Nonlocal Nonlinear Schr\"odinger Equations and Their Soliton Solutions

TL;DR

This paper derives soliton solutions for both standard and nonlocal nonlinear Schrödinger equations from the coupled AKNS system using Hirota's method, providing explicit examples and visualizations.

Contribution

It introduces a systematic approach to obtain soliton solutions for nonlocal NLS equations via reductions from the coupled AKNS system using Hirota's bilinear method.

Findings

Explicit soliton solutions for standard and nonlocal NLS equations.

Visualization of the solutions through plots of |q(t,x)|^2.

Demonstration of parameter effects on soliton behavior.

Abstract

We study standard and nonlocal nonlinear Schr\"{o}dinger (NLS) equations obtained from the coupled NLS system of equations (Ablowitz-Kaup-Newell-Segur (AKNS) equations) by using standard and nonlocal reductions respectively. By using the Hirota bilinear method we first find soliton solutions of the coupled NLS system of equations then using the reduction formulas we find the soliton solutions of the standard and nonlocal NLS equations. We give examples for particular values of the parameters and plot the function $|q(t,x)|^2$ for the standard and nonlocal NLS equations.