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We investigate the equation $$(-\Delta_{\mathbb H^n})^{\gamma} w=f(w)\quad in\mathbb H^{n},$$ where $(-\Delta_{\mathbb H^n})^\gamma$ corresponds to thefractional Laplacian on hyperbolic space for $\gamma \in (0,1)$ and $f$ is asmooth nonlinearity that typically comes from a double well potential. We provethe existence of heteroclinic connections in the following sense; a so-calledlayer solution is a smooth solution of the previous equation converging to $\pm1$ at any point of the two hemispheres $S_\pm \subset \partial_\infty \mathbbH^n$ and which is strictly increasing with respect to the signed distance to atotally geodesic hyperplane $\Pi.$ We prove that under additional conditions onthe nonlinearity uniqueness holds up to isometry. Then we provide severalsymmetry results and qualitative properties of the layer solutions. Finally, weconsider the multilayer case, at least when $\gamma$ is close to one.

Layer solutions for the fractional Laplacian on hyperbolic space: existence, uniqueness and qualitative properties