Local and global dynamics for nonlinear dispersive equations

提出的工作致力于非线性色散方程领域内的广泛问题。主要目的是为了更好地理解线性色散动力学和非线性效应之间的相互作用。在线性方面，该项目将研究波浪和薛定谔方程在弯曲背景上演化的全局时域色散特性。这也在一定程度上指向了广义相对论中爱因斯坦方程的Schwarzchild和Kerr黑洞解的推测稳定性。在非线性方面，要探讨的一个关键问题是多尺度分析。这对应于非线性色散方程，它在线性和非线性效应强烈交织的情况下演化。因此，虽然人们可能在很短的时间尺度上看到连贯的线性动力学，可能是频率相关的，但在更大的时间尺度上，这些动力学就消失了。接下来的挑战是发现在越来越长的时间尺度上保持一致的更大的模式。广义地说，色散方程的解可以被认为是在不同方向上传播的波的叠加。非线性效应允许这些波相互作用，这可能导致改变传播模式（产生新的波），或者可能导致数学家所描述的“爆炸”现象。本项目中考虑的许多方程都起源于物理理论，如广义相对论、多体量子场论、表面波传播和等离子体物理。因此，希望研究结果能对相应的物理现象有所启示。该项目涉及研究生和博士后，以及一些国内外合作者。

The proposed work is devoted to a broad range of problems within the field of nonlinear dispersive equations. The primary goal is to achieve a good understanding of the interplay between linear dispersive dynamics and nonlinear effects. On the linear side, the project will study the global-in-time dispersive properties for wave and Schroedinger equations evolving on curved backgrounds. This is also in part directed toward the conjectured stability of the Schwarzchild and Kerr black hole solutions of the Einstein equations in general relativity. On the nonlinear side, a key topic to be explored is that of multiscale analysis. This corresponds to nonlinear dispersive equations that evolve in regimes where linear and nonlinear effects are strongly interlaced. Thus while one might see coherent linear dynamics on a very short, possibly frequency dependent time scale, these are lost at larger times. The challenge is then to uncover larger patterns that remain coherent on longer and longer time scales.Broadly speaking, dispersive equations are equations whose solutions can be thought of as superpositions of waves travelling in different directions. Nonlinear effects allow these waves to interact, which may lead to modified propagation patterns (creating new waves) or possibly to the phenomenon described by mathematicians as "blow-up." Many of the equations considered in this project have their origins in physical theories such as general relativity, many-body quantum field theory, surface wave propagation, and plasma physics. For this reason, it is hoped that the results of the research may shed some light on the corresponding physical phenomena. The project involves graduate students and postdocs, as well as several domestic and international collaborators.