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

We construct a new class of entire solutions for the Allen–Cahn equation Δu+(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)=j(−1) uniformly in θ on compact subsets of (θj,θj+1), for j=0,…,2k−1.