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Full Description
The idea of devoting a complete book to this topic was born at one of the Workshops on Nonlinear and Turbulent Processes in Physics taking place reg ularly in Kiev. With the exception of E. D. Siggia and N. Ercolani, all authors of this volume were participants at the third of these workshops. All of them were acquainted with each other and with each other's work. Yet it seemed to be somewhat of a discovery that all of them were and are trying to understand the same problem - the problem of integrability of dynamical systems, primarily Hamiltonian ones with an infinite number of degrees of freedom. No doubt that they (or to be more exact, we) were led to this by the logical process of scientific evolution which often leads to independent, almost simultaneous discoveries. Integrable, or, more accurately, exactly solvable equations are essential to theoretical and mathematical physics. One could say that they constitute the "mathematical nucleus" of theoretical physics whose goal is to describe real clas sical or quantum systems. For example, the kinetic gas theory may be considered to be a theory of a system which is trivially integrable: the system of classical noninteracting particles. One of the main tasks of quantum electrodynamics is the development of a theory of an integrable perturbed quantum system, namely, noninteracting electromagnetic and electron-positron fields.
Contents
Why Are Certain Nonlinear PDEs Both Widely Applicable and Integrable?.- Summary.- Addendum.- References.- Painlevé Property and Integrability.- 1. Background.- 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.- 3. The Integrable Hénon-Heiles System: A New Result.- 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.- 2. The Burgers Type Equations.- 3. Canonical Conservation Laws.- 4. Integrable Equations.- Historical Remarks.- References.- Integrability of Nonlinear Systems and Perturbation Theory.- 1. Introduction.- 2. General Theory.- 3. Applications to Particular Systems.- Appendix I.- Appendix II.- Conclusion.- References.- What Is an Integrable Mapping?.- 1. Integrable Polynomial and Rational Mappings.- 2. Integrable Lagrangean Mappings with Discrete Time.- 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.
