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

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

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.

Suggested Citation

  • Sakkaravarthi, K. & Johnpillai, A.G. & Durga Devi, A. & Kanna, T. & Lakshmanan, M., 2018. "Lie symmetry analysis and group invariant solutions of the nonlinear Helmholtz equation," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 457-472.
    • Handle: RePEc:eee:apmaco:v:331:y:2018:i:c:p:457-472
    DOI: 10.1016/j.amc.2018.03.011
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    Cited by:

    1. Singh, Sudhir & Kaur, Lakhveer & Sakthivel, R. & Murugesan, K., 2020. "Computing solitary wave solutions of coupled nonlinear Hirota and Helmholtz equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 560(C).
    2. Chen, Haiying & Shahi, Adele & Singh, Gurpreet & Manafian, Jalil & Eslami, Baharak & Alkader, Naief Alabed, 2024. "Behavior of analytical schemes with non-paraxial pulse propagation to the cubic–quintic nonlinear Helmholtz equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 341-356.

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