Vol. 6, No. 7, 2013

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ISSN: 1948-206X (e-only)
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The classification of four-end solutions to the Allen–Cahn equation on the plane

Michał Kowalczyk, Yong Liu and Frank Pacard

Vol. 6 (2013), No. 7, 1675–1718
Abstract

An entire solution of the Allen–Cahn equation Δu = f(u), where f is an odd function and has exactly three zeros at ± 1 and 0, for example, f(u) = u(u2 1), 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 like the one-dimensional, heteroclinic solution. In this paper we consider the family of four-end solutions whose ends are almost parallel at . We show that this family can be parametrized by the family of solutions of the Toda system. As a result we obtain the uniqueness of four-end solutions with almost parallel ends. Combining this result with the classification of connected components in the moduli space of the four-end 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 π2, as well as solutions for which the angle between the asymptotic half lines of the nodal set is any θ (0,π2).

Keywords
Allen–Cahn equation, entire solutions, moduli space, Toda system, four-end solutions
Mathematical Subject Classification 2010
Primary: 35J61
Milestones
Received: 4 July 2012
Accepted: 7 January 2013
Published: 27 December 2013
Authors
Michał Kowalczyk
Departamento de Ingeníeria Matemática and Centro de Modelamiento Matemático
Universidad de Chile
Casilla 170 Correo 3
00001 Santiago
Chile
Yong Liu
Departamento de Ingeníeria Matemática and Centro de Modelamiento Matemático
Universidad de Chile
Casilla 170 Correo 3
00001 Santiago
Chile
Department of Mathematics and Physics
North China Electric Power University
Beijing 102206
China
Frank Pacard
Centre de Mathématiques Laurent Schwartz, UMR-CNRS 7640
École Polytechnique
91128 Palaiseau
France