The goal of applied mathematics is the study of equations of scientific, engineering or industrial interest in a mathematically rigorous way. Understanding such equations often requires considering a limit in which certain parameters approach zero. While some of these problems behave in a predictable manner in the limit, other interesting and important problems involve unstable behavior that becomes less and less predictable the smaller the parameter of interest becomes. One of the best illustrations of this concept is the behavior of fluid flows when the viscous drag is small, and the fluid behaves in a turbulent and chaotic way. The primary model under study in this project is the nonlinear Schrodinger equation, which is a fundamental model for the study of pulses in optical fibers. The small parameter limit for this model corresponds to the limit of ultra-short pulse propagation, which is expected to find many applications to high-speed telecommunications. This limit is known as the semiclassical limit. This project constitutes an in-depth study of initial-value problems for several partial differential equations (PDEs) in the semiclassical limit. Existing formal theories fail because they lead to model problems that are ill-posed and thus make no prediction at all for reasonable initial conditions. The goal of this project is to develop asymptotic theories that are not based on any particular ansatz and do not require unphysical conditions on the initial data. Among the problems under attack is the rigorous semiclassical analysis of the focusing nonlinear Schroedinger equation for general data, a problem that is generally considered to be one of the most important open problems in the field of integrable systems. The specific aims of the project include the development of new ansatz-free methods of asymptotic analysis --- for spectral theory and for Riemann-Hilbert problems of inverse-scattering theory --- that are insensitive to analyticity properties of the initial data, and have a "nonlinear Riemann-Lebesgue" character, directly exploiting cancellation due to oscillations where analytic deformations are impossible. These techniques, once developed, will also have important repercussions in fields that are only tangentially related, for example, the theory of orthogonal polynomials of large degree and the statistical analysis of large random matrices.
The main goal of the proposed work is to develop predictive tools that apply to extremely unstable systems when the initial conditions are rough or noisy. An example of such a system is the one governing the propagation of ultrashort data pulses in certain optical fibers, and therefor any new insights into such systems will have repercussions in the field of telecommunications. Some of the problems we propose to study admit a detailed analysis because they are idealized and can in some sense be solved exactly; however our work is also expected to lead to general methods that are applicable to less idealized systems. These techniques will also have important implications in mathematical fields that are only tangentially related, for example, the theory of orthogonal polynomials of large degree and the statistical analysis of large random matrices.This project also has an important educational aspect. We plan to involve both graduate students and postdoctoral researchers in this work, and the advanced training of the next generation of researchers is an important component of this proposal. We also plan an interdisciplinary workshop to further disseminate the results of this work beyond the boundaries of the mathematical community.
