Persistence of the Traveling Wave Solutions for the Modified Equal Width Equation with Singular Perturbation

Abstract In this paper, we investigate the persistence of solitary and periodic traveling wave solutions for the modified equal width (MEW) equation under a singular Kuramoto–Sivashinsky-type perturbation. In the traveling wave frame, the perturbed MEW equation is transformed into a singularly perturbed ordinary differential equation. By applying Fenichel’s invariant manifold theorem, we construct an invariant manifold where the system reduces to a two-dimensional nearly Hamiltonian system. We first examine the solitary and periodic wave solutions of the unperturbed MEW equation by analyzing the phase portraits and bifurcations of the corresponding traveling wave system. The existence of a limit cycle of the perturbed system is established via the Poincaré–Bendixson theorem. Furthermore, by combining the Chebyshev criterion with the Melnikov method, we prove the existence of a homoclinic loop and the uniqueness of limit cycle. These results theoretically confirm the persistence of both solitary and periodic traveling wave solutions under the perturbation. Numerical simulations are provided to support the theoretical findings.