2011/06/10 by Michal Kowalczyk, Michał Kowalczyk, Kowalczyk, Michal +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.1106.1991
openalex publication_date 2011/06/10 · 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 is an even, bistable function, 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 initiate a program to classify the four-end solutions of the Allen-Cahn equation in \R2. We show that there exists a one parameter family of solutions containing 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 is any θ∈ (0, \fracπ2). This justifies the definition of the angle map for a four-end solution u, which is the angle θ=θ(u)∈ (0, \fracπ2) between the asymptote to the nodal line in the first quadrant and the x axis. Then we show that on any connected component in the moduli space of four-end solutions the angle map is surjective onto (0,\fracπ2).