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

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