Dynamics of Traveling Waves for the Perturbed Generalized KdV Equation

This paper is devoted to the existence of traveling, solitary and periodic waves for the perturbed generalized KdV by applying geometric singular perturbation, differential manifold theory and the regular perturbation analysis of Hamiltonian systems. Under the assumptions that the distributed delay kernel is the strong general one and the average delay is sufficiently small, traveling, solitary and periodic waves are shown to exist in the perturbed system. It is further proved that the wave speed is decreasing by analyzing the ratio of Abelian integrals, and we analyze these functions by using the theory of analytic functions and algebraic geometry. Moreover, the upper and lower bounds of the limit wave speed are presented. The relationship between wavelength and wave speed of traveling waves is also established.