"Analysis and applications of nonlinear evolution equations: waves, patterns, and singularities."

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The aim of the proposed research is to understand fundamental mathematical equations -- called Partial Differential Equations -- which describe the behaviour of physical systems, such as ferromagnets, superfluids, superconductors, and lasers. The equations are non-linear, cannot be solved explicitly, and are often difficult to solve even with the help of a computer. So we aim instead to obtain a clear picture of how solutions behave -- to determine their essential qualitative features -- by using various mathematical techniques. In the case of ferromagnets, for example, we would like to determine if the commonly used mathematical models always give a fair description of the physics, or if instead they might break down ("formation of singularities"), perhaps necessitating a modified, finer model in certain circumstances. And we would like to exploit the equations to understand the long-time behaviour of the magnet, after transient effects have disappeared. A wide variety of physical systems, such as superfluids, superconductors, lasers, water waves, Bose-Einstein condensates and others, are governed by similar equations, for which the same basic questions can be asked, and which can be productively analyzed with similar techniques.