This project concerns the development of tools for the asymptotic analysis of integrable nonlinear wave equations. There are two complementary aspects of this work: asymptotic analysis in the scattering theory of linear differential equations, and asymptotic analysis of the corresponding inverse problems. The particular application problems to be studied along the way include (i) the semiclassical limit of the focusing nonlinear Schrodinger equation, (ii) the corresponding semiclassical limit of the modified nonlinear Schrodinger equation, (iii) the semiclassical limit of the sine-Gordon equation in laboratory coordinates, (iv) the continuum limit of the Ablowitz-Ladik equations, (v) singular limits for multicomponent integrable equations, and (vi) several asymptotic problems in approximation theory with applications to random matrix theory.
This work is interesting and important because it will promote understanding of singular limits leading formally to ill-posed dynamical systems. Indeed, the motivating problem of the semiclassical limit of the focusing nonlinear Schrodinger equation with general smooth but nonanalytic initial data remains one of the most important open problems in applied analysis, and the tools developed as part of this project will directly address this problem and other similar ones. Furthermore, while the basic aim of the project is the development of methods of analysis, the methods will also be applied to several specific integrable equations and also to open problems beyond the field of nonlinear waves. For example, we intend to apply specialized asymptotic methods developed in the context of nonlinear wave theory to the problem of the asymptotic analysis (in the bulk scaling limit) of the correlation functions of the normal random matrix model, with its coincident connections to the theory of Laplacian growth (also known as Hele-Shaw flow) and conformal mapping. Many of the problems addressed as part of this project have a universal character, arising in the modeling of diverse physical phenomena, and it follows that analytical techniques applicable to these problems have far-reaching consequences. This project also has an educational component, stressing the training of postdocs and graduate students through collaborative research and course development.
