The classification of four end solutions of the Allen-Cahn equation on the plane

TL;DR

This paper classifies four-ended solutions of the Allen-Cahn equation on the plane with almost parallel ends, showing they form a one-parameter family connected to solutions of the Toda system, including the saddle solution.

Contribution

It establishes a classification and uniqueness result for four-ended solutions with almost parallel ends, linking them to the Toda system and describing their moduli space.

Findings

Four-ended solutions are parametrized by the Toda system solutions.

Uniqueness of four-ended solutions with almost parallel ends is proven.

All such solutions form a one-parameter family, including the saddle solution.

Abstract

An entire solution of the Allen-Cahn equation $\Delta u=f(u)$, where $f$ has exactly three zeros at $\pm 1$ and 0, is balanced and odd, e.g. $f(u)=u(u^2-1)$, is called a $2k$-ended 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 consider the family of four ended solutions whose ends are almost parallel at $\infty$. We show that this family can be parametrized by the family of solutions of the two component Toda system. As a result we obtain the uniqueness of four ended solutions with almost parallel ends. Combining this result with the classification of connected components in the moduli space of the four ended solutions we can classify all such solutions. Thus we show that four end solutions form, up to rigid motions, a one parameter family. This family contains 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 of the nodal set is arbitrary small (almost parallel nodal sets).