Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics
This monograph is devoted to the construction and the properties of exact solutions with the help of finite dimensional invariant subspaces. These subspaces reduce evolution partial differential equations to finite dimensional dynamical systems. In the introduction, the authors explain the merit of invariant subspaces for exact solutions and in the first chapter they present basic examples (there are 41 examples in this chapter). In the second chapter invariant subspaces of maximal dimension for ordinary differential equations are described. The core of the book consists of chapters 3-5 in which invariant subspaces for nonlinear partial differential equations in one dimension are studied. The main attention is paid to thin film, Kuramoto-Sivashinsky, Kortweg-de Vries, Harry Dym, quasilinear and Boussinesq models.
Chapter 6 deals with nonlinear partial differential equations in RN. A more general notion of invariant sets leads to overdetermined dynamical systems. It is shown in chapter 7 that such systems may have solutions in some special cases. Invariant sets are connected with differential constraints and also with sign invariant operators for a given equation. These are studied in chapter 8. The last chapter is devoted to discrete operators. The book contains a considerable number of equations: the index gives 157 different equations. All chapters conclude with bibliographic comments as well as with open problems of a research character. This exhaustive monograph is addressed to advanced graduate students and to researchers in physics and engineering, as well as those who work with evolution partial differential equations, who will surely benefit from a rich and carefully chosen bibliography.