What Is Integrability? by Vladimir E. Zakharov (PDF)

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Why Are Certain Nonlinear PDEs Both Widely Applicable and Integrable?.- Summary.- 1. The Main Ideas in an Illustrative Context.- 2. Survey of Model Equations.- 3. C-Integrable Equations.- 4. Envoi.- Addendum.- References.- Painlevé Property and Integrability.- 1. Background.- 1.1 Motivation.- 1.2 History.- 2. Integrability.- 3. Riccati Example.- 4. Balances.- 5. Elliptic Example.- 6. Augmented Manifold.- 7. Argument for Integrability.- 8. Separability.- References.- Integrability.- 1. Integrability.- 2. Introduction to the Method.- 2.1 The WTC Method for Partial Differential Equations.- 2.2 The WTC Method for Ordinary Differential Equations.- 2.3 The Nature of ?.- 2.4 Truncated Versus Non-truncated Expansions.- 3. The Integrable Hénon-Heiles System: A New Result.- 3.1 The Lax Pair.- 3.2 The Algebraic Curve and Integration of the Equations of Motion.- 3.3 The Role of the Rational Solutions in the Painlevé Expansions.- 4. A Mikhailov and Shabat Example.- 5. Some Comments on the KdV Hierarchy.- 6. Connection with Symmetries and Algebraic Structure.- 7. Integrating the Nonintegrable.- References.- The Symmetry Approach to Classification of Integrable Equations.- 1. Basic Definitions and Notations.- 1.1 Classical and Higher Symmetries.- 1.2 Local Conservation Laws.- 1.3 PDEs and Infinite-Dimensional Dynamical Systems.- 1.4 Transformations.- 2. The Burgers Type Equations.- 2.1 Classification in the Scalar Case.- 2.2 Systems of Burgers Type Equations.- 2.3 Lie Symmetries and Differential Substitutions.- 3. Canonical Conservation Laws.- 3.1 Formal Symmetries.- 3.2 The Case of a Vector Equation.- 3.3 Integrability Conditions.- 4. Integrable Equations.- 4.1 Scalar Third Order Equations.- 4.2 Scalar Fifth Order Equations.- 4.3 Schrödinger Type Equations.- Historical Remarks.- References.- Integrability of Nonlinear Systems and Perturbation Theory.- 1. Introduction.- 2. General Theory.- 2.1 The Formal Classical Scattering Matrix in the Solitonless Sector of Rapidly Decreasing Initial Conditions.- 2.2 Infinite-Dimensional Generalization of Poincaré’s Theorem. Definition of Degenerative Dispersion Laws.- 2.3 Properties of Degenerative Dispersion Laws.- 2.4 Properties of Singular Elements of a Classical Scattering Matrix. Properties of Asymptotic States.- 2.5 The Integrals of Motion.- 2.6 The Integrability Problem in the Periodic Case. Action-Angle Variables.- 3. Applications to Particular Systems.- 3.1 The Derivation of Universal Models.- 3.2 Kadomtsev-Petviashvili and Veselov-Novikov Equations.- 3.3 Davey-Stewartson-Type Equations. The Universality of the Davey-Stewartson Equation in the Scope of Solvable Models.- 3.4 Applications to One-Dimensional Equations.- Appendix I.- Proofs of the Local Theorems (of Uniqueness and Others from Sect.2.3).- Appendix II.- Proof of the Global Theorem for Degenerative Dispersion Laws.- Conclusion.- References.- What Is an Integrable Mapping?.- 1. Integrable Polynomial and Rational Mappings.- 1.1 Polynomial Mapping of C: What Is Its Integrability?.- 1.2 Commuting Polynomial Mappings of ?N and Simple Lie Algebras.- 1.3 Commuting Rational Mappings of ?Pn.- 1.4 Commuting Cremona Mappings of ?2.- 1.5 Euler-Chasles Correspondences and the Yang-Baxter Equation.- 2. Integrable Lagrangean Mappings with Discrete Time.- 2.1 Hamiltonian Theory.- 2.2 Heisenberg Chain with Classical Spins and the Discrete Analog of the C. Neumann System.- 2.3 The Billiard in Quadrics.- 2.4 The Discrete Analog of the Dynamics of the Top.- 2.5 Connection with the Spectral Theory of the Difference Operators: A Discrete Analogue of the Moser-Trubowitz Isomorphism.- Appendix A.- Appendix B.- References.- The Cauchy Problem for the KdV Equation with Non-Decreasing Initial Data.- 1. Reflectionless Potentials.- 2. Closure of the Sets B(??2).- 3. The Inverse Problem.- References.

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