Quantified dynamics of nonlinear dispersive PDE

非线性色散偏微分方程的量化动力学

• 批准号: 0901582
• 负责人: Justin Holmer
• 金额: $ 15万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901582&HistoricalAwards=false
• 关键词: Quantified; dynamics; nonlinear; dispersive; PDE

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.

本计画将研究非线性色散方程，特别著重于非线性薛定谔方程、柯尔特韦格-德弗里斯方程及非线性克莱因-戈登方程。 新的技术已经出现了研究孤子的动力学和有限时间爆破解决方案，如当地维里估计，新的李雅普诺夫泛函，和浓度紧机制。 首席研究员将在调查若干科学相关问题时寻求进一步实施和发展这些技术。 本文讨论了在外加势作用下或在非线性系数随时间变化的影响下，单个或多个亮孤子和暗孤子的孤波动力学。将探讨在一个方向上产生有效降维的强限制势的能力。将制定新的标准，用于预测有限时间爆破的解决方案的初始数据，和PI将寻求构建新类型的奇异环爆破解决方案最近观察到的数值。整体解的长时间行为也将被研究。在这个项目中研究的方程作为重要的物理模型出现。 孤立子表现为局域化的稳定结构，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-爱因斯坦凝聚体中超冷原子集合的波函数。 自1995年在实验室获得诺贝尔奖以来，进一步的实验已经产生了孤子和有限时间爆破，现在有大量的物理学文献激发了首席研究员打算考虑的许多问题。 提出的问题汇集了物理学家，数值分析师和纯数学家。它们也是一个很好的教育工具，因为它们可以针对本科生和研究生水平的学生。 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，并在布朗大学教授高级主题课程。 这项研究的网络演示可以制作，例如，通过由该赠款资助的暑期本科生项目。