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An effective approach to numerical soliton solutions for the Schrödinger equation via modified cubic B-spline differential quadrature method

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  • Bashan, Ali
  • Yagmurlu, Nuri Murat
  • Ucar, Yusuf
  • Esen, Alaattin

Abstract

In this study, an effective differential quadrature method (DQM) which is based on modified cubic B-spline (MCB) has been implemented to obtain the numerical solutions for the nonlinear Schrödinger (NLS) equation. After separating the Schrödinger equation into coupled real value differential equations,we have discretized using DQM and then obtained ordinary differential equation systems. For time integration, low storage strong stability-preserving Runge–Kutta method has been used. Numerical solutions of five different test problems have been obtained. The efficiency and accuracy of the method have been measured by calculating error norms L2 and Linfinity and two lowest invariants I1 and I2. Also relative changes of invariants are given. The newly obtained numerical results have been compared with the published numerical results and a comparison has shown that the MCB-DQM is an effective numerical scheme to solve the nonlinear Schrödinger equation.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:100:y:2017:i:c:p:45-56
    DOI: 10.1016/j.chaos.2017.04.038
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    References listed on IDEAS

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    1. 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.
    2. Ömer Oruç & Alaattin Esen & Fatih Bulut, 2016. "A Haar wavelet collocation method for coupled nonlinear Schrödinger–KdV equations," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 27(09), pages 1-16, September.
    3. 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.
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    Cited by:

    1. Mohammadi, Reza, 2018. "Smooth Quintic spline approximation for nonlinear Schrödinger equations with variable coefficients in one and two dimensions," Chaos, Solitons & Fractals, Elsevier, vol. 107(C), pages 204-215.
    2. 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.

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