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

拟议研究的目的是理解基本数学方程-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，不能显式求解，即使在计算机的帮助下也很难求解。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如何表现的清晰图像-以确定其基本的定性特征。 例如，在铁磁体的情况下，我们想确定通常使用的数学模型是否总是给出物理学的公平描述，或者它们是否会崩溃（“奇点的形成”），也许在某些情况下需要修改，更精细的模型。我们希望利用这些方程来理解瞬态效应消失后磁体的长期行为。 各种各样的物理系统，如超流体、超导体、激光、水波、玻色-爱因斯坦凝聚等，都由类似的方程控制，可以提出相同的基本问题，并可以用类似的技术进行有效的分析。

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.