Spatial Patterns offers a study of nonlinear higher order model
equations that are central to the description and analysis of
spatio-temporal pattern formation in the natural sciences. Through
a unique combination of results obtained by rigorous mathematical
analysis and computational studies, the text exhibits the principal
families of solutions, such as kinks, pulses and periodic
solutions, and their dependence on critical eigenvalue parameters,
and points to a rich structure, much of which still awaits
exploration.
The exposition unfolds systematically, first focusing on a
single equation to achieve optimal transparency, and then branching
out to wider classes of equations. The presentation is based on
results from real analysis and the theory of ordinary differential
equations.
Key features:
* presentation of a new mathematical method specifically
designed for the analysis of multi-bump solutions of reversible
systems
* strong emphasis on the global structure of solution
branches
* extensive numerical illustrations of complex solutions and
their dependence on eigenvalue parameters
* application of the theory to well-known equations in
mathematical physics and mechanics, such as the Swift--Hohenberg
equation, the nonlinear SchrAdinger equation and the equation for
the nonlinearly supported beam
* includes recent original results by the authors
* exercises scattered throughout the text to help illuminate the
theory
* many research problems
The book is intended for mathematicians who wish to become
acquainted with this new area of partial and ordinary differential
equations, for mathematical physicists who wish to learn about the
theory developed for aclass of well-known higher order
pattern-forming model equations, and for graduate students who are
looking for an exciting and promising field of research.
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