Existence of traveling waves in a delayed convecting shallow water fluid model

This paper investigates a delayed shallow water fluid model that has not been studied in previous literature. Applying geometric singular perturbation theory, we prove the existence of traveling wave solutions for the model with a nonlocal weak delay kernel and local strong delay convolution kernel, respectively. When the convection term contains a nonlocal weak generic delay kernel, the desired heteroclinic orbit is obtained by using Fredholm theory and linear chain trick to prove the existence of two kink wave solutions under certain parametric conditions. When the model contains local strong delay convolution kernel and weak backward diffusion, under the same parametric conditions to the previous case, the corresponding traveling wave system can be reduced to a near-Hamiltonian system. The existence of a unique periodic wave solution is established by proving the uniqueness of zero of the Melnikov function. Uniqueness is proved by utilizing the monotonicity of the ratio of two Abelian integrals.