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Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants

Author

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  • Vyacheslav Trofimov

    (Shien-Ming Wu School of Intelligent Engineering, South China University of Technology, Guangzhou 511442, China)

  • Maria Loginova

    (The Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University GSP-1, Leninskie Gory, 119991 Moscow, Russia)

Abstract

Two 1D nonlinear coupled Schrödinger equations are often used for describing optical frequency conversion possessing a few conservation laws (invariants), for example, the energy’s invariant and the Hamiltonian. Their influence on the properties of the finite-difference schemes (FDSs) may be different. The influence of each of both invariants on the computer simulation result accuracy is analyzed while solving the problem describing the third optical harmonic generation process. Two implicit conservative FDSs are developed for a numerical solution of this problem. One of them preserves a difference analog of the energy invariant (or the Hamiltonian) accurately, while the Hamiltonian (or the energy’s invariant) is preserved with the second order of accuracy. Both FDSs possess the second order of approximation at a smooth enough solution of the differential problem. Computer simulations demonstrate advantages of the implicit FDS preserving the Hamiltonian. To illustrate the advantages of the developed FDSs, a comparison of the computer simulation results with those obtained applying the Strang method, based on either an implicit scheme or the Runge–Kutta method, is made. The corresponding theorems, which claim the second order of approximation for preserving invariants for the FDSs under consideration, are stated.

Suggested Citation

  • Vyacheslav Trofimov & Maria Loginova, 2021. "Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants," Mathematics, MDPI, vol. 9(21), pages 1-26, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2716-:d:665115
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    References listed on IDEAS

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    1. Barletti, L. & Brugnano, L. & Frasca Caccia, G. & Iavernaro, F., 2018. "Energy-conserving methods for the nonlinear Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 3-18.
    2. Twizell, E.H. & Bratsos, A.G. & Newby, J.C., 1997. "A finite-difference method for solving the cubic Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 43(1), pages 67-75.
    3. Hederi, M. & Islas, A.L. & Reger, K. & Schober, C.M., 2016. "Efficiency of exponential time differencing schemes for nonlinear Schrödinger equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 127(C), pages 101-113.
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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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