Harmonic Analysis, Mathematical Physics, and Nonlinear PDE

本项目旨在探讨数学物理中出现的方程的解的长时间行为，或证明在有限时间内会形成奇点。所考虑的方程通常承认非线性束缚态（孤子或瞬子），并且最近有许多研究致力于此类解的摄动分析。众所周知，聚焦型非线性薛定谔方程和波动方程的解可能在有限时间内爆炸（例如，如果数据的能量为负）。然而，事实证明，如果数据属于有限协维的子流形（动力系统语言中的“中心稳定”流形），则存在全局解。我们将研究是否有一个流形把一个爆炸区和一个散射区分开。我们的目标是对爆炸现象有更深入的了解。近年来，关于爆破的二维球面临界波图方程的研究取得了一些进展。它可以用一种非常精确和定量的方式来表示，这个方程的爆发是通过一个非常数谐波图的能量冒泡而发生的。此外，爆炸速率可以被先验地规定。类似的现象也出现在三加一维的半线性能量临界聚焦方程中。目前我们还不知道哪一类方程承认这种现象。科学和工程的成功很大程度上取决于其在建模和计算模拟方面对数学工具的有效使用。非线性薛定谔方程出现在光学的各种应用中，其中一个束缚态（孤子）代表一个粒子或光束，它在很长一段时间内运动而不解体。一个重要的问题是了解这些孤子的稳定性或不稳定性。也就是说，它们在小扰动下是否会持续？从理论上理解这些问题是非常困难的，需要对数学问题有新的认识。这个项目将调查这些问题，并开发出可供实践科学家和工程师使用的方法。

Harmonic Analysis, Mathematical Physics and Nonlinear PDEAbstract of Proposed ResearchWilhelm Schlag This project is explore the long time behavior, or prove that singularities will form in finite time, of solutions of equations that arise in mathematical physics. The equations under consideration typically admit nonlinear bound states (solitons or instantons) and much research has recently been devoted to the perturbative analysis of such solutions. It is well known that solutions of nonlinear Schroedinger and wave equations of the focusing type may blow up in finite time (if the energy of the data is negative, for example). It turns out, however, that global solutions exist if the data belong to a submanifold of finite co-dimension (a "center-stable" manifold in the language of dynamical systems). We shall investigate whether there is a manifold that divides a region of blow-up from one of scattering. Our goal is to obtain a deeper understanding of blow-up phenomena. Recently, progress was made for the critical wave-map equation into the two-dimensional sphere with regard to blow-up. It can be shown in a very precise and quantitative way that blow-up for this equation occurs through the bubbling off of energy via a non-constant harmonic map. Moreover, it turns out that the blow-up rate can be prescribed a priori. Similar phenomena occur for the semi-linear energy critical focusing equation in three plus one dimensions. Currently we do not understand which classes of equations admit this kind of phenomenon.Much of the success of science and engineering lies with its effective use of mathematical tools, both in terms of modeling and for computational simulation. The nonlinear Schroedinger equation arises in various applications in optics where a bound state (soliton) for represents a particle, or beam, that travels for a long time without disintegrating. An important issue is to understand the stability or instability of these solitons. That is, whether they persist under small perturbations or not? The theoretical understanding of these issues is very difficult, and is requiring new insights into mathematical problems. This project will investigate these problems and develop methods that may be used by practicing scientists and engineers.