The Semiclassical Limit of the Focusing Nonlinear Schroedinger Equation

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

NSF Award Abstract - DMS-0103909 Mathematical Sciences: The Semiclassical Limit of the Focusing Nonlinear Schroedinger Equation Abstract 0103909 Miller This project addresses the behavior of solutions of the focusing nonlinear Schroedinger equation in the singular semiclassical limit, with particular attention paid to solutions that are tied to definite given initial data. Specific research goals include (i) generalizing a steepest-descents procedure for matrix Riemann-Hilbert problems to recover asymptotics of the initial-value problem for general real-analytic and oscillatory-analytic initial data, (ii) computing rigorous spectral asymptotics for the nonselfadjoint Zakharov-Shabat operator and taking estimates of the error into account in the inverse-scattering problem, (iii) studying sets of minimal weighted Green's capacity in the upper half-plane and relating them solidly to semiclassical asymptotics, and (iv) determining the sensitivity of the asymptotics to the presence of singularities in the data and also robustness to structural perturbations. The analysis will employ numerical methods, careful asymptotic spectral analysis of a family of nonselfadjoint differential operators, and potential-theoretic aspects of functional and complex analysis. The focusing nonlinear Schroedinger equation is a ubiquitous model equation for the propagation of waves of many different kinds (water waves, light waves, etc.) in the simultaneous presence of nonlinear effects that can "self-amplify" the waves and "dispersion" which can pull the waves apart. In particular, it is a tested and accepted model for the transmission of lightwave pulses along certain types of glass optical fibers. This project will produce new understanding of this model equation relevant to situations where the coefficient of the dispersive term in the equation is relatively small, or alternatively, nonlinear processes dominate the evolution of broad disturbances for short times. "Dispersion-shifted" optical fibers currently being installed in many modern telecommunication systems provide an environment where the effects of dispersion and nonlinearity are present in precisely such a skewed proportion. The results of this project will be likely to influence the analysis and design of the next generation of high-speed optical telecommunications systems.