  We demonstrate for the first time the possibility for explicit construction
in a discrete Hamiltonian model of an exact solution of the form $\exp(-|n|)$,
i.e., a discrete peakon. These discrete analogs of the well-known, continuum
peakons of the Camassa-Holm equation [Phys. Rev. Lett. {\bf 71}, 1661 (1993)]
are found in a model different from their continuum siblings. Namely, we
observe discrete peakons in Klein-Gordon-type and nonlinear Schr\"odinger-type
chains with long-range interactions. The interesting linear stability
differences between these two chains are examined numerically and illustrated
analytically. Additionally, inter-site centered peakons are also obtained in
explicit form and their stability is studied. We also prove the global
well-posedness for the discrete Klein-Gordon equation, show the instability of
the peakon solution, and the possibility of a formation of a breathing peakon.
