Symmetries and recursion operators for classical and supersymmetric differential equations. (English) Zbl 0952.35001
Mathematics and its Applications (Dordrecht). 507. Dordrecht: Kluwer Academic Publishers. xvi, 384 p. (2000).
The book presents a self-contained exposition of the theory of recursion operators, which are closely related to hidden symmetries and Lie-Bäcklund transformations of integrable equations. They allow one to construct infinite hierarchies of integrable models, starting with a simple classical integrable equation, such as the Korteweg - de Vries, Burgers, nonlinear Schrödinger, or massive Thirring equations. In the book, the theory is presented from an abstract geometric point of view. Applications to the above-mentioned classical equations are included too. Recursion operators and hierarchies for supersymmetric generalizations of the classical integrable equations (i.e., mixed Boson-Fermion models related to them) are also presented in a detailed form.
Reviewer: B.A.Malomed (Tel Aviv)
MSC:
35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |
37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |
35Q55 | NLS equations (nonlinear Schrödinger equations) |
37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
35Q53 | KdV equations (Korteweg-de Vries equations) |