Existence of solitary waves and periodic waves for a perturbed generalized BBM equation

The existence of solitary waves and periodic waves for a perturbed generalized BBM equation is established by using geometric singular perturbation theory. Attention goes to perturbations of the Hamiltonian vector field with an elliptic Hamiltonian of degree four, exhibiting a cuspidal loop. It is proven that the wave speed c0(h)c0(h) is decreasing on h∈[0,1/12]h∈[0,1/12] by analyzing the ratio of Abelian integrals. The upper and lower bounds of the limit wave speed are given. Moreover, the relation between the wave speed and the wavelength of traveling waves is obtained.