Direct Scattering of the Focusing Nonlinear Schr\"odinger Equation with Step-like Oscillatory Initial Data

TL;DR

This paper develops a scattering theory framework for the nonlinear Schrödinger equation with step-like oscillatory initial data, connecting inverse scattering to Riemann-Hilbert problems and soliton-gas solutions.

Contribution

It formulates the direct and inverse scattering problems for step-like elliptic traveling wave initial data and establishes their analytic properties and solvability.

Findings

Formulated the direct scattering problem for step-like initial data.

Established the Riemann-Hilbert problem for the inverse scattering.

Linked the formulation to soliton-gas initial data.

Abstract

In this manuscript we set up the direct and inverse scattering problems for step-like traveling-wave solutions of the nonlinear Schr\"odinger equation. Specifically, we consider initial data $u(x,0)$ satisfying $u(x,0)\to u_0^\ell(x)$ as $x\to-\infty$ and $u(x,0)\to u_0^r(x)$ as $x\to+\infty$, where $u_0^\ell(x)$ and $u_0^r(x)$ are elliptic traveling waves. Under suitable assumptions on the initial data we formulate the direct scattering problem and establish analytic properties of the scattering data. We then formulate the inverse problem as a Riemann--Hilbert problem and prove its solvability. Finally, we observe that this Riemann--Hilbert formulation is a special case of the one arising for full soliton-gas initial data.