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Symmetries and integrability. (English) Zbl 0639.35075

Integrable nonlinear evolution equations in one-spatial and one-temporal dimensions possess a remarkably rich algebraic structure: Infinitely many symmetries and conserved quantities, existence of a bi-Hamiltonian formulation etc. A certain operator \(\Phi\), called recursion operator, plays a central role in investigating the above algebraic properties. Recently the above theory has been extended to equations in two spatial and one temporal dimensions. In particular, the multidimensional analogue of the operator \(\Phi\) has been found and its bi-Hamiltonian factorization has been established. The above and other aspects of integrable equations including mastersymmetries, are reviewed. Furthermore, a definition of integrability is proposed based on symmetry considerations.
Reviewer: A.S.Fokas

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35A30 Geometric theory, characteristics, transformations in context of PDEs
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
35G20 Nonlinear higher-order PDEs
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