Moduli space theory for the allen-cahn equation in the plane

In this paper we study entire solutions of the Allen-Cahn equation Δu - F'(u) = 0, where F is an even, bistable function. We are particularly interested in the description of the moduli space of solutions which have some special structure at infinity. The solutions we are interested in have their zero set asymptotic to 2k, k ≥ 2 oriented affine half-lines at infinity and, along each of these affine half-lines, the solutions are asymptotic to the one-dimensional heteroclinic solution: such solutions are called multiple-end solutions, and their set is denoted by M_2k. The main result of our paper states that if u∈M_2k is nondegenerate, then locally near u the set of solutions is a smooth manifold of dimension 2k. This paper is part of a program whose aim is to classify all 2k-ended solutions of the Allen-Cahn equation in dimension 2, for k ≥ 2.