The classification of four-end solutions to the Allen–Cahn equation on the plane

Abstract

Abstract. An entire solution of the Allen-Cahn equation ∆u = f (u), where f has exactly three zeros at ±1 and 0, is balanced and odd, 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 like the one dimensional, heteroclinic solution. In this paper we consider the family of four ended solutions whose ends are almost parallel at ∞. We show that this family can be parametrized by the family of solutions of the two component Toda system. As a result we obtain the uniqueness of four ended solutions with almost parallel ends. Combining this result with the classification of connected components in the moduli space of the four ended 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 π 2 as well as solutions for which the angle between the asymptotic half lines of the nodal set is arbitrary small (almost parallel nodal sets).

End-to-end construction for the Allen–Cahn equation in the plane

“…In [9] and [10], it is proven that M even 4 is diffeomorphic to R and also that the angle map is surjective. In particular, for each α ∈ (− Let us now turn to one of the most important step in this paper, namely the construction of the approximate solution.…”

“…The interested reader will find a similar decomposition for (x ∂ y − y ∂ x ) v. To proceed with the analysis of the kernel of L v , recall that it follows from the results in [3], [9] and [10], that M Observe that φ s is symmetric with respect to both the x−axis and the y−axis since elements of M even 4 also are. Hence, to understand the decomposition of φ s , it is enough to restrict our attention to Q where we can write…”

“…When L = 2, thanks to [6], [9] and [10], we have a rather complete description of the set of 4-ended solutions M 4 . This description is of particular importance in the present paper and we will return to it in Section 3.…”

Saddle solutions for bistable symmetric semilinear elliptic equations

“…This kind of solutions of (1.1) has been studied assuming (W ) and (W 2) in [20,31,34,35] (see also [19]). In particular, in [31] Gui proved that any four ends solution is, modulo roto-translations, even in its variables and monotonic in x and y on the first quadrant.…”

“…In particular, in [31] Gui proved that any four ends solution is, modulo roto-translations, even in its variables and monotonic in x and y on the first quadrant. In [35] it is shown that the set of even four end solutions constitutes a continuum containing the saddle solution.…”