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

TL;DR

This paper classifies four-end solutions to the Allen-Cahn equation in the plane, establishing a family of solutions with varying angles between nodal lines and analyzing the structure of their moduli space.

Contribution

It introduces a classification of four-end solutions, constructs a one-parameter family, and studies the angle map's surjectivity in the moduli space.

Findings

Existence of a one-parameter family of solutions including the saddle solution.

The angle between asymptotic nodal lines can vary continuously in (0, π/2).

The angle map is surjective onto (0, π/2) for connected components of the solution space.

Abstract

An entire solution of the Allen-Cahn equation $\Delta u=F'(u)$, where $F$ is an even, bistable function, is called a $2k$-end solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$ looks like the one dimensional, heteroclinic solution. In this paper we initiate a program to classify the four-end solutions of the Allen-Cahn equation in $\R^2$. We show that there exists a one parameter family of solutions containing the saddle solution, for which the angle between the nodal lines is $\frac{\pi}{2}$, as well as solutions for which the angle between the asymptotic half lines is any $\theta\in (0, \frac{\pi}{2})$. This justifies the definition of the angle map for a four-end solution $u$, which is the angle $\theta=\theta(u)\in (0, \frac{\pi}{2})$ between the asymptote to the nodal line in the first quadrant and the x axis. Then we show that on any connected component in the moduli space of four-end solutions the angle map is surjective onto $(0,\frac{\pi}{2})$.