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

拟议研究的目的是理解基本的数学方程-称为偏微分方程-描述物理系统的行为，如铁磁体，超流体，超导体和激光。这些方程是非线性的，不能明确地求解，即使借助计算机也很难求解。因此，我们的目标是通过使用各种数学技术来获得解决方案行为的清晰图像，以确定其基本的定性特征。

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.