Lie symmetry analysis and group invariant solutions of the nonlinear Helmholtz equation

TL;DR

This paper applies Lie symmetry analysis to the nonlinear Helmholtz equation to find symmetry reductions, integrable ODEs, explicit solutions, and numerical wave train solutions, revealing new insights into beam propagation in nonlinear waveguides.

Contribution

It provides the first detailed symmetry analysis and explicit solutions for the nonlinear Helmholtz equation under non-paraxial approximation, including comparisons with the nonlinear Schrödinger equation.

Findings

Symmetry reductions lead to Painlevé integrable ODEs.

Explicit traveling wave and solitary wave solutions are constructed.

Numerical analysis reveals multi-peak nonlinear wave trains.

Abstract

We consider the nonlinear Helmholtz (NLH) equation describing the beam propagation in a planar waveguide with Kerr-like nonlinearity under non-paraxial approximation. By applying the Lie symmetry analysis, we determine the Lie point symmetries and the corresponding symmetry reductions in the form of ordinary differential equations (ODEs) with the help of the optimal systems of one-dimensional subalgebras. Our investigation reveals an important fact that in spite of the original NLH equation being non-integrable, its symmetry reductions are of Painlev\'e integrable. We study the resulting sets of nonlinear ODEs analytically either by constructing the integrals of motion using the modified Prelle-Singer method or by obtaining explicit travelling wave-like solutions including solitary and symbiotic solitary wave solutions. Also, we carry out a detailed numerical analysis of the reduced equations and obtain multi-peak nonlinear wave trains. As a special case of the NLH equation, we also make a comparison between the symmetries of the present NLH system and that of the standard nonlinear Schr\"odinger equation for which symmetries are long available in the literature.