Providing exact solutions of more than 100 nonlinear equations and models, this book begins with classical as well as more recent examples of interesting solutions on linear invariant subspaces for nonlinear operators. In the remainder of the book, the authors develop several techniques for constructing exact solutions of various nonlinear PDEs, including gas dynamics models, free-boundary problems, Green-Naghdi equations, and quasilinear pseudo-parabolic (magma) equations. Using exact solutions, they describe the evolution properties related to singularity blow-up or extinction phenomena, finite interface propagation and regularity, and the oscillatory, changing sign behavior of weak solutions near interfaces.

Galaktionov, Victor A.; Svirshchevskii, Sergey R.

Introduction. Linear Invariant Subspaces in Quasilinear Equations. Invariant Subspaces and Modules. Parabolic Equations in One Dimension. Odd-Order One-Dimensional Equations. Quasilinear Wave and Boussinesq Models in One Dimension. Applications to Partial Differential Equations in IRN. Partially Invariant Subspaces, Invariant Sets, and Generalized Separation of Variables. Sign-Invariants for Second-Order Parabolic Equations and Exact Solutions. Invariant Subspaces for Discrete Operators, Moving Mesh Methods, and Lattices. References. List of Frequently Used Abbreviations. Index.