Nonlinear integrable equations. Recursion operators, group-theoretical and Hamiltonian structures of soliton equations
type: Item
instance of: scholarly article
title: Nonlinear integrable equations. Recursion operators, group-theoretical and Hamiltonian structures of soliton equations
zbMATH Open document ID: 0628.35001
published in: Lecture Notes in Physics
publication date: 1987-01-01T00:00:00Z
review text: As the author states in the preface, the inverse spectral transform (IST) is now one of the most powerful tools for investigation of nonlinear differential equations. A great number of various nonlinear equations of physical significance have been studied in detail by the IST method. These nonlinear integrable equations possess many remarkable properties such as soliton solutions, infinite sets of integrals of motion, infinite symmetry groups, Bäcklund transformations, hierarchies of Hamiltonian structures, etc. Different versions of the IST method allow one to investigate various properties of soliton equations and their solutions. The present work is devoted to the recursion operator method which enables one to study and treat the recursion, group-theoretical and Hamiltonian structures of soliton equations from a common point of view. The recursion operator method gives the possibility to represent in a compact form the hierarchies of integrable equations connected with a given spectral problem. To construct their general Bäcklund transformations and infinite symmetry groups and to find the hierarchies of Hamiltonian structures for these equations. This method effectively works for the variety of one- and two-dimensional spectral problems. The book consists of 12 chapters with the following headings: Introduction; General ideas of the recursion operator method and adjoint representation; Linear matrix bundle; Generalizations of the linear bundle; Bäcklund-Cologero group and general integrable equations under reductions; Quadratic bundle with \({\mathbb{Z}}_ 2\) grading; Polynomial and rational bundles; General differential spectral problem; Generalization and reduction of differential and integro-differential spectral problems; Two-dimensional matrix spectral problem; Two-dimensional differential spectral problem; Towards the general theory of recursion structures of nonlinear evolution equations.
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 58C40
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 18F15
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35A30
zbMATH DE Number: 4021692
zbMATH Keywords: inverse spectral transform
zbMATH Keywords: nonlinear integrable equations
zbMATH Keywords: soliton
zbMATH Keywords: integrals of motion
zbMATH Keywords: infinite symmetry groups
zbMATH Keywords: Bäcklund transformations
zbMATH Keywords: hierarchies of Hamiltonian structures
zbMATH Keywords: recursion operator
author: B. Konopelchenko
reviewed by: Jürgen Appell
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4021692