2011/09/26 by Frank Pacard, Pacard, Frank, Michał Kowalczyk +4
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Meromorphic and Entire Functions #Nonlinear Partial Differential Equations #math.AP
paper · pdf · doi:10.48550/arxiv.1109.5604
openalex publication_date 2011/09/26 · arxiv created 2011/09/29 · arxiv updated 2011/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An entire solution of the Allen-Cahn equation Δu=f(u), where f has exactly three zeros at ± 1 and 0, is balanced and odd, e.g. f(u)=u(u2-1), 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 ∞. 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 \fracπ2 as well as solutions for which the angle between the asymptotic half lines of the nodal set is arbitrary small (almost parallel nodal sets).