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Hamiltonians of the Generalized Nonlinear Schrödinger Equations

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  • Nikolay A. Kudryashov

    (Moscow Engineering Physics Institute, National Research Nuclear University MEPhI, 31 Kashirskoe Shosse, 115409 Moscow, Russia
    National Research Center: “Kurchatov Center”, 1 Akademika Kurchatova Sq., 123098 Moscow, Russia)

Abstract

Some types of the generalized nonlinear Schrödinger equation of the second, fourth and sixth order are considered. The Cauchy problem for equations in the general case cannot be solved by the inverse scattering transform. The main objective of this paper is to find the conservation laws of the equations using their transformations. The algorithmic method for finding Hamiltonians of some equations is presented. This approach allows us to look for Hamiltonians without the derivative operator and it can be applied with the aid of programmes of symbolic calculations. The Hamiltonians of three types of the generalized nonlinear Schrödinger equation are found. Examples of Hamiltonians for some equations are presented.

Suggested Citation

  • Nikolay A. Kudryashov, 2023. "Hamiltonians of the Generalized Nonlinear Schrödinger Equations," Mathematics, MDPI, vol. 11(10), pages 1-12, May.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:10:p:2304-:d:1147469
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    References listed on IDEAS

    as
    1. Fang, Yin & Wu, Gang-Zhou & Kudryashov, Nikolay A. & Wang, Yue-Yue & Dai, Chao-Qing, 2022. "Data-driven soliton solutions and model parameters of nonlinear wave models via the conservation-law constrained neural network method," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    2. Hu, Xiang & Yin, Zhixiang, 2022. "A study of the pulse propagation with a generalized Kudryashov equation," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    3. Sun, Zhengjie, 2022. "A conservative scheme for two-dimensional Schrödinger equation based on multiquadric trigonometric quasi-interpolation approach," Applied Mathematics and Computation, Elsevier, vol. 423(C).
    4. Wu, Gang-Zhou & Fang, Yin & Kudryashov, Nikolay A. & Wang, Yue-Yue & Dai, Chao-Qing, 2022. "Prediction of optical solitons using an improved physics-informed neural network method with the conservation law constraint," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    5. Nikolay A. Kudryashov, 2022. "Optical Solitons of the Generalized Nonlinear Schrödinger Equation with Kerr Nonlinearity and Dispersion of Unrestricted Order," Mathematics, MDPI, vol. 10(18), pages 1-9, September.
    Full references (including those not matched with items on IDEAS)

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