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Stability satisfied numerical approximates to the non-analytical solutions of the cubic Schrödinger equation

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  • Korkmaz, Alper

Abstract

The time dependent complex Schrödinger equation with cubic non linearity is solved by constructing differential quadrature algorithm based on sinc functions. Reduction the equation to a coupled system of real equations enables to approximate the space derivative terms by the proposed method. The resultant ordinary differential equation system is integrated with respect to the time variable by using various explicit methods of lower and higher orders. Some initial boundary value problems containing some analytical and non-analytical initial data are solved for experimental illustrations. The computational errors between the analytical and numerical solutions are measured by the discrete maximum error norm in case the analytical solutions exist. The two conserved quantities are calculated by using the numerical results in all cases. The matrix stability analysis is implemented to control the time step size.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:331:y:2018:i:c:p:210-231
    DOI: 10.1016/j.amc.2018.03.035
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
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