Asymptotic Dynamics of Nonlinear Wave and Dispersive Equations

自然科学和工程中的许多波传播现象都可以用非线性波动方程和频散方程来模拟。虽然线性波动方程理论预测波随着时间的推移而传播和衰减，但一旦考虑到非线性效应，情况就会发生急剧变化。特别是，线性和非线性效应可能会平衡，产生所谓的孤子解决方案，其形状持续存在，拒绝分散。人们普遍认为，大多数非线性波和色散方程的解与“通用”的初始数据最终应该分解成一个有限数量的孤子加上一个辐射项趋于零。这个项目集中在两个主题，发挥了重要作用，在寻求了解这一宏伟的图片如何波传播超时。首席研究员（PI）将开发新的方法和技术，用于研究强非线性相互作用下孤子的渐近稳定性。渐近稳定性指的是这样一种现象，即如果一个孤立子受到一点推动，它可能会摆动一段时间，但最终会恢复到与它开始时类似的形式。此外，PI将研究在几种新的状态下具有一般随机初始数据的非线性波动方程解的长期动力学。更具体地说，PI将使用调和分析和向量场技术，以及谱理论和概率论的工具，以实现以下目标：（1）编制了一个程序，得到了一维变系数非线性Klein-Gordon方程小解的精确渐近性，这与物理学中众多场论中的“扭结”孤子的渐近稳定性问题有关;（2）发展了一种调制方法来证明某些准线性几何波动方程中某些孤子的（余维）稳定性（3）开创了几何波动方程和长周期波动方程解的长时间动力学研究的先河。随机初始数据的范围非线性。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准。

Many wave propagation phenomena in the natural sciences and in engineering can be modeled by nonlinear wave and dispersive equations. While the theory of linear wave equations predicts that waves spread out and decay as time goes by, once nonlinear effects are taken into account, this changes drastically. In particular, linear and nonlinear effects may balance out to create so-called soliton solutions, whose shapes persist and refuse to disperse. It is widely believed that solutions to most nonlinear wave and dispersive equations with "generic" initial data should eventually decompose into a finite number of solitons plus a radiative term that goes to zero. This project concentrates on two themes that play an important role in the quest to understand this grand picture how waves propagate overtime. The principal investigator (PI) will develop new methods and techniques for the study of the asymptotic stability of solitons in the presence of strong nonlinear interactions. Asymptotic stability refers to the phenomenon that if a soliton gets pushed a little bit, it may wiggle for a while, but ultimately return to a form similar to the one it began with. Further, the PI will investigate the long-time dynamics of solutions to nonlinear wave equations with generic randomized initial data in several novel regimes.More specifically, the PI will use harmonic analysis and vector field techniques together with tools from spectral theory and probability theory to work toward the following goals: (1) carry out a program to obtain precise asymptotics of small solutions to one-dimensional Klein-Gordon equations with variable coefficient nonlinearities, which are related to asymptotic stability questions for "kink" solitons in numerous field theories in physics; (2) develop a modulational approach for proving the (co-dimensional) stability of certain solitons arising in some quasilinear geometric wave equations; and (3) initiate the study of the long-time dynamics of solutions to geometric wave equations and to wave equations with long-range nonlinearities for random initial data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.