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Towards classification of multiple-end solutions to the Allen-Cahn equation in $\Bbb R^2$.

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.
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https://hal.archives-ouvertes.fr/hal-00851585
Contributor : Carole Juppin <>
Submitted on : Thursday, August 15, 2013 - 12:12:16 AM
Last modification on : Thursday, March 5, 2020 - 6:26:00 PM

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  • HAL Id : hal-00851585, version 1

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Frank Pacard, Michal Kowalczyk, Liu Yong. Towards classification of multiple-end solutions to the Allen-Cahn equation in $\Bbb R^2$.. Netw. Heterog. Media, 2012, 7 (4), pp.837-855. ⟨hal-00851585⟩

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