Analytical and numerical solution of a coupled KdV–MKdV system

In this work a two-fold compound elementary Darboux transformations (DTs) are newly used to produce two-parameters explicit solutions for a coupled KdV–MKdV system. We consider a second order differential equation as a spectral problem with 2 × 2 matrix coefficients. A second covariant (with respect to DTs) equation is selected to form a Lax pair of a coupled KdV–MKdV system, under correspondent reduction constraint. This reduction gives an automorphism that relates two pairs of solutions of the spectral equation corresponding to different values of the spectral parameters. We use this result in the compound elementary DTs to produce explicit solutions to the coupled KdV–MKdV system being the compatibility condition of Lax pair under this reduction. Effects of parameters on the solution (reality, singularity) are analyzed. A numerical method of solution (difference scheme) of a Cauchy problem for the coupled KdV–MKdV system is also introduced. We analyze stability and prove the convergence of the scheme which gives the conditions and the appropriate choice of the grid sizes. The scheme is tested by numerical simulation of the explicit solutions evaluation.