What Is Integrability? - Brossura

Brossura

ISBN 10: 3642887058 ISBN 13: 9783642887055

Casa editrice: Springer, 2012

Riassunto

This monograph deals with integrable dynamic systems with an infinite number of degrees of freedom. Leading scientists were invited to discuss the notion of integrability with two main points in mind: 1. a presentation of the various recently elaborated methods for determining whether a given system is integrable or not; 2. to understand the increasingly more important role of integrable systems in modern applied mathematics and theoretical physics.

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Contenuti

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.

Dettagli dell'edizione

• Editore: Springer
• Data di pubblicazione: 2012
• Lingua: Inglese
• ISBN 10: 3642887058
• ISBN 13: 9783642887055
• Rilegatura: Copertina flessibile
• Numero di pagine: 344
• Redattore: Zakharov Vladimir E.
• Contatto del produttore: non disponibile
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