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

K. Sakkaravarthi, A.G. Johnpillai, A. Durga Devi, T. Kanna and M. Lakshmanan

Applied Mathematics and Computation, 2018, vol. 331, issue C, 457-472

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 Painlevé 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 Schrödinger equation for which symmetries are long available in the literature.

Keywords: Lie symmetry analysis; Nonlinear Helmholtz equation; Symmetry reduction; Painlevé analysis; Modified Prelle–Singer method; Periodic and solitary waves (search for similar items in EconPapers)
Date: 2018
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:331:y:2018:i:c:p:457-472

DOI: 10.1016/j.amc.2018.03.011

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