Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations: 507 (Mathematics and Its Applications, 507) - Hardcover

Krasil'shchik, I.S.; Kersten, P.H.

Hardcover

ISBN 10: 0792363159 ISBN 13: 9780792363156

Publisher: Springer, 2000

Synopsis

To our wives, Masha and Marian Interest in the so-called completely integrable systems with infinite num­ ber of degrees of freedom was aroused immediately after publication of the famous series of papers by Gardner, Greene, Kruskal, Miura, and Zabusky [75, 77, 96, 18, 66, 19J (see also [76]) on striking properties of the Korteweg-de Vries (KdV) equation. It soon became clear that systems of such a kind possess a number of characteristic properties, such as infinite series of symmetries and/or conservation laws, inverse scattering problem formulation, L - A pair representation, existence of prolongation structures, etc. And though no satisfactory definition of complete integrability was yet invented, a need of testing a particular system for these properties appeared. Probably one of the most efficient tests of this kind was first proposed by Lenard [19]' who constructed a recursion operator for symmetries of the KdV equation. It was a strange operator, in a sense: being formally integro-differential, its action on the first classical symmetry (x-translation) was well-defined and produced the entire series of higher KdV equations; but applied to the scaling symmetry, it gave expressions containing terms of the type J u dx which had no adequate interpretation in the framework of the existing theories. It is not surprising that P. Olver wrote "The de­ duction of the form of the recursion operator (if it exists) requires a certain amount of inspired guesswork. . . " [80, p.

About this title

Synopsis

This is a detailed exposition of algebraic and geometrical aspects related to the theory of symmetries and recursion operators for nonlinear partial differential equations (PDE), both in classical and in super, or graded, versions. It contains an original theory of Frolicher-Nijenhuis brackets which is the basis for a special cohomological theory naturally related to the equation structure. This theory gives rise to infinitesimal deformations of PDE, recursion operators being a particular case of such deformations. Efficient computational formulas for constructing recursion operators are deduced and, in combination with the theory of coverings, lead to practical algorithms of computations. Using these techniques, previously unknown recursion operators (together with the corresponding infinite series of symmetries) are constructed. In particular, complete integrability of some superequations of mathematical physics (Korteweg-de Vries, nonlinear Schrodinger equations, etc.) is proved. It should be of interest to mathematicians and physicists specializing in geometry of differential equations, integrable systems and related topics.

Edition details

• Publisher: Springer
• Publication date: 2000
• Language: English
• ISBN 10: 0792363159
• ISBN 13: 9780792363156
• Binding: Hardcover
• Number of pages: 400