Direct and Inverse Methods in Nonlinear Evolution Equations: Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, September 5–12, 1999 (Lecture Notes in Physics (632))

Many physical phenomena are described by nonlinear evolution equation. Those that are integrable provide various mathematical methods, presented by experts in this tutorial book, to find special analytic solutions to both integrable and partially integrable equations. The direct method to build solutions includes the analysis of singularities à la Painlevé, Lie symmetries leaving the equation invariant, extension of the Hirota method, construction of the nonlinear superposition formula. The main inverse method described here relies on the bi-hamiltonian structure of integrable equations. The book also presents some extension to equations with discrete independent and dependent variables. The different chapters face from different points of view the theory of exact solutions and of the complete integrability of nonlinear evolution equations. Several examples and applications to concrete problems allow the reader to experience directly the power of the different machineries involved. front-matter Chapter 1 1 Introduction 2 Various levels of integrability for PDEs, de.nitions 3 Importance of the singularities: a brief survey of the theory of Painlev ́e 4 The Painlev ́e test for PDEs in its invariant version 4.2 The WTC part of the Painlev ́e test for PDEs 4.3 The various ways to pass or fail the Painlev ́e test for PDEs 5 Ingredients of the “singular manifold method” 5.1 The ODE situation 5.2 Transposition of the ODE situation to PDEs 5.3 The singular manifold method as a singular part transformation 5.4 The degenerate case of linearizable equations 5.5 Choices of Lax pairs and equivalent Riccati pseudopotentials 6 The algorithm of the singular manifold method 6.1 Where to truncate, and with which variable? 7 The singular manifold method applied to one-family PDEs 7.1 Integrable equations with a second order Lax pair 7.2 Integrable equations with a third order Lax pair 7.3 Nonintegrable equations, second scattering order 7.4 Nonintegrable equations, third scattering order 8 Two common errors in the one-family truncation 8.1 The constant level term does not de.ne a BT 8.2 The WTC truncation is suitable i. the Lax order is two 9 The singular manifold method applied to two-family PDEs 9.1 Integrable equations with a second order Lax pair 9.2 Integrable equations with a third order Lax pair 9.3 Nonintegrable equations, second and third scattering order 10 Singular manifold method versus reduction methods 11 Truncation of the unknown, not of the equation 12 Birational transformations of the Painlev ́e equations 13 Conclusion, open problems Chapter 2 1 Introduction: The tensorial approach and the birth of the method of Poisson pairs 1.1 The Miura map and the KdV equation 1.2 Poisson pairs and the KdV hierarchy 1.3 Invariant submanifolds and reduced equations 1.4 The modi.ed KdV hierarchy 2 The method of Poisson pairs 3 A .rst class of examples and the reduction technique 3.1 Lie–Poisson manifolds 3.2 Polynomial extensions 3.3 Geometric reduction 3.4 An explicit example 3.5 A more general example 4 The KdV theory revisited 4.1 Poisson pairs on a loop algebra 4.2 Poisson reduction 4.3 The GZ hierarchy 4.4 The central system 4.5 The linearization process 4.6 The relation with the Sato approach 5 Lax representation of the reduced KdV .ows 5.1 Lax representation 5.2 First example 5.3 The generic stationary submanifold 5.4 What more? 6 Darboux–Nijenhuis coordinates and separability 6.1 The Poisson pair 6.2 Passing to a symplectic leaf 6.3 Darboux–Nijenhuis coordinates 6.4 Separation of variables Chapter 3 1 Introduction 2 Integrability by the singularity approach 3 B ̈acklund transformation: de.nition and example 4 Singularity analysis of nonlinear di.erential equations 4.1 Nonlinear ordinary di.erential equations 4.2 Nonlinear partial di.erential equations 5 Lax Pair and Darboux transformation 5.1 Second order scalar scattering problem 5.2 Third order scalar scattering problem 5.3 A third order matrix scattering problem 6 Di.erent truncations in Painlev ́e analysis 7 Method for a one-family equation 8 Nonlinear superposition formula 9 Results for PDEs possessing a second order Lax pair 9.1 First example: KdV equation 9.2 Second example: MKdV and sine-Gordon equations 10 PDEs possessing a third order Lax pair 10.1 Sawada-Kotera, KdV5, Kaup-Kupershmidt equations 10.2 Painlev ́e test 10.3 Truncation with a second order Lax pair 10.4 Truncation with a third order Lax pair 10.5 B ̈acklund transformation 10.6 Nonlinear superposition formula for Sawada-Kotera 10.7 Nonlinear superposition formula for Kaup-Kupershmidt 10.8 Tzitz ́eica equation Chapter 4 1 Introduction 2 Soliton solutions 2.1 The Burgers equation 2.2 The Korteweg-de Vries equation 2.3 The nonlinear Schr ̈odinger equation 2.4 The Toda equation 2.5 Painlev ́e equations 2.6 Di.erence vs di.erential 3 Multidimensional equations 3.1 The Kadomtsev-Petviashvili equation 3.2 The two-dimensional Toda lattice equation 3.3 Two-dimensional Toda molecule equation 3.4 The Hirota-Miwa equation 4 Sato theory 4.1 Micro-di.erential operators 4.2 Introduction of an in.nite number of time variables 4.3 The Sato equation 4.4 Generalized Lax equation 4.5 Structure of tau functions 4.6 Algebraic identities for tau functions 4.7 Vertex operators and the KP bilinear identity 4.8 Fermion analysis based on an in.nite dimensional Lie algebra 5 Extensions of the bilinear method 5.1 q-discrete equations 5.2 Special function solution for soliton equations 5.3 Ultra discrete soliton system 5.4 Trilinear equations Chapter 5 1 Introduction 2 The symmetry group of a system of di.erential equations 2.1 Formulation of the problem 2.2 Prolongation of vector .elds and the symmetry algorithm 2.3 First example: Variable coe.cient KdV equation 2.4 Symmetry reduction for the KdV 2.5 Second example: Modi.ed Kadomtsev-Petviashvili equation 3 Classi.cation of the subalgebras of a .nite dimensional Lie algebra 3.1 Formulation of the problem 3.2 Subalgebras of a simple Lie algebra 3.3 Example: Maximal subalgebras of o(4, 2) 3.4 Subalgebras of semidirect sums 3.5 Example: All subalgebras of sl(3,R) classi.ed under the group SL(3,R) 3.6 Generalizations 4 The Clarkson-Kruskal direct reduction method and conditional symmetries 4.1 Formulation of the problem 4.2 Symmetry reduction for Boussinesq equation 4.3 The direct method 4.4 Conditional symmetries 4.5 General comments 5 Concluding comments 5.1 References on nonlinear superposition formulas 5.2 References on continuous symmetries of di.erence equations back-matter