Towards classification of multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$

Abstract

An entire solution of the Allen-Cahn equation $\Delta u=f(u)$, where $f$ is an odd function and has exactly three zeros at $\pm 1$ and $0$, e.g. $f(u)=u(u^2-1)$, 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 of such 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 lines are precisely the straight lines $y=\pm x$. We describe the connected components of the moduli space of $4$-ended solutions. Finally we establish a uniqueness result which gives a complete classification of these solutions. It says that all $4$-ended solutions are continuous deformations of the saddle solution.