Persistence of solitary wave solutions for the delayed regularized long wave equation under Kuramoto–Sivashinsky perturbation and Marangoni effect

Persistence of solitary wave solutions of the regularized long wave equation with small perturbations are investigated by the geometric singular perturbation theory and bifurcation theory. Two different kinds of the perturbations are considered in this paper: one is the Kuramoto–Sivashinsky perturbation, the other is the Marangoni effects. Indeed, the solitary wave persists under small perturbations. Furthermore, the different perturbations do affect the proper wave speed c ensuring the persistence of the solitary waves. Finally, numerical simulations are utilized to confirm the theoretical results.