Stability and Long-Time Behavior of Hamiltonian Partial Differential Equations

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This project studies the stability and long time behavior of distinguished solutions to Hamiltonian partial differential equations of mathematical physics, viewed as dynamical systems on an infinite dimensional space. The project focuses on two different methods for investigating the stability of solitary waves. The first technique is to use linearization of the equation written in evolution form and to prove stability based on convexity of a certain function of the wave speed. The second technique is to use variational methods to simultaneously establish both existence and stability of solitary waves. The project emphasizes study of stability in the nonlinear Schrodinger equations, dispersion managed nonlinear Schrodinger equations, and discrete nonlinear Schrodinger equations that are related to the study of diffraction managed waveguide arrays. The project will also investigate existence and regularity of attractors for Benjamin-Bona-Mahoney and Camassa-Holm type equations.Most important physical phenomena are described by partial differential equations, some of which can be viewed as dynamical systems on an infinite dimensional space. The goal of this work is to analyze, via this viewpoint, the qualitative behavior of solutions to a class of partial differential equations that arise in numerous physical situations. A primary example under study in this work is the envelope equation describing an electromagnetic wave propagating in an optical waveguide. As a consequence of a balance between nonlinear and dispersive effects, the equation possesses pulse-like solutions called optical solitons. Research in the field of optical solitons is one of the most exciting areas of research in nonlinear science, since self-guided waves (solitons) are ideal candidates for use in the next generation of optical communication networks. This research will deepen our understanding of the waves and coherent structures in nonlinear models of this phenomenon and others of physical importance. The project will also enable undergraduate and graduate students to be involved in modern interdisciplinary mathematics.