  In this paper we study the existence of multiple-layer solutions to the
elliptic Allen-Cahn equation in hyperbolic space: \[ -\Delta_{\mathbb H}
u+F'(u)=0; \] 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.
