Stability and Long Time Behavior for Infinite-Dimensional Dynamical Systems

这项研究是关于了解偏微分方程解的稳定性和长期行为的。具体地说，首席研究人员考虑用作控制海洋中的水流运动和脉冲在光纤中传播的模型的方程。孤立波和周期波通常可以在物理系统和实验室实验中观察到，它们是这些应用特别感兴趣的解决方案。解的稳定性意味着它们在环境的微小变化、材料的杂质、模型的缺陷等情况下是持久的。研究结果可用于解决工程中的重要问题，如优化问题。这个项目解决了偏微分方程中的各种问题，专注于研究相干结构的稳定性和长时间行为，如周期波和孤立波以及更复杂的激发态。主要的研究人员使用了无限维动力系统的观点，它利用了偏微分方程和常微分方程组之间的类比。方法是观察其时间演化发生在适当定义的无限维函数空间上的系统，并使用常微分方程式对象，如不变流形和吸引子，以及更微妙的联系。该提案由两部分组成。第一部分集中讨论了几种模型的波线性稳定性，包括一维和更高维的波稳定性。首席研究人员通过一种同样处理空间周期波和孤立波的方法找到了频谱稳定性。这里的重要例子是Boussinesq系统和短脉冲方程，以及Sine-Gordon和Klein-Gordon方程。这些问题在频谱和线性稳定性水平上提出了许多挑战，但几乎没有人知道它们的渐近稳定性。在第二部分中，项目重点研究了这些模型的波浪的长时间行为和非线性稳定性。令人感兴趣的是线性和非线性稳定性之间的关系，特别是对于波动方程，其中半群的生成元是其谱不容易计算的算子矩阵。这些方程的共同特征是它们支持类孤子解，如基态或激发态以及孤立波或周期行波或驻波。

This research project is on understanding the stability and long-time behavior of special solutions for partial differential equations. Specifically, the principal investigator considers equations used as models governing the motion of currents in the oceans and the propagation of pulses in optical fibers. Solitary and periodic waves, often observable in physical systems and lab experiments, are solutions of particular interest for these applications. The stability of solutions implies that they are persistent through small changes in the environment, impurities of the materials, imperfections of the model, etc. Results of the research can be used to address important problems, such as optimization, in engineering. This project addresses a variety of problems in partial differential equations, focusing on the study of stability and long-time behavior of coherent structures, such as periodic and solitary waves and more complicated excited states. The principal investigator uses the point of view of infinite-dimensional dynamical systems, which takes advantage of the analogy between partial and ordinary differential equations. The approach is to look at systems whose time evolution occurs on appropriately defined infinite-dimensional function spaces and use ordinary differential equation objects such as invariant manifolds and attractors, as well as more subtle connections. The proposal consists of two parts. The first part focuses on the linear stability of waves for several models, both in one and higher spatial dimensions. The principal investigator finds the spectral stability with a method that treats both spatially periodic and solitary waves equally well. Important examples here are the Boussinesq system and the short-pulse equation, and also the sine-Gordon and Klein-Gordon equations. These problems present many challenges at the spectral and linear stability level, but virtually nothing is known for their asymptotic stability. In the second part, the project focuses on the long-time behavior and nonlinear stability of waves for these models. Of interest is the relation between linear and nonlinear stability, particularly for wave equations where the generators for the semi-groups are operator matrices whose spectrum is not easy to compute. The common feature of these equations is that they support soliton-like solutions, such as ground or excited states and solitary or periodic traveling or standing waves.