Exact solitary wave solutions of the nonlinear Schr\"odinger equation with a source

TL;DR

This paper derives exact rational solitary wave solutions of the nonlinear Schrödinger equation with a source, revealing various soliton types including bright, dark, and singular solutions through elliptic function connections.

Contribution

It introduces a fractional transformation method to find explicit solutions of the NLSE with a source, encompassing trigonometric, hyperbolic, and rational forms, which was not previously established.

Findings

Derived explicit bright and dark soliton solutions.

Identified conditions for singular solitons.

Connected solutions to elliptic functions.

Abstract

We use a fractional transformation to connect the traveling wave solutions of the nonlinear Schr\"odinger equation (NLSE), phase-locked with a source, to the elliptic functions satisfying, $f^{\prime\prime}\pm af\pm \lambda f^{3}=0$. The solutions are {\it necessarily} of the rational form, containing both trigonometric and hyperbolic types as special cases. Bright, and dark solitons, respectively, for attractive and repuslive type of nonlinearities, as also singular solitons, are obtained in suitable range of parameter values.