Existence of Periodic Waves in a Perturbed Generalized BBM Equation

Abstract

In this paper, a perturbed quintic BBM equation with weak backward diffusion and dissipation effects is investigated. By applying geometric singular perturbation theory and analyzing the perturbations of a Hamiltonian system with a hyper-elliptic Hamiltonian of degree six, we prove the existence of isolated periodic wave solutions with certain wave speed in an open interval. It is also shown that isolated periodic wave solutions persist for any energy parameter [Formula: see text] in an open interval under small perturbation. Furthermore, we prove that the wave speed [Formula: see text] of periodic wave is strictly monotonically increasing with respect to [Formula: see text] by analyzing Abelian integral having three generating elements. Moreover, the upper and lower bounds of the limiting wave speed are obtained. Our analysis is mainly based on Melnikov theory, Chebyshev criteria, and symbolic computation, which may be useful for other problems.