Dynamics of the modified Rotation-Camassa-Holm Equation with backward diffusion term

Considered herein is a modified Rotation-Camassa-Holm equation with backward diffusion term, which is regarded as a model describing the velocity dynamics of shallow water waves. We first investigate the equilibrium point types of the unperturbed R-CH system and acquire the homoclinic solution of the unperturbed system via analysis of phase plane and Hamilton function. Furthermore, the existence of solitary wave solutions for the perturbed Rotation-Camassa-Holm equation with a local and nonlocal delay convolution kernel are investigated, respectively. Our refined results are based on the method of dynamical system, including the geometric singular perturbation theory, invariant manifold and Melnikov method, which is the novel aspect.