Constructing new nonlinear evolution equations with supersymmetry

TL;DR

This paper extends the factorisation method in supersymmetric quantum mechanics to nonlinear systems, enabling the construction of new nonlinear evolution equations and revealing potential dualities among soliton solutions.

Contribution

It introduces a novel supersymmetry-based method for nonlinear quantum systems, allowing the creation of new equations from known solutions and expanding analytical tools in the field.

Findings

Extended factorisation method to nonlinear systems

Constructed new nonlinear evolution equations

Potential discovery of dualities among soliton solutions

Abstract

The factorisation method commonly used in linear supersymmetric quantum mechanics is extended, such that it can be applied to nonlinear quantum mechanical systems. The new method is distinguishable from the linear formalism, as the superpotential is forced to become eigenfunction-dependent. An example solution is given for the nonlinear Schroedinger equation and its supersymmetric partner equation. This method allows new nonlinear evolution equations to be constructed from the solutions of known nonlinear equations, and has the potential to be a useful tool for mathematicians and physicists working in the field of nonlinear systems, allowing the discovery of previously unknown `dualities' amongst soliton solutions and their respective equations.