The existence of solitary waves of singularly perturbed mKdV–KS equation

Abstract

We study a sort of nonlinear reaction diffusion equation based on the modified Korteweg–de Vries (mKdV) equation with a higher order singularly perturbing term as the Kuramoto–Sivashinsky (KS) equation, called mKdV–KS equation. Special attention is paid to the question of the existence of solitary wave solutions. Based on the analogue between solitary wave solution and homoclinic orbits of the associated ordinary differential equations, from geometric singular perturbation point of view, we prove that solitary wave persists when the perturbation parameter is suitably small. This argument does not require an explicit expression for the original mKdV solitary wave solution.