Multiple-end solutions to the Allen–Cahn equation in R2

We construct a new class of entire solutions for the Allen–Cahn equation u + (1 − u2)u = 0, in R2(∼ C). Given k 1, we find a family of solutions whose zero level sets are, away from a compact set, asymptotic to 2k straight lines (which we call the ends). These solutions have the property that there exist θ0 < θ1 < ··· < θ2k = θ0 +2π such that limr→+∞u(reiθ ) = (−1)j uniformly in θ on compact subsets of (θj , θj+1), for j = 0, . . . , 2k − 1. © 2009 Published by Elsevier Inc