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A high-order nonlinear Schrödinger equation as a variational problem for the averaged Lagrangian of the nonlinear Klein–Gordon equation

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Abstract

We use Whitham’s averaged Lagrangian method extended with the multiple-scale formalism to derive a sixth-order nonlinear Schrödinger equation for the complex amplitude of the envelope of the slowly modulated wave trains whose evolution is governed by the nonlinear Klein–Gordon equation with polynomial nonlinearity. Such a high-order nonlinear Schrödinger equation is obtained as a variational problem for the Lagrangian averaged over the rapid oscillations of the carrier wave train. As compared to classical Whitham’s approach, we take into account the derivatives of the complex amplitude with respect to slow timescales and long coordinate in the averaged Lagrangian. The coefficients of the high-order nonlinear Schrödinger equation derived in this work identically coincide with those derived by us earlier by the method of multiple scales applied directly to the original nonlinear Klein–Gordon equation. Finally, we consider the sine-Gordon equation as a partial case of the nonlinear Klein–Gordon equation and demonstrate one example of the numerical solution to the corresponding sixth-order nonlinear Schrödinger equation in the form of the envelope quasi-soliton.

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Acknowledgements

Dr. Ivan Gandzha is grateful to Wolfram Research for providing him with a free contemporary licence for the use of Mathematica 12.

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  1. Institute of Physics, National Academy of Sciences of Ukraine, Prosp. Nauky 46, Kyiv, 03028, Ukraine

    Ivan S. Gandzha & Yuri V. Sedletsky

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  1. Ivan S. Gandzha
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Correspondence to Ivan S. Gandzha.

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Gandzha, I.S., Sedletsky, Y.V. A high-order nonlinear Schrödinger equation as a variational problem for the averaged Lagrangian of the nonlinear Klein–Gordon equation. Nonlinear Dyn 98, 359–374 (2019). https://doi.org/10.1007/s11071-019-05197-x

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