Traveling Waves in a Generalized KdV Equation with Arbitrarily High-Order Nonlinearity and Different Distributed Delays

This paper focuses on the existence of periodic and solitary wave solutions in a generalized KdV equation with an arbitrarily high-order convection term which introduces a time delay in the nonlinearity. For the equation with two different local generic delay kernels, by applying geometric singular perturbation theory and analyzing the perturbation of a hyper-elliptic Hamiltonian system of arbitrary higher degree, we respectively prove the existence of one or two periodic wave solutions with certain wave speed in an open interval, depending on the degree. The existence of solitary wave solutions with certain wave speeds is also established by Melnikov's method. Our results demonstrate that distributed delays and the degree of nonlinear term can influence the existence and number of traveling wave solutions with particular wave speeds.