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

Michał Kowalczyk; Yong Liu; Frank Pacard; Michał Kowalczyk; Yong Liu; Frank Pacard

doi:10.3934/nhm.2012.7.837

Networks and Heterogeneous Media

2012, Volume 7, Issue 4: 837-855. doi: 10.3934/nhm.2012.7.837

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

1.
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago
2.
Departamento de Ingeniería Matemática and CMM, (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago
3.
Centre de Mathématiques Laurent Schwartz, École Polytechnique, UMR-CNRS 7640, 91128 Palaiseau

Received: 01 May 2012 Revised: 01 October 2012
Primary: 35B08, 35Q80.

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.
- Allen-Cahn equation,
- multiple-end solutions,
- moduli spaces,
- classification of solutions
Citation: Michał Kowalczyk, Yong Liu, Frank Pacard. Towards classification of multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$[J]. Networks and Heterogeneous Media, 2012, 7(4): 837-855. doi: 10.3934/nhm.2012.7.837

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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.

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