Moduli space theory for the Allen-Cahn equation in the plane

Abstract:

In this paper we study entire solutions of the Allen-Cahn equation $\Delta u-F^{\prime }(u)=0$, 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 $2k$, $k\geq 2$ 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 $\mathcal M_{2k}$. The main result of our paper states that if $u \in \mathcal M_{2k}$ 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 $k \geq 2$.