Traveling Wave Solutions for Two Perturbed Nonlinear Wave Equations with Distributed Delay

Abstract

Traveling wave solutions are a class of invariant solutions which are critical for shallow water wave equations. In this paper, traveling wave solutions for two perturbed KP-MEW equations with a local delay convolution kernel are examined. The model equation is reduced to a planar near-Hamiltonian system via geometric singular perturbation theorem, and the qualitative properties of the corresponding unperturbed system are analyzed by using dynamical system approach. The persistence of the bounded traveling wave solutions for the perturbed KP-MEW equations with delay is investigated. By using a criterion for the monotonicity of ratio of two Abelian integrals and Melnikov’s method, the existence of kink (anti-kink) wave solutions and periodic wave solutions of the model equation are established. The result shows that the delayed KP-MEW equations with positive perturbation and the one with negative perturbation exhibit completely diverse dynamical properties. These new findings greatly enrich the understanding of dynamical properties of the traveling wave solutions of perturbed nonlinear wave equations with local delay convolution kernel. Numerical experiments further confirm and illustrate the theoretical results.