Multiple-layer solutions to the Allen-Cahn equation on hyperbolic space

In this paper we study the existence of multiple-layer solutions to the elliptic Allen-Cahn equation in hyperbolic space: $-Δ_{\mathbb H} u+F'(u)=0;$ here $F$ is a nonnegative double-well potential with nondegenerate minima. We prove that for any collection of widely separated, non-intersecting hyperplanes in ${\mathbb H}$, there is a solution to this equation which has nodal set very close to this collection of hyperplanes. Unlike the corresponding problem in $\RR^n$, there are no constraints beyond the separation parameter.

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