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和色散方程来模拟。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，这种情况就会发生巨大的变化。具体地说，线性和非线性效应可能会平衡，从而产生所谓的孤子解，其形状持续存在，拒绝分散。人们普遍认为，具有“一般”初始数据的大多数非线性波动和色散方程的解最终应该分解为有限数量的孤子加上一个接近于零的辐射项。这个项目集中在两个主题上，这两个主题在理解这幅宏伟的图景中发挥着重要作用，波是如何超时传播的。首席研究员(PI)将发展新的方法和技术来研究存在强非线性相互作用的孤子的渐近稳定性。渐近稳定性是指，如果一个孤子受到一点推动，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。更具体地说，PI将使用调和分析和矢量场技术，结合谱理论和概率论的工具，致力于以下目标：(1)执行一个程序来获得一维变系数非线性Klein-Gordon方程小解的精确渐近性，这与物理中许多场论中的“扭结”孤子的渐近稳定性问题有关；(2)发展一种调制方法来证明某些拟线性几何波动方程中的某些孤子的(共维)稳定性；以及(3)开始研究几何波动方程和具有长程非线性的随机初始数据波动方程的解的长期动力学。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.