Singly Periodic Solutions of the Allen-Cahn Equation and the Toda Lattice

Abstract

The Allen-Cahn equation - Delta u = u - u (3) in DOUBLE-STRUCK CAPITAL R-2 has family of trivial singly periodic solutions that come from the one dimensional periodic solutions of the problem -u '' =u - u (3). In this paper we construct a non-trivial family of singly periodic solutions to the Allen-Cahn equation. Our construction relies on the connection between this equation and the infinite Toda lattice. We show that for each one-soliton solution to the infinite Toda lattice we can find a singly periodic solution to the Allen-Cahn equation, such that its level set is close to the scaled one-soliton. The solutions we construct are analogues of the family of Riemann minimal surfaces in DOUBLE-STRUCK CAPITAL R-3.