  Periodic waves in the fractional Korteweg-de Vries equation have been
previously characterized as constrained minimizers of energy subject to fixed
momentum and mass. Here we characterize these periodic waves as constrained
minimizers of the quadratic form of energy subject to fixed cubic part of
energy and the zero mean. This new variational characterization allows us to
unfold the existence region of travelling periodic waves and to give a sharp
criterion for spectral stability of periodic waves with respect to
perturbations of the same period. The sharp stability criterion is given by the
monotonicity of the map from the wave speed to the wave momentum similarly to
the stability criterion for solitary waves.
