Analytical and numerical solution of a coupled KdV-MKdV system

TL;DR

This paper develops analytical Darboux transformation techniques and a numerical difference scheme to solve the coupled KdV-MKdV system, analyzing solution properties and stability.

Contribution

It introduces a new analytical approach using Darboux transformations and a stable numerical scheme for the coupled KdV-MKdV system.

Findings

Two-parameter solutions with reductions are obtained.

The numerical scheme is proven stable and convergent.

Numerical simulations confirm the analytical solutions.

Abstract

The matrix 2x2 spectral differential equation of the second order is considered on x in ($-\infty,+\infty$). We establish elementary Darboux transformations covariance of the problem and analyze its combinations. We select a second covariant equation to form Lax pair of a coupled KdV-MKdV system. The sequence of the elementary Darboux transformations of the zero-potential seed produce two-parameter solution for the coupled KdV-MKdV system with reductions. We show effects of parameters on the resulting solutions (reality, singularity). A numerical method for general coupled KdV-MKdV system is introduced. The method is based on a difference scheme for Cauchy problems for arbitrary number of equations with constants coefficients. We analyze stability and prove the convergence of the scheme which is also tested by numerical simulation of the explicit solutions.