This research project will use rigorous asymptotic analysis to answer open questions concerning the following three aspects of integrable nonlinear wave equations: (A) The asymptotic behavior of initial value problems for multicomponent integrable systems in one spatial and one temporal variable, including the n-wave model and the Markov n-wave model for Burgers turbulence. (B) Understanding and classifying critical behavior in boundary regions in systems such as the semiclassical sine-Gordon and long-time nonlinear Schrodinger equations, and uncovering fundamental properties of associated Painleve functions. (C) Asymptotic analysis of models in two spatial and one temporal variables, such as the Davey-Stewartson II and Kadomtsev-Petviashvili II equations, requiring the d-bar approach to inverse scattering. These three projects will make use of and further develop recent advances in asymptotic analysis, scattering theory, and the study of orthogonal polynomials and random matrices.
Nonlinear wave equations are powerful mathematical models for energy transmission in fluid dynamics, solid state physics, and other physical systems. Specific applications of the models considered include the evolution of cosmological structures, the onset of wave collapse in fiber-optic cables and superconducting Josephson junctions, the accumulation of fluid vortices, and Hele-Shaw flow. Solutions to these equations are typically too complex to describe in full generality, yet salient features often become clear in certain asymptotic limits for entire families of initial conditions. This universality is an indication of the reasonableness of the equation as a good physical model. Incorporating the effect of interaction terms in multicomponent systems, analyzing critical transitions to help completely describe global behavior, and developing the mathematical tools necessary to study systems in two spatial dimensions will help build a fuller picture of the behavior of general nonlinear wave equations and their applicability as physical models.
