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

### Frank Pacard

Institut Universitaire de France et Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau, France### Michał Kowalczyk

Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile### Yong Liu

Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile

## Abstract

We are interested in entire solutions of the Allen–Cahn equation $Δu−F_{′}(u)=0$ 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 $θ∈(0,π/2)$, 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 $θ$, $π−θ$, $π+θ$ and $2π−θ$ 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 $k⩾2$.

## Cite this article

Frank Pacard, Michał Kowalczyk, Yong Liu, The space of 4-ended solutions to the Allen–Cahn equation in the plane. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 5, pp. 761–781

DOI 10.1016/J.ANIHPC.2012.04.003