Dispersive equations with broken symmetries

这个项目的主要目的是进一步了解对称性破缺的色散方程解的长时间行为。具体而言，主要研究者认为全球适定性，散射，孤子形成，有限时间爆破问题的方程，如以下：非捕获域外的非线性薛定谔，Gross-Pitaevskii，非线性薛定谔结合聚焦和散焦非线性，广义Korteweg-de弗里斯。这些方程的共同特征是每个方程都有一个或多个对称性破缺，如空间平移、缩放或伽利略不变性。这意味着，虽然解可以集中在空间中的任意位置（或长度尺度或频率位置），但演化并不简单地取决于集中的位置（或尺度或频率位置）。在这种情况下可能出现的一个引人注目的新现象是，最小化对称性破缺方程的解序列可以收敛到同一方程的解，但在不同的几何形状，甚至完全不同的方程的解。在这个项目中要研究的偏微分方程作为各种物理领域的有效模型出现，例如浅水中的波浪模型，克尔介质中的激光、陷阱中的玻色-爱因斯坦气体和超导体。虽然它们是用于工程目的的过度简化的模型，但对这些例子的研究使人们能够专注于对称性被打破时出现的新现象学。在这样的扰动下，结果的稳定性对于与现实世界的系统建立有意义的联系至关重要;事实上，工程限制意味着现实世界的系统永远不会有完美的对称性。相反，稳定性的失败意味着不能依靠计算机模拟来准确预测实验现象。

The main thrust of this project is to further the understanding of the long-time behavior of solutions to dispersive equations with broken symmetries. Specifically, the principal investigator considers global well-posedness, scattering, soliton formation, and finite-time blowup questions for equations such as the following: nonlinear Schrodinger outside nontrapping domains, Gross-Pitaevskii, nonlinear Schrodinger with combined focusing and defocusing nonlinearities, generalized Korteweg-de Vries. The common feature of these equations is that each has one or more broken symmetries such as spatial-translation, scaling, or Galilei invariance. By this is meant that, while the solution can concentrate at arbitrary positions in space (or length scales or frequency locations), the evolution depends nontrivially on the position (or scale or frequency location) of concentration. One of the remarkable new phenomena that can occur in this setting is that minimizing sequences of solutions to an equation with broken symmetries can converge to a solution of the same equation but in a different geometry, or even to a solution of an entirely different equation.The partial differential equations to be investigated in this project arise as effective models in various areas of physics, such as models for waves in shallow water, laser light in Kerr media, Bose-Einstein gases in a trap, and superconductors. While they are oversimplified models for engineering purposes, the study of these examples allows one to focus on the new phenomenology that arises when symmetries are broken. Stability of results under such perturbations is essential to make a meaningful connection to real-world systems; indeed, engineering limitations mean that real-world systems never have perfect symmetry. Conversely, failure of stability implies that computer simulations cannot be relied upon to accurately predict experimental phenomena.