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- W2080558006 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." @default.
- W2080558006 created "2016-06-24" @default.
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- W2080558006 date "2012-01-01" @default.
- W2080558006 modified "2023-09-23" @default.
- W2080558006 title "Towards classification of multiple-end solutions to the Allen-Cahn equation in $mathbb{R}^2$" @default.
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- W2080558006 doi "https://doi.org/10.3934/nhm.2012.7.837" @default.
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