Quantified dynamics of nonlinear dispersive PDE

非线性色散波动方程，特别是非线性薛定谔方程、非线性Klein-Gordon方程和Korteweg-de Vries方程，可以显示出称为孤子的相干和稳定解，以及在有限时间内经历爆破的解。首席研究人员将证明严格的结果，这些结果给出了在许多科学相关的背景下这些方程的解的定量描述。这些结果描述了小的随机外场对孤子或爆破解的动力学的影响，多个孤子之间的相互作用力对其动力学的影响，快速振荡的非线性系数对孤子的稳定影响，以及强限制外场可以降低动力学的有效维度的程度。这位首席研究员以前已经开发了基于辛投影的新技术，以及它与这些方程的哈密顿结构的关系，以研究相关问题。他将继续改进和扩展他的方法来解决这些更复杂的问题。这个项目中研究的方程是作为重要的物理模型出现的。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发了这个项目将考虑的许多问题。这项工作将纯数学、数值分析和物理学结合在一起。首席研究员与美国和海外几个机构的教职员工合作，并监督几名研究生与该项目相关的工作。该项目的某些方面可以进行调整，以便于本科生使用，特别是那些涉及计算机模拟和制作网络演示的部分。

Nonlinear dispersive wave equations,in particular the nonlinear Schrodinger, nonlinear Klein-Gordon, and Korteweg-de Vries equations, can exhibit coherent and stable solutions called solitons, as well as solutions that experience blow-up in finite time. The principal investigator will prove rigorous results that give quantitative descriptions of solutions to these equations in a number of scientifically relevant contexts. Among these are results describing the effect of a small random external field on the dynamics of solitons or blow-up solutions, the effect of interaction forces between multiple solitons on their dynamics, the stabilizing influence of a rapidly oscillating nonlinear coefficient on a soliton, and the extent to which a strong confining external field can reduce the effective dimension of the dynamics. The principal investigator has previously developed new techniques based on symplectic projection, and its relation to the Hamiltonian structure of these equations, to study related problems. He will continue to refine and extend his methods to address these more sophisticated problems.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 Schrodinger 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 this project will consider. This work brings together pure mathematics, numerical analysis, and physics. The principal investigator collaborates with faculty at several institutions in the U.S. and overseas and supervises several graduate students on work related to the project. Certain aspects of the project can be adapted to be accessible to undergraduates, especially those components involving computer simulation and the production of web demonstrations.