|
|||||
 |
|
|
|
|
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 , where is an odd function and has exactly three zeros at and , for example, , is called a -end solution if its nodal set is asymptotic to half lines, and if along each of these half lines the function 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 , as well as solutions for which the angle between the asymptotic half lines of the nodal set is any . |
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 |
|
|
