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 $\mathrm{\Delta }u-F^{\prime }(u)=0$ 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 $\theta$, $\pi -\theta$, $\pi +\theta$ 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 $k⩾2$.