Quantified dynamics of nonlinear dispersive PDE

本项目将研究非线性色散方程，特别是非线性薛定谔方程、Korteweg-de Vries方程和非线性Klein-Gordon方程。研究孤子动力学和有限时间爆破解的新技术已经出现，如局部维里估计、新的Lyapunov泛函和集中紧致性机制。首席调查员将寻求在调查几个与科学有关的问题时进一步实施和发展这些技术。我们将考虑单个和多个明、暗孤子在外势或含时非线性系数影响下的孤波动力学。将探索在一个方向上的强限制势产生有效降维的能力。新的初值准则将被用来预测解的有限时间爆破，PI将寻求构造最近在数值上观察到的新类型的奇异环爆破解。我们还将研究整体解的长期行为。本项目中所研究的方程作为重要的物理模型出现。孤子表现为良好的局域稳定结构，而爆破与波的急剧聚焦和物理模型的最终崩溃有关。例如，非线性薛定谔方程是玻色-爱因斯坦凝聚体中一组超冷原子的波函数。自从1995年在实验室实现诺贝尔奖以来，进一步的实验已经产生了孤立子和有限时间爆炸，现在有大量的物理文献激励着首席研究人员打算考虑的许多问题。提出的问题将物理学家、数值分析师和纯数学家聚集在一起。它们也是一个很好的教育工具，因为它们可以为本科生和研究生量身定做。为了激发这些学生的兴趣，首席研究员将在本科生研讨会上发表演讲，并在布朗大学教授一门高级主题课程。例如，这项研究的网络演示可以通过由这笔资金资助的夏季本科生项目来进行。

This project will study nonlinear dispersive equations, with particular emphasis on the nonlinear Schroedinger equation, Korteweg-de Vries equation, and nonlinear Klein-Gordon equation. New techniques have emerged for studying the dynamics of solitons and finite-time blow-up solutions, such as local virial estimates, new Lyapunov functionals, and concentration compactness machinery. The principal investigator will seek to implement and develop these techniques further in the investigation of several scientifically relevant problems. The dynamics of solitary waves in the presence of an external potential or under the influence of a time-dependent nonlinear coefficient for single and multiple bright and dark solitons will be considered. The ability of a strong confining potential in one direction to produce an effective reduction in dimensionality will be explored. New criteria on initial data for predicting finite-time blow-up of solutions will be developed, and the PI will seek to construct new types of singular ring blow-up solutions recently observed numerically. Long-time behavior of global solutions will also be studied.The equations studied in this project arise as important physical models. Solitons appear as well-localized stable structures, while blow-up is associated with a sharp focusing of a wave and ultimate break-down of the physical model. For example, the nonlinear Schroedinger equation is the wave function for a collection of ultracold atoms in a Bose-Einstein condensate. Since its Nobel-prize winning realization in the laboratory in 1995, further experiments have produced solitons and finite-time blow-up, and there is now a large body of physics literature motivating many of the problems the principal investigator intends to consider. The problems proposed bring together physicists, numerical analysts, and pure mathematicians. They also serve as an excellent educational tool, since they can be tailored to students at the undergraduate and graduate level. To stimulate interest among such students, the principal investigator will give talks in the undergraduate seminar and teach an advanced topics course at Brown University. Web demonstrations of this research could be produced, for example, through a summer undergraduate project funded by this grant.