The classification of four-end solutions to the allen-cahn equation on the plane

An entire solution of the Allen-Cahn equation δu=f(u), where f is an odd function and has exactly three zeros at ± 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 onedimensional, heteroclinic solution. In this paper we consider the family of four-end solutions whose ends are almost parallel at 1. 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 ε (0, π/2).