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.