Stability in nonlinear wave equations

非线性波动方程的稳定性

• 批准号: 0707800
• 负责人: Eduard-Wilhelm Kirr
• 金额: $ 19.36万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707800&HistoricalAwards=false
• 关键词: Stability; nonlinear; wave; equations

The project studies the long time behavior of solutions of nonlinear wave equations. It focuses on finding minimal hypotheses under which the solitary wave solutions of Schrödinger, Hartree, and Klein-Gordon equations are asymptotically stable. In addition, the work will investigate the decomposition of solutions into solitary and scattered waves (asymptotic completeness), as well as energy transfer among these waves. The research will not only develop new dispersive estimates for the semigroup of operators generated by the linearization around solitary waves whose shape is changing in time, but will also develop mathematical techniques to translate these estimates into results on the long time dynamics of the full, nonlinear equations.Wave equations are ubiquitous in quantum mechanics, statistical physics, optics, and acoustics, and they model certain phenomena in fluid dynamics, mechanics, and biology. Nonlinear wave equations support solitary waves, which are the building blocks for modeling such diverse phenomena as Bose-Einstein condensates, the propagation of information through optical fibers, and the formation of tsunamis. The detailed dynamics of these physical systems are still not fully understood. This project aims to create a mathematical theory for these phenomena and, more generally, for the long time propagation and interaction of solitary waves. It will focus on specific models from statistical physics and nonlinear optics, but the theory is expected to be sufficiently robust to apply to wave equations arising from models in other areas of science and engineering.

本项目研究非线性波动方程解的长时间行为。 它着重于寻找最小的假设下，孤立波解的薛定谔，哈特里，和克莱因-戈登方程是渐近稳定的。 此外，工作将调查分解成孤立波和散射波的解决方案（渐近完整性），以及这些波之间的能量转移。 这项研究不仅将为形状随时间变化的孤立波线性化产生的算子半群开发新的色散估计，而且还将开发数学技术，将这些估计转化为完整的非线性方程的长时间动力学结果。波动方程在量子力学，统计物理，光学和声学中无处不在，他们模拟流体动力学、力学和生物学中的某些现象。 非线性波动方程支持孤立波，孤立波是模拟玻色-爱因斯坦凝聚、信息通过光纤传播和海啸形成等多种现象的基石。 这些物理系统的详细动力学仍然没有完全理解。 该项目旨在为这些现象以及更普遍的孤波的长时间传播和相互作用建立数学理论。它将侧重于统计物理和非线性光学的特定模型，但该理论预计将足够强大，以适用于其他科学和工程领域的模型所产生的波动方程。