Existence of periodic and solitary waves of a Boussinesq equation under perturbations

In this paper, we consider a Boussinesq equation containing weak backward diffusion, delay in the convection term, dissipation and Marangoni effect. By applying geometric singular perturbation theory, a locally invariant manifold diffeomorphic to the critical manifold is established. For Boussinesq equation with delay and weak backward diffusion, the monotonicity of ratio of Abelian integrals is analyzed by utilizing the Picard–Fuchs equation. The conditions on existence of a unique periodic wave and solitary waves are obtained as well as the bound of wave speed. For Boussinesq equation with weak backward diffusion, dissipation and Marangoni effect, the corresponding Melnikov function containing three generic elements is given. The parametric conditions on existence of a unique and two periodic waves are derived respectively. Furthermore, the existence of a unique solitary wave is proved under some parametric conditions.