The space of 4-ended solutions to the Allen–Cahn equation in the plane

Abstract

We are interested in entire solutions of the Allen–Cahn equation \mathrm{\Delta }u−F^{′}(u) = 0 which have some special structure at infinity. In this equation, the function F is an even, double well potential. The solutions we are interested in have their zero set asymptotic to 4 half oriented affine lines at infinity and, along each of these half affine lines, the solutions are asymptotic to the one dimensional heteroclinic solution: such solutions are called 4 -ended solutions . The main result of our paper states that, for any \theta \in (0,\pi / 2) , there exists a 4 -ended solution of the Allen–Cahn equation whose zero set is at infinity asymptotic to the half oriented affine lines making the angles θ , \pi −\theta , \pi + \theta and 2\pi −\theta with the x -axis. 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 .