A method to solve nonlinear Schr\"odinger equation using Riccati equation

TL;DR

This paper introduces a novel method that connects the nonlinear Schrödinger equation to the Riccati equation, enabling the derivation of exact solutions, including vortex solutions, for equations on lines and planes.

Contribution

It presents a new approach linking the nonlinear Schrödinger equation with the Riccati equation to find exact solutions and generalize known solutions.

Findings

Derived new exact solutions for nonlinear Schrödinger equation

Generalized existing solutions on a line

Obtained vortex solutions on a plane

Abstract

A method to find exact solutions to nonlinear Schr\"odinger equation, defined on a line and on a plane, is found by connecting it with second order linear ordinary differential equation. The connection is essentially made using Riccati equation. Generalisation of several known solutions is found using this method, in case of nonlinear Schr\"odinger equation defined on a line. This method also yields non-singular and singular vortex solutions, when applied to nonlinear Schr\"odinger equation on a plane.