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A finite-difference method for solving the cubic Schrödinger equation

Author

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  • Twizell, E.H.
  • Bratsos, A.G.
  • Newby, J.C.

Abstract

A family of finite-difference methods is used to transform the initial/boundary-value problem associated with the nonlinear Schrödinger equation into a first-order, linear, initial-value problem. Numerical methods are developed by replacing the time and space derivatives by central-difference replacements. The resulting finite-difference methods are analysed for local truncation, errors, stability and convergence. The results of a number of numerical experiments are given.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:43:y:1997:i:1:p:67-75
    DOI: 10.1016/S0378-4754(96)00056-0
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    Cited by:

    1. Bashan, Ali & Yagmurlu, Nuri Murat & Ucar, Yusuf & Esen, Alaattin, 2017. "An effective approach to numerical soliton solutions for the Schrödinger equation via modified cubic B-spline differential quadrature method," Chaos, Solitons & Fractals, Elsevier, vol. 100(C), pages 45-56.
    2. Dereli, Yılmaz & Irk, Dursun & Dağ, İdris, 2009. "Soliton solutions for NLS equation using radial basis functions," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1227-1233.
    3. Korkmaz, Alper, 2018. "Stability satisfied numerical approximates to the non-analytical solutions of the cubic Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 210-231.
    4. Zou, Guang-an & Wang, Bo & Sheu, Tony W.H., 2020. "On a conservative Fourier spectral Galerkin method for cubic nonlinear Schrödinger equation with fractional Laplacian," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 168(C), pages 122-134.
    5. 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.

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