The Existence Of Solitary Waves Of Singularly Perturbed KdV-mKdV Equation

This paper is concerned with the persistence of solitary wave solutions of perturbed KdV-mKdV equation. Based on the relation between solitary wave solutions and homoclin-ic orbits of associated ordinary differential equations, using geometric singular perturbation theory and Melnikov theory, we prove that solitary wave persists when the perturbation pa-rameter ε is sufficiently small. Moreover, we give the numerical simulation for ε sufficiently small and large. When ε is large, the homoclinic orbits will break and the corresponding wave is oscillatory. Besides we analyze the limited system and present the closed orbits and heteroclinic orbits for ε=0 and ε sufficiently small. In the numerical simulation, we see that perturbed KdV-mKdV equation exists bell, anti-bell, kink and anti-kink shaped solitary wave and also periodic wave.