Traveling waves in a generalized Bogoyavlenskii coupled system under perturbation of distributed delay and weak dissipation

Abstract

In this paper, we focus on the traveling waves for a perturbed generalized Bogoyavlenskii coupled system with delay convection term and weak dissipation, including periodic waves, solitary waves, kink and anti-kink waves. The corresponding traveling wave equation can be transformed into a 4-dimensional dynamical system, which is regarded as a singularly perturbed system and is reduced to a near-Hamiltonian form. We construct a locally invariant manifold diffeomorphic to the critical manifold with normally hyperbolicity and then give the existence conditions of traveling waves by the geometric singular perturbation theory as well as the number of periodic waves by analyzing the monotonicity of ratios of Abelian integrals. Moreover, the monotonicity of the wave speed is provided as well as its supremum and infimum. Numerical simulations are in complete agreement with the theoretical predictions.