Nearly integrable infinite-dimensional Hamiltonian systems / Sergej B. Kuksin

This book is devoted to nonlinear Hamiltonian perturbations of integrable (linear and nonlinear) Hamiltonian systems of large and infinite dimension.

The main part of the book deals with perturbations of linear Hamiltonian equations, depending on a finite-dimensional parameter. However, it turns out that the problem of persistence quasiperiodic solutions of a nonlinear integrable system can be reduced to the same problem for a parameter-depending linear equation.

The goal of this book is to formulate the main theorem in a way to simplify its nonlinear applications and to prove that “many” quasiperiodic solutions of the unperturbed integrable system, which describes a conservative physical system with one spatial dimension, persist under perturbations. The main theorem gives some explanations to the recurrence effect in spatially one-dimensional systems. It proves that in some strict sense the one-dimensional world “is not very ergodic”. A rather expanded discussion of the main theorem and its applications is given in the introduction. (Zentralblatt)