Stability of Nonlinear Waves in Dissipative and Dispersive PDE

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The focus of the proposed research is stability and long-time dynamics of travelling wave solutions of nonlinear partial differential equations that arise in many areas of science and engineering. This proposal will develop a systematic analysis of spectral, linearized, and nonlinear stability and long-time dynamics of spatially periodic traveling structures in a variety of models. In particular, stability of uni-directional and multiply periodic traveling waves with respect to uni-lateral as well as higher-dimensional transverse perturbations will be considered. Furthermore, stability of such traveling structures to non-localized perturbations inducing global phase shifts will be investigated.The main thrust of the project is a study of stability and behavior of traveling waves in many equations arising, for example, as models of water waves and thin films, and in plasma physics and engineering. Such solutions travel at constant speed without changing shape, and often form fundamental building blocks for more complicated solutions of the model equations. Studying their stability to small perturbations (i.e. whether the wave is quickly restored to the orginal wave form) is of practical importance, since waves that are unstable do not naturally manifest themselves in physical situations, except possibly as transient phenomena. This proposal aims to provide researchers with practical and efficient "rules of thumb" to ascertain stability of mathematical solutions arising in physical models. A central component of the proposal is the educational and professional training of undergraduate and graduate students, who will receive interdisciplinary instruction through seminars, conferences, and special topics courses.