Stability and Long Time Behavior for Infinite-Dimensional Dynamical Systems

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

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.