Solitary wave solution in a perturbed simplified modified Camassa–Holm equation

This paper investigates the persistence of solitary wave solution in a perturbed simplified modified Camassa–Holm equation. The perturbation terms are the backward diffusion and dissipation, which come from the Kuramoto–Sivashinsky equation. Firstly, on the basis of the classical Camassa–Holm equation, the bifurcated phase portraits of homoclinic orbit (or cuspidal loop) and solitary wave solution for unperturbed simplified modified Camassa–Holm equation are obtained by dynamic system method. And the persistence of solitary wave solutions with suitable wave speed $c$ under Kuramoto–Sivashinsky perturbation is proved by using the geometric singular perturbation approach and Melnikov integral. Different from the study of the cuspidal loop in previous work, the simple zero point of Melnikov integral is obtained by calculating its explicit expression. At last, the results are verified by numerical simulation.