Multiple-end Solutions to the Allen-cahn Equation in R

We construct new class of entire solutions of 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 exists θ0 < θ1 < . . . < θ2k = θ0 + 2π such that limr→+∞ u(re iθ) = (−1)j uniformly in θ on compacts of (θj , θj+1), for j = 0, . . . 2k−1, they complement the solutions with dihedral symmetry which have been obtained in [14].