers was overwhelming: many hundreds, maybe thousands of them. All this output followed one single work by Gardner, Green, Kruskal, and Mizura on the Korteweg-de Vries equation (KdV), which had seemed to be merely an unassuming equation of mathematical physics describing waves in shallow water. Besides its obvious practical use, this theory is attractive also because it satisfies the aesthetic need in a beautiful formula which is so inherent to mathematics. The second edition is up-to-date and differs from the first one considerably. One third of the book is completely new and the rest is refreshed and edited. Preface to the Second Edition......Page 6 Contents......Page 8 Introduction to the First Edition......Page 13 1.1 Differential Algebra A......Page 19 1.2 Space of Functionals A......Page 20 1.3 Ring of Pseudodifferential Operators......Page 21 1.4 Lax Pairs. GD Hierarchies of Equations......Page 24 1.5 First Integrals (Constants of Motion)......Page 26 1.6 Compatibility of the Equations of a Hierarchy......Page 27 1.7 Soliton Solutions......Page 28 1.8 Resolvent. Adler Mapping......Page 30 2.1 Finite-Dimensional Case......Page 35 2.2 Hamilton Mapping......Page 40 2.3 Variational Principles......Page 41 2.4 Symplectic Form on an Orbit of the Coadjoint Representation of a Lie Group......Page 45 2.5 Purely Algebraic Treatment of the Hamiltonian Structure......Page 48 2.6 Examples......Page 51 3.1 Lie Algebra V Dual Space Q1 and Module Q0......Page 57 3.2 Proof of Theorem 3.1.2......Page 60 3.3 Poisson Bracket......Page 65 3.4 Reduction to the Submanifold Un-1 = 0......Page 68 3.5 Variational Derivative of the Resolvent......Page 69 3.6 Hamiltonians of the GD Hierarchies......Page 71 3.7 Theory of the KdV-Hierarchy (n = 2) Independent of the General Case......Page 72 4 1 Miura Transformation. The Kupershmidt-Wilson Theorem......Page 79 4.2 Modified KdV Equation. Backlund Transformations......Page 83 4.3 More on Modified GD Equations......Page 84 5.1 Definition of the KP Hierarchy......Page 87 5.2 Reduction of the KP Hierarchy to GD......Page 89 5.3 First Integrals and Soliton Solutions......Page 91 5.4 Hamiltonian Structure......Page 93 5.5 Resolvent......Page 96 5.6 Hamiltonians of the KP Hierarchy......Page 99 6.1 Dressing......Page 101 6.2 Baker Function......Page 102 6.3 Shift Operator and T-Function......Page 106 6.4 Resolvent and Baker Function. Fay Identities......Page 112 6.5 Vertex Operators......Page 115 6.6 T-Function and Fock Representation......Page 118 6A Appendix. List of Useful Formulas for the Faa di Bruno Polynomials......Page 123 7.1 Additional Symmetries......Page 125 7.2 Generating Function for Additional Symmetries......Page 129 7.3 String Equation......Page 131 8.1 Infinite-Dimensional Grassmannian......Page 135 8.2 Modified Definition of the Grassmannian T-Function......Page 140 8.3 Algebraic-Geometrical Solutions of Krichever......Page 144 8A Appendix. Abel Mapping and the 0-Function......Page 149 9.1 Hierarchy of Equations Generated by a First-Order Matrix Differential Operator......Page 153 9.2 Hamiltonian Structure......Page 159 9.3 Hamiltonians of the AKNS-D Hierarchy......Page 163 9.4 GD Hierarchies as Reductions of the Matrix Hierarchies (Drinfeld-Sokolov Reduction)......Page 166 9A Appendix. Extension of the Algebra A to an Algebra Closed with Respect to the Indefinite Integration......Page 174 10.1 Single-Pole Matrix Hierarchy......Page 177 10.2 Single-Pole Hierarchy. Presentation not Depending on a Distinguished Operator 1......Page 183 10.3 Multi-Pole (General Zakharov-Shabat) Hierarchy......Page 185 10.4 Example: Principal Chiral Field Equation......Page 189 10.5 Grassmannian......Page 190 11.1 Isomonodromic Deformations......Page 199 11.2 General Matrix Hierarchy......Page 207 12.1 Segal-Wilson's T-Function for AKNS-D......Page 215 12.2 Tau Functions for More General Matrix Hierarchies......Page 221 13.1 Modified GD (Cont'd)......Page 225 13.2 Modified KP and Constrained KP......Page 227 13.3 Discrete KP......Page 232 13.4 q-KP......Page 236 14.1 Introduction. More About the Modified KP......Page 239 14.2 Stabilizing Chain......Page 243 14.3 Solutions to the Chain......Page 246 14.4 Solutions in the Form of Series in Schur Polynomials. Stabilization......Page 249 14.5 From the Stabilizing Chain to the Kontsevich Integral......Page 251 15.1 Tensors with Respect to Diffeomorphisms and the AGD-Algebra......Page 263 15.2 Another Construction of Primary Fields......Page 274 16.1 The Ring of Functions on the Phase Space of the Equation......Page 281 16.2 Characteristics of the First Integrals......Page 284 16.3 Hamiltonian Structure......Page 285 16.4 Stationary Equations of the KdV Hierarchy ([GD79])......Page 290 16.5 Integration after Liouville......Page 296 16.6 Return to the Original Variables......Page 301 17.1 First Integrals......Page 307 17.2 Hamiltonian Structure of Stationary Equations......Page 315 17.3 Action-Angle Variables......Page 320 17A Appendix. Genus of the Riemann Surfaces and the Newton Diagram......Page 324 18.1 Baker Function. Return to Original Variables......Page 329 18.2 Rotation of the n-Dimensional Rigid Body......Page 335 19.1 Introduction......Page 341 19.2 Variational Bi-Complex......Page 343 19.3 Exactness of the Bi-Complex......Page 348 19.4 Variational Derivative......Page 354 19.5 Lagrangian-Hamiltonian Formalism......Page 358 19.6 Variational Bi-Complex of a Differential Equation. First Integrals......Page 362 19.7 Poisson Bracket......Page 368 19.8 Relationship with the Single-Time Formalism......Page 369 20.1 KP-Hierarchy......Page 375 20.2 The Zakharov-Shabat Equation with Rational Dependence on the Spectral Parameter......Page 380 20.3 Principal Chiral Field......Page 396 20.4 Lagrangians of the nth Reduced KP (GD) Hierarchy......Page 404 Bibliography......Page 409 Index......Page 419 The theory of soliton equations and integrable systems has developed rapidly during the last 30 years with numerous applications in mechanics and physics. For a long time, books in this field have not been written but the flood of papers was overwhelming: many hundreds, maybe thousands of them. All this output followed one single work by Gardner, Green, Kruskal, and Mizura on the Korteweg-de Vries equation (KdV), which had seemed to be merely an unassuming equation of mathematical physics describing waves in shallow water.Besides its obvious practical use, this theory is attractive also because it satisfies the aesthetic need in a beautiful formula which is so inherent to mathematics.The second edition is up-to-date and differs from the first one considerably. One third of the book (five chapters) is completely new and the rest is refreshed and edited. The theory of soliton equations and integrable systems has developed rapidly with numerous applications in mechanics and physics. This study explores this branch of science, and seeks to revive an interest in the basic principles of mathematics.