Asymptotic Methods for Singularity Perturbed Nonlinear Systems

The University Of Central Florida Board Of Trustees, Orlando FL

Investigators

Abstract

The aim of the project is to advance the development of analytic tools to study various asymptotic and singularly perturbed problems, including the semiclassical (or zero dispersion) limit of the focusing nonlinear Schroedinger equation (NLS) and exponentially small effects in singularly perturbed dynamical systems. Numerical experiments for the focusing NLS reveal the formation of regions of violent and disorganized oscillations. This is in drastic contrast to the case of the defocusing NLS, which is reasonably well understood by now. The method of Riemann-Hilbert problems will be used to describe the complicated behavior of the focusing NLS. The second part of the project is connected with the phenomenon of breaking of homoclinic and heteroclinic connections in singularly perturbed and discretized systems. A number of aspects of nonlinear optics, such as the stationary profiles of beams propagating in nonlinear media and propagation of certain pulses in optical fibers, can be described by the NLS equation and its various asymptotic regimes. We will concentrate on one of the most difficult asymptotic problems for this equation. The second part of the project investigates the phenomenon of transition between integrability and chaos in dynamical systems.