Stability and Long Time Behavior for Infinite-Dimensional Dynamical Systems

本研究项目是关于理解偏微分方程特解的稳定性和长时性的。具体来说，首席研究员考虑了用于控制海洋中洋流运动和光纤中脉冲传播的模型的方程。在物理系统和实验室实验中经常观察到的孤立波和周期波是这些应用中特别感兴趣的解决方案。溶液的稳定性意味着它们在环境的微小变化、材料的杂质、模型的缺陷等方面都是持久的。研究结果可用于解决工程中的重要问题，如优化。本项目解决了偏微分方程中的各种问题，重点研究了相干结构的稳定性和长期行为，如周期波和孤立波以及更复杂的激发态。主要研究者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之间的类比。方法是观察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。第一部分着重于几种模型的波的线性稳定性，包括一维和更高的空间维度。首席研究员发现光谱稳定性的方法，既可以很好地处理空间周期性和孤立波。重要的例子有Boussinesq系统和短脉冲方程，还有sin - gordon方程和Klein-Gordon方程。这些问题在谱稳定性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的长期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，其中半群的发生器是算子矩阵，其谱不易计算。这些方程的共同特点是它们支持类孤子解，如基态或激发态，孤波或周期性行波或驻波。

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.