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Discrete peakons

A. Comech·J. Cuevas·P. G. Kevrekidis
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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. {\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.

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A. Comech, J. Cuevas, P. G. Kevrekidis. 2005-02-01. Discrete peakons. https://doi.org/10.1016/j.physd.2005.05.019

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