Numerical computation of dark solitons of a nonlocal nonlinear Schr\"odinger equation

TL;DR

This paper introduces a numerical method to compute dark solitons in nonlocal nonlinear Schrödinger equations, providing simulations that explore their shapes and properties related to physical parameters and validating theoretical conditions.

Contribution

It develops a numerical approach for approximating dark solitons in nonlocal nonlinear Schrödinger equations and investigates their properties through simulations.

Findings

Dark solitons can be numerically approximated for various nonlocal potentials.

The shape of dark solitons depends on parameters like the dispersion relation and speeds.

Numerical results support the theoretical conditions for soliton existence.

Abstract

The existence and decay properties of dark solitons for a large class of nonlinear nonlocal Gross-Pitaevskii equations with nonzero boundary conditions in dimension one has been established recently in [de Laire and S. L\'opez-Mart\'inez, Comm. Partial Differential Equations, 2022]. Mathematically, these solitons correspond to minimizers of the energy at fixed momentum and are orbitally stable. This paper provides a numerical method to compute approximations of such solitons for these types of equations, and provides actual numerical experiments for several types of physically relevant nonlocal potentials. These simulations allow us to obtain a variety of dark solitons, and to comment on their shapes in terms of the parameters of the nonlocal potential. In particular, they suggest that, given the dispersion relation, the speed of sound and the Landau speed are important values to understand the properties of these dark solitons. They also allow us to test the necessity of some sufficient conditions in the theoretical result proving existence of the dark solitons.