Asymptotic and qualitive analysis of nonlinear partial differential equations

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Partial differential equations provide a mathematical representation of the relationship between the temporal and spatial variation of an unknown function or set of functions. Such equations arise in a natural way in numerous settings including fluid motion, chemical reactions, developmental biology, and spatial ecology. In general such equations involve nonlinear effects corresponding to, for example, underlying saturation effects and interactions between model components. Such effects while making the models more realistic, also make the solution of such equations much more challenging and indeed exact closed form solutions are typically not accessible except in certain idealized circumstances. Therefore alternative methodologies based on approximation and analysis of qualitative features in certain limiting circumstances become essential in understanding the nature of the models, their solutions and in turn the underlying physical system. Within this context, this proposal considers two classes of nonlinear partial differential equations and the analytic and computational methodologies used to study them. First I intend to continue an ongoing study of resonant solutions of nonlinear wave equations which underly waves in shallow fluids such as atmospheric flows over topography. The key goals involve both the study of solutions in generalized circumstances, as well as the development of the analytic methodology which in turn may be applicable to a wider range of problems. Second is the study of a widely applicable and increasingly prevalent class of equations which arise in the context of chemical and biological applications involving the combination of diffusive and reaction processes. The focus of this aspect of the study will be to study in general the robustness of certain solution features such as pattern formation, waves and long time behaviour under the influence of perturbative effects. In addition with the growing complexity of the models, numerical approximation of solutions is becoming an increasingly crucial analytic tool. As such this study will also investigate and develop more efficient computational methodologies suited for models of this type.