Local and global dynamics for nonlinear dispersive equations

所提出的工作致力于非线性色散方程领域内的一系列问题。主要目标是很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，该项目将研究在弯曲背景下发展的波和薛定谔方程的全局时间色散特性。这在一定程度上也是针对广义相对论中爱因斯坦方程的施瓦茨柴尔德和科尔黑洞解的猜想稳定性。在非线性方面，需要探索的一个关键课题是多尺度分析。这对应于在线性和非线性效应强烈交错的区域中发展的非线性色散方程。因此，虽然人们可能会在非常短的、可能与频率相关的时间尺度上看到相干的线性动力学，但这些动力学在较大的时间内会丢失。接下来的挑战是发现在越来越长的时间尺度上保持连贯的更大的模式。广义地说，色散方程是其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，这可能会导致传播模式的改变(产生新的波)，或者可能导致数学家所说的“爆炸”现象。这个项目中考虑的许多方程都起源于物理理论，如广义相对论、多体量子场理论、表面波传播和等离子体物理。因此，希望这项研究的结果能对相应的物理现象有所帮助。该项目涉及研究生和博士后，以及几名国内外合作者。

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.