The Semiclassical Modified Nonlinear Schrödinger Equation I: Modulation Theory and Spectral Analysis
J. C. DiFranco and P. D. Miller
Department of Mathematics, University of Michigan, Ann Arbor
Abstract:
We study an integrable modification of the focusing nonlinear Schrödinger equation from the point of view of semiclassical asymptotics. In particular, (i) we establish several important consequences of the mixed-type limiting quasilinear system including the existence of maps that embed the limiting forms of both the focusing and defocusing nonlinear Schrödinger equations into the framework of a single limiting system for the modified equation, (ii) we obtain bounds for the location of discrete spectrum for the associated spectral problem that are particularly suited to the semiclassical limit and that generalize known results for the spectrum of the nonselfadjoint Zakharov-Shabat spectral problem, and (iii) we present a multiparameter family of initial data for which we solve the associated spectral problem in terms of special functions for all values of the semiclassical scaling parameter. We view our results as part of a broader project to analyze the semiclassical limit of the modified nonlinear Schrödinger equation via the noncommutative steepest descent procedure of Deift and Zhou, and we also present a self-contained development of a Riemann-Hilbert problem of inverse-scattering that differs from those given in the literature and that is well-adapted to semiclassical asymptotics.