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

We construct a new class of entire solutions for the Allen-Cahn equation Δ 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 < ⋯ < θ_{2 k} = θ₀ + 2 π such that lim_{r → + ∞} u (r e^{i θ}) = (- 1)^j uniformly in θ on compact subsets of (θ_j, θ_{j + 1}), for j = 0, ..., 2 k - 1.