Dynamics on traveling wave solutions for Broer-Kaup equation with distributed delay

This paper investigates the traveling wave solutions of the Broer-Kaup equation with distributed delay. Using geometric singular perturbation theory and the Melnikov function method, a qualitative analysis on the traveling wave equation is carried out. By restricting the system to a locally invariant manifold, the delayed traveling wave equation is reduced to a near-Hamiltonian system. A translation transformation is applied to simplify the near-Hamiltonian planar system. Through involution mapping criterion, the monotonicity of the ratio of two Abelian integrals is established, which ensures that the Melnikov function possesses a unique simple zero. Based on Poincaré and heteroclinic bifurcation theory, the sufficient conditions on the persistence of periodic and kink (anti-kink) wave solutions are derived. Moreover, we present numeric simulations to illustrate the given results.