Existence of Kink Waves and Periodic Waves for a Perturbed Defocusing mKdV Equation

The existence of kink waves and periodic waves for a perturbed defocusing mKdV equation is established by using geometric singular perturbation theory. In addition, by analyzing the perturbation of the Hamiltonian vector field with an elliptic Hamiltonian of degree four, a two saddle cycle is exhibited. It is proven that the wave speed is decreasing on by analyzing the ratio of Abelian integrals and the limit of wave speed is given. Furthermore, the relationship between the wave speed and the wavelength of traveling waves is obtained.