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Konopelchenko B.G. Nonlinear Integrable Equations: Recursion Operators, Group-Theoretical and Hamiltonian Structures of Soliton Equations

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Konopelchenko B.G. Nonlinear Integrable Equations: Recursion Operators, Group-Theoretical and Hamiltonian Structures of Soliton Equations
Springer, 1987. — 370 p.
The inverse spectral transform (IST) method discovered two decades ago 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, Backlund 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 Backlund 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.
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