Soliton solutions of the non-linear Schroedinger equation with nonlocal Coulomb and Yukawa-interactions

TL;DR

This paper investigates soliton solutions in a non-linear Schrödinger equation with nonlocal Coulomb and Yukawa interactions, demonstrating their existence through numerical methods and confirming results with analytical approximations.

Contribution

The study provides the first numerical evidence of solitonic solutions in a nonlocal non-linear Schrödinger equation with Coulomb and Yukawa interactions, aligning with analytical Gaussian approximations.

Findings

Solitonic solutions exist for all nonlocal coupling constants.

Numerical results agree with Gaussian-based analytical predictions.

The soliton height depends on the coupling constant as predicted.

Abstract

We study the non-linear Schroedinger equation in (1+1) dimensions in which the nonlinear term is taken in the form of a nonlocal interaction of the Coulomb or Yukawa-type. We solve the equation numerically and find that, for all values of the nonlocal coupling constant, and in all cases, the equation possesses solitonic solutions. We show that our results, for the dependence of the height of the soliton on the coupling constant, are in good agreement with the predictions based on an analytic treatment in which the soliton is approximated by a gaussian.