  The present paper is the first part of a project devoted to the fractional
nonlinear Schr\"{o}dinger (fNLS) equation. It is concerned with the existence
and numerical generation of the solitary-wave solutions. For the first point,
some conserved quantities of the problem are used to search for solitary-wave
solutions as relative equilibria. From the relative equilibrium condition, a
result of existence via the Concentration-Compactness theory is derived.
Several properties of the waves, such as the regularity and the asymptotic
decay in some cases, are derived from the existence result. Some other
properties, such as the monotone behaviour and the speed-amplitude relation,
will be explored computationally. To this end, a numerical procedure for the
generation of the profiles is proposed. The method is based on a Fourier
pseudospectral approximation of the differential system for the profiles and
the use of Petviashvili's iteration with extrapolation.
