Rogue waves on the periodic wave background in the focusing nonlinear Schrodinger equation

We present exact solutions for rogue waves arising on the background of periodic waves in the focusing nonlinear Schrodinger equation. The exact solutions are obtained by characterizing the Lax spectrum related to the periodic waves and by using the one-fold Darboux transformation. The magnification factor of the rogue waves is computed in the closed analytical form. We relate the rogue wave solutions to the modulation instability of the periodic wave background.