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Exact Solutions and Invariant Subspaces of Nonlinear Partial
Differential Equations in Mechanics and Physics is the first book
to provide a systematic construction of exact solutions via linear
invariant subspaces for nonlinear differential operators. Acting as
a guide to nonlinear evolution equations and models from physics
and mechanics, the book focuses on the existence of new exact
solutions on linear invariant subspaces for nonlinear operators and
their crucial new properties. This practical reference deals with
various partial differential equations (PDEs) and models that
exhibit some common nonlinear invariant features. It begins with
classical as well as more recent examples of solutions on invariant
subspaces. In the remainder of the book, the authors develop
several techniques for constructing exact solutions of various
nonlinear PDEs, including reaction-diffusion and gas dynamics
models, thin-film and Kuramoto-Sivashinsky equations, nonlinear
dispersion (compacton) equations, KdV-type and Harry Dym models,
quasilinear magma equations, and Green-Naghdi equations. Using
exact solutions, they describe the evolution properties of blow-up
or extinction phenomena, finite interface propagation, and the
oscillatory, changing sign behavior of weak solutions near
interfaces for nonlinear PDEs of various types and orders. The
techniques surveyed in Exact Solutions and Invariant Subspaces of
Nonlinear Partial Differential Equations in Mechanics and Physics
serve as a preliminary introduction to the general theory of
nonlinear evolution PDEs of different orders and types.
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