The continuum and discrete fractional nonlinear Schr\"odinger equations
(fDNLS) represent new models in nonlinear wave phenomena with unique
properties. In this paper, we focus on various aspects of localization
associated to fDNLS featuring modulational instability, asymptotic construction
of onsite and offsite solutions, and the role of Peierls-Nabarro barrier. In
particular, the localized onsite and offsite solutions are constructed using
the map approach. Under the long-range interaction characterized by the
L\'{e}vy index α>0, the phase space of solutions is infinite-dimensional
unlike that of the well-studied nearest-neighbor interaction. We show that an
orbit corresponding to this spatial dynamics translates to an approximate
solution that decays algebraically. We also show as α→∞, the discrepancy between local and nonlocal dynamics becomes negligible
on a compact time interval, but persists on a global time scale. Moreover it is
shown that data of small mass scatter to free solutions under a sufficiently
high nonlinearity, which proves the existence of strictly positive excitation
threshold for ground state solutions of fDNLS