We are interested in entire solutions of the Allen–Cahn equation which have some special structure at infinity. In this equation, the function F is an even, double well potential. The solutions we are interested in have their zero set asymptotic to 4 half oriented affine lines at infinity and, along each of these half affine lines, the solutions are asymptotic to the one dimensional heteroclinic solution: such solutions are called 4*-ended solutions*. The main result of our paper states that, for any , there exists a 4-ended solution of the Allen–Cahn equation whose zero set is at infinity asymptotic to the half oriented affine lines making the angles θ, , and with the x-axis. This paper is part of a program whose aim is to classify all 2k-ended solutions of the Allen–Cahn equation in dimension 2, for .
Cite this article
Frank Pacard, Michał Kowalczyk, Yong Liu, The space of 4-ended solutions to the Allen–Cahn equation in the plane. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 5, pp. 761–781DOI 10.1016/J.ANIHPC.2012.04.003