Existence, Stability, and Dynamics of Nonlinear Waves

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

This research project focuses on stability and behavior of special classes of solutions of mathematical equations modeling a variety of physical phenomena including optical communication, shallow or stratified fluid flow, and inclined thin film flow. Emphasis is placed on the stability of nonlinear waves exhibiting a spatially periodic structure, which form fundamental building blocks for more complicated solutions in many applications. The stability of such structures (i.e. their ability to retain their form when disturbed) is of great practical importance, as waves that are not stable do not naturally manifest themselves in physical applications, except possibly for transient phenomena. The aim of this project is to provide researchers a mathematically rigorous theory by which they may distinguish between mathematical solutions that are stable, and hence have a possibility of being manifested in reality, and those that are not. This project will also include undergraduate and graduate students in research projects. The methodologies and techniques developed through this work will be incorporated into seminars, reading courses, and special topics courses appropriate for students from a variety of scientific disciplines. This project focuses on the existence, stability, and dynamics of special classes of traveling wave solutions in models arising naturally in mathematical physics and fluid mechanics. The aim is to systematically develop a linear and nonlinear stability analysis of spatially periodic standing or traveling structures in both Hamiltonian dispersive partial differential equations, in which energy is conserved, as well as hyperbolic-parabolic systems of conservation and balance laws, where energy is partially dissipated due to, for example, viscous effects. In the dispersive context, the research project will consider a variety of model equations with nonlocal descriptions of dispersion and will develop techniques and methodologies capable of treating not only low-frequency phenomena, which is in the realm of many classical theories, but also high-frequency behaviors of solutions, which are beyond the regime of validity of classical results. In the dissipative context, the research project will develop a systematic investigation of roll-waves, which are commonly-observed hydrodynamic instabilities arising naturally in many applications, such as fluid flow in conduits. Particular attention will be placed on understanding connections between the recently-developed viscous theories of such waves and more well-known inviscid theories.