Direct search for exact solutions to the nonlinear Schroedinger equation

TL;DR

This paper identifies symmetry groups of the nonlinear Schrödinger equation and constructs exact solutions using transformation ansätze, revealing a bifurcation phenomenon and diverse solution types.

Contribution

It computes the Lie symmetry algebra for the nonlinear Schrödinger equation and develops methods to generate exact solutions with various functional forms.

Findings

Identified a five-dimensional Lie symmetry algebra.

Constructed explicit solutions with constant, trigonometric, exponential, and rational amplitudes.

Demonstrated a bifurcation phenomenon in the solution space.

Abstract

A five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schroedinger equation, which, together with a reflection invariance, generates two five-parameter solution groups. Three ansaetze of transformations are secondly analyzed and used to construct exact solutions to the nonlinear Schroedinger equation. Various examples of exact solutions with constant, trigonometric function type, exponential function type and rational function amplitude are given upon careful analysis. A bifurcation phenomenon in the nonlinear Schroedinger equation is clearly exhibited during the solution process.