Localized exact solutions of $\mathcal{PT}$ symmetric nonlinear Schr\"odinger equation with space and time modulated nonlinearities

TL;DR

This paper derives localized exact solutions for a nonlinear Schrödinger equation with space-time modulated nonlinearities and external potentials, including non-Hermitian PT-symmetric cases, using canonical transformations.

Contribution

It introduces a method to find localized exact solutions of the NLSE with complex, space-time dependent nonlinearities and PT-symmetric potentials, expanding solution techniques for such complex systems.

Findings

Exact localized solutions for NLSE with space-time modulated nonlinearities

Solutions include non-Hermitian PT-symmetric potentials

Method applicable to various external potential configurations

Abstract

Using canonical transformations we obtain localized (in space) exact solutions of the nonlinear Schr\"odinger equation (NLSE) with space and time modulated nonlinearity and in the presence of an external potential depending on space and time. In particular we obtain exact solutions of NLSE in the presence of a number of non Hermitian $\mathcal{PT}$ symmetric external potentials.