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Numerical Simulation of Modified Kortweg-de Vries Equation by Linearized Implicit Schemes

Applied Mathematics 2020年第11期11卷 1139-1161页

作者： M. S. Ismail Fakhirah Alotaibi Department of Math King Abdulaziz University Jeddah Saudi Arabia

In this paper, we are going to derive four numerical methods for solving the Modified Kortweg-de Vries (MKdV) equation using fourth Pade approximation for space direction and Crank Nicolson in the time direction. Two nonlinear schemes and two linearized schemes are presented. All resulting schemes will be analyzed for accuracy and stability. The exact solution and the conserved quantities are used to highlight the efficiency and the robustness of the proposed schemes. interaction of two and three solitons will be also conducted. The numerical results show that the interaction behavior is elastic and the conserved quantities are conserved exactly, and this is a good indication of the reliability of the schemes which we derived. A comparison with some existing is presented as well.

关键词： MKdV Equation Pade Approximation Nonlinear Numerical Schemes Linearly Implicit Schemes Fixed Point Method interaction of solitons

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Computational Methods for Three Coupled Nonlinear Schrödinger Equations

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Applied Mathematics 2016年第17期7卷 2110-2131页

作者： M. S. Ismail S. H. Alaseri Department of Math King Abdulaziz University Jeddah Kingdom of Saudi Arabia

In this work, we will derive numerical schemes for solving 3-coupled nonlinear Schrödinger equations using finite difference method and time splitting method combined with finite difference method. The resulting schemes are highly accurate, unconditionally stable. We use the exact single soliton solution and the conserved quantities to check the accuracy and the efficiency of the proposed schemes. Also, we use these methods to study the interaction dynamics of two solitons. It is found that both elastic and inelastic collision can take place under suitable parametric conditions. We have noticed that the inelastic collision of single solitons occurs in two different manners: enhancement or suppression of the amplitude.

关键词： Three Coupled Nonlinear Schrodinger Equations Finite Difference Method Time Splitting Method interaction of solitons

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