  Periodic waves in the modified Korteweg-de Vries (mKdV) equation are
revisited in the setting of the fractional Laplacian. Two families of solutions
in the local case are given by the sign-definite dnoidal and sign-indefinite
cnoidal solutions. Both solutions can be characterized in the general
fractional case as global minimizers of the quadratic part of the energy
functional subject to the fixed $L^4$ norm: the sign-definite (sign-indefinite)
solutions are obtained in the subspace of even (odd) functions. Morse index is
computed for both solutions and the spectral stability criterion is derived. We
show numerically that the family of sign-definite solutions has a generic fold
bifurcation for the fractional Laplacian of lower regularity and the family of
sign-indefinite solutions has a generic symmetry-breaking bifurcation both in
the fractional and local cases.
