Traveling waves in a quintic BBM equation under both distributed delay and weak backward diffusion

Abstract

In this paper, we focus on the existence of periodic and solitary waves for a quintic Benjamin–Bona–Mahony (BBM) equation with distributed delay and diffused perturbation. The corresponding traveling wave equation is transformed into a three-dimensional dynamical system, which is regarded as a singularly perturbed system for small perturbation parameter. By geometric singular perturbation theory, the three-dimensional dynamical system can be reduced to a near-Hamiltonian planar system and a locally invariant manifold diffeomorphic to the critical manifold with normally hyperbolicity can be constructed. Further, the existence of periodic wave and solitary wave are established by proving the persistence of periodic and homoclinic orbits of the near-Hamiltonian planar system for sufficiently small perturbation, and their existence conditions for the delayed quintic BBM equation have also been obtained. The uniqueness of periodic wave is established by analyzing the monotonicity of ratios of Abelian integrals form a Chebyshev set. Moreover, the monotonicity of wave speed is proved, and the supremum and infimum of wave speed are gotten. Numerical simulations are in complete agreement with the theoretical predictions.