Exact solutions of the inhomogeneous nonlinear Schr\"odinger equation through supersymmetric potentials

TL;DR

This paper develops a method using supersymmetric quantum mechanics to find exact stationary solutions of the inhomogeneous nonlinear Schrödinger equation by constructing supersymmetric partner potentials, demonstrated with a Pösch-Teller potential example.

Contribution

It introduces a novel algorithm connecting supersymmetric quantum mechanics with the INLSE to derive exact solutions, expanding analytical tools for nonlinear wave equations.

Findings

Constructed exact solutions for INLSE using supersymmetric partner potentials.

Linked INLSE solutions with Schrödinger equation via Lie symmetries.

Demonstrated method with Pösch-Teller potential example.

Abstract

By employing supersymmetric quantum mechanics, we present a general algorithm to construct supersymmetric partner potentials and hence derive exact stationary solutions of the inhomogeneous nonlinear Schr\"odinger equation (INLSE). This is possible due to the connection between the INLSE and the nonlinear Schr\"odinger equation (NLSE), which can be established from a treatment based on Lie point symmetries and is related with Schr\"odinger equation, under certain conditions. As an illustrative example, we construct exact solutions for the INLSE through a P\"osch-Teller potential with a single bound state.