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

Abstract

An entire solution of the Allen–Cahn equation Δu=f(u), where f is an odd function and has exactly three zeros at ±1 and 0, for example, f(u)=u(u2−1), 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 ∞. 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 π∕2, as well as solutions for which the angle between the asymptotic half lines of the nodal set is any θ∈(0,π∕2).