Numerical inverse scattering transform for the defocusing nonlinear Schr\"odinger equation with box-type initial conditions on a nonzero background

TL;DR

This paper develops a numerical method to solve the defocusing nonlinear Schrödinger equation with box-type initial conditions on a nonzero background, using the inverse scattering transform and steepest descent method, demonstrating accuracy in different asymptotic regimes.

Contribution

It introduces a numerical approach for the inverse scattering transform applied to the defocusing NLS with box initial data on a nonzero background, including contour deformation techniques.

Findings

Accurate numerical solutions in different asymptotic regions.

Method handles nonzero background and box-type initial conditions.

No discrete spectrum (solitons) considered in this work.

Abstract

We present a method to solve numerically the Cauchy problem for the defocusing nonlinear Schr\"{o}dinger (NLS) equation with a box-type initial condition (IC) having a nontrivial background of amplitude $q_o>0$ as $x\to \pm \infty$ by implementing numerically the corresponding Inverse Scattering Transform (IST). The Riemann--Hilbert problem associated to the inverse transform is solved numerically by means of appropriate contour deformations in the complex plane following the numerical implementation of the Deift-Zhou nonlinear steepest descent method. In this work, the box parameters are chosen so that there is no discrete spectrum (i.e., no solitons). The numerical method is demonstrated to be accurate within the two asymptotic regimes corresponding to two different regions of the $(x,t)$-plane depending on whether $|x/(2t)| < q_o$ or $|x/(2t)| > q_o$, as $t \to \infty$.