The semiclassical modified nonlinear Schr\"odinger equation II: asymptotic analysis of the Cauchy problem. The elliptic region for transsonic initial data

Abstract

We begin a study of a multi-parameter family of Cauchy initial-value problems for the modified nonlinear Schr\"odinger equation, analyzing the solution in the semiclassical limit. We use the inverse scattering transform for this equation, along with the steepest descent method of Deift and Zhou. The initial conditions are selected both to allow all relevant scattering data to be calculated without approximation and also to place the governing equation in a transsonic state in which the quantum fluid dynamical system formally approximating it is of hyperbolic type for some $x$ and of elliptic type for other $x$. Our main result is a global approximation theorem valid in a maximal space-time region connected to the elliptic part of the initial data.