Fundamental Solutions for the Coupled KdV System and Its Stability

In this paper, we establish exact solutions for the non-linear coupled KdV equations. The exp-function method is used to construct the solitary travelling wave solutions for these equations. The numerical adaptive moving mesh PDEs (MMPDEs) method is also implemented in order to solve the proposed coupled KdV equations. The achieved results may be applicable to some plasma environments, such as ionosphere plasma. Some numerical simulations compared with the exact solutions are provided to illustrate the validity of the proposed methods. Furthermore, the modulational instability is analyzed based on the standard linear-stability analysis. The depiction of the techniques are straight, powerful, robust and can be applied to other nonlinear systems of partial differential equations.