Dynamics of traveling waves to a shallow-water wave equation with Marangoni effect

This paper is concerned with the existence and asymptotic behavior of traveling waves to a class of shallow-water wave equations with Marangoni effect based on Camassa-Holm equation. First, by transforming the high-order equations into ordinary differential equations, we clarify the existence of traveling waves for unperturbed equations. Then, we prove the existence of traveling waves of the equation with Marangoni perturbation by geometric singular perturbation, invariant manifold theory, and Melnikov method. Finally, we discuss the asymptotic behavior of traveling waves to both unperturbed and perturbed equations.