Stability in Discrete and Continuous Dynamical Systems

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

The PI will apply the techniques of modern analysis to some outstanding open problems in the stability theory of breathers in the discrete Schr\"odinger equation and Klein-Gordon equation context as well as special solutions for dispersive partial differential equations. The project concentrates on the basic questions concerning stability as well as the existence and the precise description of stable manifolds for solutions, with a few unstable directions. While the stable scenarios have received a great deal of attention in the last twenty years or so, the behavior close to unstable solutions has been less well-studied. One reason is that unstable structures present themselves in a more challenging environment in terms of the mathematical techniques that must be used. This is especially true in the presence of a marginally stable spectrum, resonant edges of the essential spectrum and in the low dimensional cases, all of which will be of primary interest in this project. The project will focus on the study of nonlinear dispersive equations, which model mathematically important processes, such as propagation of light in optical medium. Some of these models arise in the study of quantum mechanical systems and nonlinear optics, while others find their roots in fluid dynamics. These problems can also present themselves as a control problem in stabilization theory. Roughly speaking, if one starts close to an unstable configuration, how does one make only small adjustments along the way, in order to stay close to the initial configuration? Better mathematical description of the behavior of the solutions of these equations, especially their asymptotic behavior in time and space, will greatly improve our understanding of the underlying physics and it will help in the development of technologies that use them.