A finite difference scheme for smooth solutions of the general mKdV equation

TL;DR

This paper introduces a finite difference scheme for a generalized mKdV equation family, capable of preserving classical solutions and demonstrating soliton interactions through numerical simulations.

Contribution

A novel finite difference scheme that maintains the global classical solutions of the generalized mKdV equations under certain conditions.

Findings

The scheme effectively captures soliton propagation and interaction.

Numerical results confirm the scheme's stability and accuracy.

The method applies to a broad class of nonlinear dispersive equations.

Abstract

We consider a generalization of the mKdV model of shallow water out-flows. This generalization is a family of equations with nonlinear dispersion terms containing, in particular, KdV, mKdV, Benjamin-Bona-Mahony, Camassa-Holm, and Degasperis-Procesi equations. Nonlinear dispersion, generally speaking, implies instability of classical solutions and wave breaking in a finite time. However, there are special conditions under which the general mKdV equation admits classical solutions that are global in time. We have created an economic finite difference scheme that preserves this property for numerical solutions. To illustrate this we demonstrate some numerical results about propagation and interaction of solitons.