Rogue waves on the periodic background are considered for the nonlinear
Schrodinger (NLS) equation in the focusing case. The two periodic wave
solutions are expressed by the Jacobian elliptic functions dn and cn. Both
periodic waves are modulationally unstable with respect to long-wave
perturbations. Exact solutions for the rogue waves on the periodic background
are constructed by using the explicit expressions for the periodic
eigenfunctions of the Zakharov-Shabat spectral problem and the Darboux
transformations. These exact solutions labeled as rogue periodic waves
generalize the classical rogue wave (the so-called Peregrine's breather). The
magnification factor of the rogue periodic waves is computed as a function of
the wave amplitude (the elliptic modulus). Rogue periodic waves constructed
here are compared with the rogue wave patterns obtained numerically in recent
publications.