Mathematical Sciences: Stability,Instability and SingularityFormation in Nonlinear Partial Differential Equations

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Nonlinear partial differential equations arising in models of physical phenomena and in geometry will be the focus of this research. Studies on the stability and instability of solitary traveling waves arising in laser propagation, plasma waves and water waves will be carried out. Solitary waves are localized finite energy solutions of nonlinear equations that result from a balance of dispertion and a focusing nonlinearity. Important questions to be addressed in this work are concerned with the stability of solitary waves and their role in the general structure of solutions. The model equation to be analyzed is the nonlinear Schroedinger equation. Many of the phenomena which hold for this equation also hold for other nonlinear dispersive systems which arise as envelope equations. These include the generalized Korteweg-de Vries, Benjamin-Ono, Intermediate Long Wave and Benjamin-Bona-Mahoney equations. The ground state solitary wave solution plays a critical role in that it is an interface between stable and unstable solutions. Two goals of this work are to determine the sense in which blowing up solutions are attracted to the ground state and to quantify the loss of compactness in these functions. Work related to differential geometry concerns the motion of curves governed by their curvature. Flows of this type arise in the mathematical description of propagating fronts. The initial expectation was that one could obtain geodesics as the asymptotic limit of some gradient flow on the space of curves. This has turned out to be impossible in general. Efforts will be made to understand the global behavior of these flows and to determine the final form of flows on spaces of curves lying in unstable manifolds.