Semiclassical Soliton Ensembles for the Focusing Nonlinear Schroedinger Equation

TL;DR

This paper develops a new steepest descent method for matrix Riemann-Hilbert problems to analyze the semiclassical limit of the focusing nonlinear Schrödinger equation, providing explicit asymptotics for a sequence of solutions.

Contribution

It introduces a generalized steepest descent approach and characterizes the complex phase function via a finite-gap ansatz and critical point theory.

Findings

Explicit strong asymptotics for solutions in certain regions

Characterization of the complex phase function via finite-gap ansatz

Connection to variational principles in zero-dispersion limits

Abstract

We present a new generalization of the steepest descent method introduced by Deift and Zhou for matrix Riemann-Hilbert problems and use it to study the semiclassical limit of the focusing nonlinear Schroedinger equation with real analytic, even, bell-shaped initial data. We provide explicit strong locally uniform asymptotics for a sequence of exact solutions corresponding to initial data that has been modified in an asymptotically small sense. We call this sequence of exact solutions a semiclassical soliton ensemble. Our asymptotics are valid in regions of the (x,t) plane where a certain scalar complex phase function can be found. We characterize this complex phase function directly by a finite-gap ansatz and also via the critical point theory of a certain functional; the latter provides the correct generalization of the variational principle exploited by Lax and Levermore in their study of the zero-dispersion limit of the Korteweg-de Vries equation.