Semiclassical limit of the focusing NLS: Whitham equations and the Riemann-Hilbert Problem approach

TL;DR

This paper integrates Whitham theory and Riemann-Hilbert problem methods to analyze the semiclassical limit of the focusing nonlinear Schrödinger equation, enhancing understanding of slowly modulated nonlinear wave solutions.

Contribution

It combines two prominent approaches to study the semiclassical focusing NLS, clarifying their interrelations and potential for broader applications.

Findings

Unified framework for Whitham and Riemann-Hilbert methods

Insights into the modulation of N-phase solutions

Potential for improved analysis of semiclassical fNLS behavior

Abstract

The main goal of this paper is to put together: a) the Whitham theory applicable to slowly modulated $N$-phase nonlinear wave solutions to the focusing nonlinear Schr\"odinger (fNLS) equation, and b) the Riemann-Hilbert Problem approach to particular solutions of the fNLS in the semiclassical (small dispersion) limit that develop slowly modulated $N$-phase nonlinear wave in the process of evolution. Both approaches have their own merits and limitations. Understanding of the interrelations between them could prove beneficial for a broad range of problems involving the semiclassical fNLS.