Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations
I.S. Krasil'shchik and P.H.M. Kersten

Mathematics and its applications, Volume 507,
Kluwer Academic Publishers, Dordrecht

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 Frölicher–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 Schrödinger 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.

1. Classical symmetries.
2. Higher symmetries and conservation laws.
3. Nonlocal theory.
4. Brackets.
5. Deformations and recursion operators.
6. Super and graded theories.
7. Deformations of supersymmetric equations.
8. Symbolic computations in differential geometry.
Bibliography. Index.