Critical Nonlinear Dispersive Equations

本专题将研究色散偏微分方程。这些方程模拟了许多不同的现象，其中包括各种波的传播，如水波和激光。这些方程也模拟了粒子物理学中的某些现象。本研究课题旨在了解这类方程及其相关系统的长时间行为，在此奋进中将研究许多问题。本研究将主要围绕三个著名的色散偏微分方程：波动方程、薛定谔方程和Korteweg de弗里斯方程。很多是未知的聚焦薛定谔和Korteweg德弗里斯（KdV）的问题，特别是质量临界问题。PI打算研究基态质量以上质量的聚焦，质量临界薛定谔问题，以及基态质量以下大质量的聚焦gKdV问题。PI还计划将最近的工作与I-方法扩展到波和Klein-Gordon方程。最后，PI将研究超双曲薛定谔方程。

This project will study dispersive partial differential equations. Such equations model many different phenomena, among them the propagation of various kinds of waves, such as water waves and laser light. These equations also model certain phenomena in particle physics. This project attempts to understand long-time behavior of such equations and related systems.There are a number of problems that will be studied in the course of this endeavor. The study will mainly revolve around the three well-known dispersive partial differential equations: wave, Schrodinger, and Korteweg de Vries. A great deal is unknown for the focusing Schrodinger and Korteweg de Vries (KdV) problems, particularly for the mass-critical problem. The PI intends to study the focusing, mass-critical Schrodinger problem for mass above the mass of the ground state, as well as the focusing gKdV problem for large mass below the mass of the ground state. The PI also plans to extend recent work with the I-method to the wave and Klein-Gordon equations. Finally, the PI will study the ultra hyperbolic Schrodinger equation.