Nonlinear integrable equations. Recursion operators, group-theoretical and Hamiltonian structures of soliton equations
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.
scientific article; zbMATH DE number 4021692