A Self-Dual Bogomol'nyi Formulation of the Nonlinear Schr\"odinger Equation

TL;DR

This paper derives a self-dual formulation of the 1+1 dimensional nonlinear Schrödinger equation from a higher-dimensional model, enabling the discovery of new soliton solutions including those influenced by background sources.

Contribution

It introduces a novel self-dual formulation of the NLSE via dimensional reduction from a 2+1 dimensional self-dual Chern-Simons model, expanding the solution space.

Findings

Recovery of known soliton solutions from the Bogomol'nyi bound

Discovery of new soliton solutions with background sources

Extension of solution methods for the NLSE

Abstract

We obtain a self-dual formulation of the conventional nonlinear Schr\"odinger equation (NLSE) in the 1+1 dimension by studying the dimensional reduction of the self-dual Chern-Simons nonlinear Schr\"odinger model (NLSM) in the 2+1 dimension. It is found that this self-dual formulation allows us to find not only the well-known soliton solutions from the Bogomol'nyi bound and the Galilean boost, but also other soliton solutions in the presence of the background sources.