Towards classification of multiple-end solutionsto the Allen-Cahn equation in ℝ²

An entire solution of the Allen-Cahn equation Δu = f(u), where f is an odd function and has exactly three zeros at ±1 and 0, e.g. f(u)=u(u²-1), is called a 2k-ended solution if its nodal set is asymptotic to 2k half lines, and if along each of these half lines the function u looks (up to a multiplication by -1) like the one dimensional, odd, heteroclinic solution H, of H″= f(H). In this paper we present some recent advances in the theory of the multiple-end solutions. We begin with the description of the moduli space ofsuch solutions. Next we move on to study a special class of these solutions with just four ends. A special example is the saddle solutions U whose nodal linesare precisely the straight lines y = ±x. We describe the connected components of the moduli space of 4-ended solutions. Finally we establish a uniqueness result which gives a complete classication of these solutions. It says that all 4-ended solutions are continuous deformations of the saddle solution.