  In the present chapter, we explore the possibility of a Frenkel-Kontorova
(discrete sine-Gordon) model to bear interactions that decay algebraically with
space, inspired by the continuum limit of the corresponding fractional
derivative. In such a setting, we revisit the realm of discrete breathers
including onsite, intersite and out-of-phase ones and identify their power-law
spatial decay, as well as explore their corresponding stability analysis, by
means of Floquet multipliers. The relevant stability is also explored
parametrically as a function of the frequency and connected to stability
criteria for breather dependence of energy vs. frequency. Finally, by suitably
perturbing the breathers, we also generate moving waveforms and explore their
radiation and potential robustness.
