End-to-end construction for the Allen–Cahn equation in the plane

Abstract

In this paper, we construct a wealth of bounded, entire solutions of the Allen–Cahn equation in the plane. The asymptotic behavior at infinity of these solutions is determined by 2L half affine lines, in the sense that, along each of these half affine lines, the solution is close to a suitable translated and rotated copy of a one dimensional heteroclinic solution. The solutions we construct belong to a smooth 2L-dimensional family of bounded, entire solutions of the Allen–Cahn equation, in agreement with the result of del Pino (Trans Am Math Soc 365(2):721–766, 2010) and, in some sense, they provide a description of a collar neighborhood of part of the compactification of the moduli space of 2L-ended solutions for the Allen–Cahn equation. Our construction is inspired by a construction of minimal surfaces by Traizet.