Abstract
In this paper we are going to derive two numerical methods for solving the coupled nonlinear Schrodinger-Boussinesq equation. The first method is a nonlinear implicit scheme of second order accuracy in both directions time and space; the scheme is unconditionally stable. The second scheme is a nonlinear implicit scheme of second order accuracy in time and fourth order accuracy in space direction. A generalized method is also derived where the previous schemes can be obtained by some special values of l. The proposed methods will produced a coupled nonlinear tridiagonal system which can be solved by fixed point method. The exact solutions and the conserved quantities for two different tests are used to display the robustness of the proposed schemes.
Highlights
In this work we are going to derive a highly a accurate schemes for the coupled nonlinear SchrödingerBoussinesq equations (CSBE) where i=iut − uxx + uv= 0, x ∈ R, t > 0 (1)( ) ( ) vtt − vxx + α v2 xx + σ vxxxx − u2 = 0, xx (2)−1, u ( x, t ) represents the complex short wave amplitude, v ( x, t ) represents the long waveHow to cite this paper: Ismail, M.S. and Ashi, H.A. (2016) A Compact Finite Difference Schemes for Solving the Coupled Nonlinear Schrodinger-Boussinesq Equations
Few numerical methods exist in literature for solving the CSBE
The conservation of the conserved quantities for the proposed system using the numerical methods presented in this work is a good indication for the efficiency and robustness these methods
Summary
In this work we are going to derive a highly a accurate schemes for the coupled nonlinear SchrödingerBoussinesq equations (CSBE). −1 , u ( x, t ) represents the complex short wave amplitude, v ( x, t ) represents the long wave. (2016) A Compact Finite Difference Schemes for Solving the Coupled Nonlinear Schrodinger-Boussinesq Equations. Equations (1) and (2) were considered as a model of the interactions between short and intermediate long waves, and were originated in describing the dynamics of Langmuir soliton formation, the interaction in plasma [1]-[4]. Numerical solution of coupled nonlinear Schrödinger equation using different methods can be found in [5]-[8]. Zhang et al [9] derived a conservative difference scheme to solve the CSBE. A multi-symplectic scheme for solving the CSBE is developed in [10]
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