Spatial Patterns

A study of nonlinear higher order model equations central to the description and analysis of spatio-temporal pattern formation in the natural sciences. Unique combination of results obtained by rigorous mathematical analysis and computational studies. Text exhibits the principal families of solutions--kinks, pulses and periodic solutions, and their dependence on critical eigenvalue parameters. Exposition first focuses on a single equation to achieve optimal transparency, then branches out to wider classes of equations. Includes many exercises, open problems, recent original results, and applications to mathematical physics and mechanics. Intended for mathematicians, mathematical physicists, and graduate students.

"The book is on the one hand written for mathematicians and mathematical physicists, who want to learn about this fascinating subject, and on the other hand also accessible to graduate students. One finds a large amount of exercises and open problems that can serve as a starting point for further research . . . The authors have produced a well-written book, which gives a good picture of what is known about the canonical equation."

Quantum Information and Computation

"The book is very well written in a very clear and readable style, which makes it accessible to a nonspecialist or graduate student. There are a large number of exercises, which fill in details of proofs or provide illuminating examples or straightforward generalisations as well as a good number of open problems. There are also a large number of numerically computed graphs of branching curves and bifurcation curves throughout the book, which provide insights into the mathematically formulated results. The book is a valuable contribution to the literature, for both the specialist and the nonspecialist reader."

Mathematical Reviews

Klappentext
This monograph is divided into two parts. The first part presents unifying properties of a family of the new higher order model equations, as well as a new mathematical way of studying them. Topics include the stationary solutions of the Swift-Hohenberg equation and the extended Fisher-Kolmogorov equation which are well established fourth non-linear model equations of parabolic type. In the second part, the exposition centres around applications of the theory presented earlier. Exercises for self-study or classroom use are included.

Inhalt
1 Introduction.- 1.1 Model equations.- 1.2 The FisherKolmogorov equation.- 1.3 An overview.- 1.4 Methods.- 1.5 About the presentation.- 1.6 Bibliographical notes.- I The Symmetric Bistable Equation.- 2 Real Eigenvalues.- 3 Estimates.- 4 Periodic Solutions.- 5 Kinks and Pulses.- 6 Chaotic Solutions.- 7 Variational Problems.- II Related Equations.- 8 The Asymmetric Double-Well Potential.- 9 The SwiftHohenberg Equation.- 10 Waves in Nonlinearly Supported Beams.- References.