# Multiple end solutions to the Allen-Cahn equation in ${ℝ}^{2}$

Michał Kowalczyk; Yong Liu; Frank Pacard

•  Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS) Universidad de Chile Casilla 170 Correo 3 Santiago Chile
•  Centre de Mathématiques Laurent Schwartz and Institut Universitaire de France École Polytechnique 91128 Palaiseau France
• Volume: 7, Issue: 4, page 1-19
• ISSN: 2266-0607

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## Abstract

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An entire solution of the Allen-Cahn equation $\Delta u=f\left(u\right)$, where $f$ is an odd function and has exactly three zeros at $±1$ and $0$, e.g. $f\left(u\right)=u\left({u}^{2}-1\right)$, is called a $2k$ end 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 ${H}^{\text{'}\text{'}}=f\left(H\right)$. 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 this solutions with just four ends. A special example is the saddle solutions $U$ whose nodal lines are precisely the straight lines $y=±x$. We describe completely connected components of the moduli space of four end solutions. Finally we establish a uniqueness result which gives a complete classification of these solutions. It says that all four end solutions are continuous deformations of the saddle solution.

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