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Conservation laws of femtosecond pulse propagation described by generalized nonlinear Schrödinger equation with cubic nonlinearity

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  • Trofimov, Vyacheslav A.
  • Stepanenko, Svetlana
  • Razgulin, Alexander

Abstract

We derive conservation laws for so-called generalized nonlinear Schrödinger equation (GNLSE), which describes a propagation of super-short femtosecond pulse in a medium with cubic nonlinear response in the framework of slowly-evolving-wave approximation (SEWA). We take into account the beam diffraction, the pulse spreading due to second order dispersion, the pulse self-steepening, as well as mixed derivatives of the pulse envelope. Such nonlinear interaction of the laser pulse with a medium is widely investigated by many authors because various substances manifest a cubic nonlinear response of medium in various laser systems. However, until present time the conservation laws (integrals of motion) of the GNLSE are absent. For their deriving we propose a novel transform of the GNLSE. It results in an equation containing neither the derivative of a term describing the nonlinear response of medium nor mixed derivatives of a complex amplitude. In new variables, the femtosecond pulse propagation is described by three equations containing only the linear differential operators. Using this transform, the conservation laws for a problem under consideration are found out. We claim that for avoiding a non-physical modulation instability of a laser pulse propagation it is necessary to satisfy to a spectral invariant at the frequency, which is singular one in the Fourier space. This frequency is inherent to the GNLSE. The conservation laws allow developing the conservative finite-difference schemes that preserve difference analogs of these laws.

Suggested Citation

  • Trofimov, Vyacheslav A. & Stepanenko, Svetlana & Razgulin, Alexander, 2021. "Conservation laws of femtosecond pulse propagation described by generalized nonlinear Schrödinger equation with cubic nonlinearity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 366-396.
    • Handle: RePEc:eee:matcom:v:182:y:2021:i:c:p:366-396
    DOI: 10.1016/j.matcom.2020.11.009
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    References listed on IDEAS

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    1. Brugnano, L. & Frasca Caccia, G. & Iavernaro, F., 2015. "Energy conservation issues in the numerical solution of the semilinear wave equation," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 842-870.
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    Cited by:

    1. Vyacheslav Trofimov & Maria Loginova & Mikhail Fedotov & Daniil Tikhvinskii & Yongqiang Yang & Boyuan Zheng, 2022. "Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation," Mathematics, MDPI, vol. 10(11), pages 1-24, June.

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