Harmonic Analysis, Mathematical Physics, and Nonlinear PDE

调和分析，数学物理和非线性偏微分方程拟议的研究摘要威廉·施拉格这个项目是探索长期的行为，或证明奇点将在有限时间内形成，在数学物理中出现的方程的解决方案。所考虑的方程通常承认非线性束缚态（孤子或瞬子）和许多研究最近一直致力于微扰分析这样的解决方案。众所周知，非线性薛定谔方程和聚焦型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，如果数据属于有限余维的子流形（动力系统语言中的“中心稳定”流形），则存在全局解。我们将研究是否存在一个流形，它将一个爆破区域与一个散射区域分开。我们的目标是更深入地了解爆破现象。近年来，临界波映象方程在爆破方面的研究取得了一些进展。它可以以一种非常精确和定量的方式表明，这个方程的爆破是通过一个非常数的调和映射，通过能量的冒泡而发生的。此外，事实证明，爆破速率可以预先规定。类似的现象发生在三维加一维的半线性能量临界聚焦方程。目前我们还不知道哪类方程允许这种现象。科学和工程的成功很大程度上取决于它在建模和计算模拟方面有效地使用了数学工具。非线性薛定谔方程出现在光学中的各种应用中，其中束缚态（孤子）表示长时间行进而不分裂的粒子或光束。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持续存在？ 对这些问题的理论理解是非常困难的，需要对数学问题有新的见解。该项目将调查这些问题，并开发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.