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Analytical and numerical solutions are obtained for coupled nonlinear partial differential equation by the well-known Laplace decomposition method. We combined Laplace transform and Adomain decomposition method and present a new approach for solving coupled Schrödinger-Korteweg-de Vries (Sch-KdV) equation. The method does not need linearization, weak nonlinearity assumptions, or perturbation theory. We compared the numerical solutions with corresponding analytical solutions.

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In the paper, the coupled 1D nonlinear Schrödinger system (CNLS) is considered as the model equation for wave-wave interaction in ionic media. A finite difference scheme is derived for the model equations.A new six-point scheme, which is equivalent to the multi-symplectic integrator, is derived. The numerical simulation is also presented for the model equations.