Nonlinear Waves and Stability in Partial Differential Equations

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9704924 Pego Nonlinearity helps to create wave phenomena in a variety of important systems of partial differential equations that arise in science and engineering. The focus of the proposed research is on developing and improving methods for analyzing the stability of waves in several systems of physical interest. These include 1) Solitary waves in nonlinear dispersive media, including lattice dynamics and water waves; 2) A recently discovered class of localized nonradial solutions of nonlinear Schrodinger equations in 2+1 dimensions; and 3) Internal waves in fluids near the liquid-vapor critical point. The methods under development involve improving the use of: a) Evans functions to analyze eigenvalue problems in two dimensions with symmetries; b) singular perturbation theory for resolvent operators, to study how stability results for integrable systems persist in nonintegrable systems for lattice dynamics and water waves; c) infinite-dimensional center manifold theory in ill-posed systems, to study the existence of traveling water waves in three dimensions; d) zero-Mach-number asymptotic analysis of low-velocity flows, to study hydrodynamic phenomena near the critical point, specifically: damping rates of internal waves about a strongly stratified equilibrium, and possible capillary effects in one-phase flows of near-critical fluids. The general goal of the first part of this research is to understand how "robust" are nonlinear wave phenomena. Nonlinear waves in rare, so-called "integrable" systems can be very well understood due to what seems miraculous -- they can be solved in closed form. But most realistic systems are not integrable, so one needs to know what phenomena depend on integrability and what do not. An important infrastructural technology where nonlinear waves are important and are not completely understood is long-distance communication via optical fiber. The second part of the work was motivated by physics experiments carried out on the spa ce shuttle; also, the use of supercritical fluids in materials processing is extensive and growing. The behavior of flows of such fluids near the critical point is unusual and little understood, and this has led to costly failures in experimental design in the past. Fundamental investigations are needed to build the knowledge base about such flows that can serve as the foundation for the development of applications.