Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations (Paperback)

I.S. Krasil'shchik

ISBN 10: 904815460X / ISBN 13: 9789048154609

Published by Springer, Dordrecht, 2010

About this Item

Paperback. This book 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. Audience: The book will be of interest to mathematicians and physicists specializing in geometry of differential equations, integrable systems and related topics. To our wives, Masha and Marian Interest in the so-called completely integrable systems with infinite numA 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 deA duction of the form of the recursion operator (if it exists) requires a certain amount of inspired guesswork. . . " [80, Shipping may be from our Sydney, NSW warehouse or from our UK or US warehouse, depending on stock availability.

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Bibliographic Details

• Title: Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations (Paperback)
• Publisher: Springer, Dordrecht
• Publication Date: 2010
• Language: English
• ISBN 10: 904815460X
• ISBN 13: 9789048154609
• Binding: Paperback
• Edition: 1st Edition
• Condition: new

About this title

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.

Review

This book is probably the most informative and highly helpful work on the subject of structural aspects of composites, and merits an honored place in the study of every student and researcher associated with composites. Practicing composite structures engineers dealing with composites cannot fail to find much of interest and motivate them in this volume. It demands a place in their libraries. Indeed a worthwhile investment that continues the level of excellence associated with the earlier edition. – Current Engineering Practice, vol. 47, 2004.