A Comech, J Cuevas and PG Kevrekidis (pdf copy 500 Kb)

Abstract:

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. 71, 1661 (1993)] are found in a model different from their continuum siblings. Namely, we observe discrete peakons in Klein-Gordon-type and nonlinear Schrödinger-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.