Abstract:

In paper[1] (Indiana Univ Math J, 2001, 50:443-507), the authors studied the existence and stability of multiple-pulse homoclinic solutions in a generalized Gierer-Meinhardt equation. However, in this paper, a general integral measuring the distance of the stable and unstable manifolds of the critical manifold of the layer system, i.e., the Melnikov integral, was not computed explicitly. So we have two aims in this manuscript. Firstly, we give an elementary method to solve a second-order nonlinear conservative system and hence obtain the explicit representation of the homoclinic orbit. Secondly, we substitute the explicit representation of the homoclinic orbit into the the Melnikov integral. By computing such a general integral, we obtain a more explicitly parametric condition on the existence of multiple-pulse homoclinic solutions in such a generalized Gierer-Meinhardt equation.

Key words: Generalized Gierer-Meinhardt equation, Multi-pulse homoclinic orbit, Melnikov function