| 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 Schrödinger 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 a class of well-known higher order pattern-forming model equations, and for graduate students who are looking for an exciting and promising field of research. |
