  In this paper, the existence and orbital stability of the periodic standing
waves solutions for the nonlinear fractional Schrodinger (fNLS) equation with
cubic nonlinearity is studied. The existence is determined by using a
minimizing constrained problem in the complex setting and we it is showed that
the corresponding real solution is always positive. The orbital stability is
proved by combining some tools regarding positive operators, the oscillation
theorem for fractional Hill operators and a Vakhitov-Kolokolov condition, well
known for Schrodinger equations. We then perform a numerical approach to
generate periodic standing wave solutions of the fNLS equation by using the
Petviashvili's iteration method. We also investigate the Vakhitov-Kolokolov
condition numerically which cannot be obtained analytically for some values of
the order of the fractional derivative.
