The Gierer & Meinhardt system: The breaking of homoclinics and multi-bump ground states

In this paper we study ground-states of the Gierer & Meinhardt system on the line, namely solutions of the problem u - u + u(2)/v = 0, sigma (-2)v - v + u(2) = 0, u, v > 0, u(+/-infinity) = 0 = v(+/-infinity). We prove that given any number N, there exists a solution to this problem exhibiting exactly N bumps in its u-component, separated from each other at a distance O(\ log sigma\), whenever sigma is sufficiently small. These bumps resemble the shape of the unique solution of U - U + U-2 = 0, 0 < U(+/-) = 0, U'(0) = 0.