Traveling wave solutions for a Gardner equation with distributed delay under KS perturbation

To investigate the effect of distributed delay, diffusion and dissipation on traveling wave solutions, we consider the KS perturbation of a Gardner equation with distributed delay. By applying geometric singular perturbation theory, the traveling wave system of singularly perturbed Gardner equation is reduced to a regularly perturbed planar system. Chebyshev criterion is used to prove the monotonicity of ratio of Abelian integrals. Translation transformation is introduced to simplify the traveling wave system of the singular perturbed equation. Conditions on the existence of periodic wave, kink wave and solitary wave solutions are given by the bifurcation theory. Final numerical simulations show that the amplitude and wavelength of the persisted periodic wave change monotonously following monotonous variation of the wave speed.