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

Abstract

An entire solution of the Allen–Cahn equation $\Delta u=f\left(u\right)$, where $f$ is an odd function and has exactly three zeros at $\pm 1$ and $0$, for example, $f\left(u\right)=u\left(u^2-1\right)$, 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 consider the family of four-end solutions whose ends are almost parallel at $\infty$. We show that this family can be parametrized by the family of solutions of the Toda system. As a result we obtain the uniqueness of four-end solutions with almost parallel ends. Combining this result with the classification of connected components in the moduli space of the four-end 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 $\pi ∕2$, as well as solutions for which the angle between the asymptotic half lines of the nodal set is any $\theta \in \left(0,\pi ∕2\right)$.