The present proposal consists of two major topics: A) semiclassical (small dispersion) limit of the focusing Nonlinear Schroedinger Equation (NLS) and related problems, and; B) persistence of integrable dynamics, including homoclinic/heteroclinic solutions, of a system undergoing a singular perturbation. A) Numerical experiments of Bronski and McLauchlin revealed the formation of a region of violent and disorganized oscillations in the small dispersion limit of the focusing NLS. Using the method of Riemann-Hilbert Problem (RHP), we found a way to track the evolution of our initial data into the region of ``violent and disorganized oscillations" through the evolution of a hyperelliptic surface associated with the problem. Changes of the genus of this surface correspond to phase transitions in the evolution of our initial data. The pure radiational case was studied completely in the joint work with Venakides and Zhou. Here we propose to study phase transitions in the most difficult case that includes both solitons and radiation. B) It has been recently proved (Tovbis, Pelinovsky) that persistence of homoclinic/heteroclinic solutions to singularly perturbed 5th Kortveg - de Vries (KdV) equation can be expressed in terms of the Stokes' constants of certain (leading order) rescaled system. The proposed goal is to develop the corresponding technique to other types of solutions and systems, including stationary and moving travelling waves on lattices for NLS and other models.
It is well known that only a tiny portion of nonlinear systems used to model real world problems allows for explicit form mathematical solutions, leaving approximate methods and computer simulations to be the most used tools. Asymptotic methods play a prominent role among approximate methods, as it is often easier to study a system in some asymptotic limit (say, infinite time) and then to allow some ``small" corrections for large but finite values of time. This approach was used for centures, for example, in celestian mechanics. However, ``small" corrections are not necessarily small if the original and the limiting systems exibit qualitatively different behavior. For example, a weakly coupled system of 2nd order nonlinear oscillators generically has a chaotic behavior whereas the limiting system of uncoupled oscillators has no chaos. Such systems, called singularly perturbed, are the most difficult subject in the asymptotic analysis. The current proposal is focused on two still developing methods for singularly perturbed nonlinear problems: the method of RHP and of asymptotics beyond all orders. The first method will be used to find small dispersion limit of the focusing NLS - the problem that was open for many years. The NLS is the most frequently used model for nonlinear waves phenomena in physics, engineering, etc. The second method will be used to find when travelling waves solutions to KdV, NLS and some other integrable models survive singular perturbations, including discretizations. Solitons and travelling waves on lattices (discretized models) are rapidly becoming a very important topic, for example, in fiber optics (including periodic optical structures), design of computational numerical methods, etc.
