Periodic standing waves in the focusing nonlinear Schrödinger equation: Rogue waves and modulation instability

We present exact solutions for rogue waves arising on the background of periodic standing waves in the focusing nonlinear Schrödinger equation. The exact solutions are obtained by characterizing the Lax spectrum related to the periodic standing 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 background of the periodic standing waves.