zbMATH — the first resource for mathematics

The classification of four-end solutions to the Allen-Cahn equation on the plane. (English) Zbl 1287.35031
Summary: An entire solution of the Allen-Cahn equation \(\Delta u=f(u)\), where \(f\) is an odd function and has exactly three zeros at \(\pm1\) and \(0\), for example, \(f(u)=u(u^2-1)\), 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 \(\infty\). 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 \(\pi/2\), as well as solutions for which the angle between the asymptotic half lines of the nodal set is any \(\theta\in(0,\pi/2)\).

35J61 Semilinear elliptic equations
PDF BibTeX Cite
Full Text: DOI