The space of 4-ended solutions to the Allen-Cahn equation in the plane

We are interested in entire solutions of the Allen-Cahn equation Δu- F^′(u)=0 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 θ ∈ (0,π/2), 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, π-θ, π+θ and 2π-θ 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 k ≥2.