Integrability

ISBN-13: 9783642099908 / Angielski / Miękka / 2010 / 339 str.

Alexander Mikhailov

Opis

The principal aim of the book is to give a comprehensive account of the variety of approaches to such an important and complex concept as Integrability. Dev- oping mathematical models, physicists often raise the following questions: whether the model obtained is integrable or close in some sense to an integrable one and whether it can be studied in depth analytically. In this book we have tried to c- ate a mathematical framework to address these issues, and we give descriptions of methods and review results. In the Introduction we give a historical account of the birth and development of the theory of integrable equations, focusing on the main issue of the book - the concept of integrability itself. A universal de nition of Integrability is proving to be elusive despite more than 40 years of its development. Often such notions as "- act solvability" or "regular behaviour" of solutions are associated with integrable systems. Unfortunately these notions do not lead to any rigorous mathematical d- inition. A constructive approach could be based upon the study of hidden and rich algebraic or analytic structures associated with integrable equations. The requi- ment of existence of elements of these structures could, in principle, be taken as a de nition for integrability. It is astonishing that the nal result is not sensitive to the choice of the structure taken; eventually we arrive at the same pattern of eq- tions.

Spis treści

Symmetries of Differential Equations and the Problem of Integrability.- Number Theory and the Symmetry Classification of Integrable Systems.- Four Lectures: Discretization and Integrability. Discrete Spectral Symmetries.- Symmetries of Spectral Problems.- Normal Form and Solitons.- Multiscale Expansion and Integrability of Dispersive Wave Equations.- Painlevé Tests, Singularity Structure and Integrability.- Hirota’s Bilinear Method and Its Connection with Integrability.- Integrability of the Quantum XXZ Hamiltonian.

Blurb

This is a unique collection of lectures on integrability, intended for graduate students or anyone who would like to master the subject from scratch, and written by leading experts in the field including Fields Medallist Serge Novikov. Since integrable systems have found a wide range of applications in modern theoretical and mathematical physics, it is important to recognise integrable models and, ideally, to obtain a global picture of the integrable world. The main aims of the book are to present a variety of views on the definition of integrable systems; to develop methods and tests for integrability based on these definitions; and to uncover beautiful hidden structures associated with integrable equations.