arXiv 1109.56042011-09-29math.AP

平面上Allen-Cahn方程四端解的分类

The classification of four end solutions of the Allen-Cahn equation on the plane

Frank Pacard, Michal Kowalczyk, Yong Liu

摘要：

Allen-Cahn方程$Δu=f(u)$的完整解，其中$f$在$\pm 1$和0处正好有三个零，是平衡的和奇数的，例如$f(u)=u(u^2-1)$，如果它的节点集渐近于$2k$射线，并且如果沿着这些射线中的每一条，函数$u$看起来像一维异宿解，则称为$2k$结束解。本文考虑在$\infty$上端点几乎平行的四端点解族。我们表明，这个家庭可以参数化的两个组件户田系统的解决方案的家庭。结果得到了具有几乎平行端点的四端点解的唯一性。将这一结果与四端解的模空间中连通分量的分类相结合，我们可以对所有这类解进行分类。因此，我们表明，四个端部解决方案的形式，刚性运动，一个参数的家庭。这个家庭包含鞍解，其中的角度之间的节线是$\fracπ{2}$以及解决方案的渐近射线之间的角度的节点集是任意小（几乎平行的节点集）。

英文摘要：

An entire solution of the Allen-Cahn equation $Δu=f(u)$, where $f$ has exactly three zeros at $\pm 1$ and 0, is balanced and odd, 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 like the one dimensional, heteroclinic solution. In this paper we consider the family of four ended solutions whose ends are almost parallel at $\infty$. We show that this family can be parametrized by the family of solutions of the two component Toda system. As a result we obtain the uniqueness of four ended solutions with almost parallel ends. Combining this result with the classification of connected components in the moduli space of the four ended solutions we can classify all such solutions. Thus we show that four end solutions form, up to rigid motions, a one parameter family. This family contains the saddle solution, for which the angle between the nodal lines is $\fracπ{2}$ as well as solutions for which the angle between the asymptotic half lines of the nodal set is arbitrary small (almost parallel nodal sets).

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