Observation of modulation instability and rogue breathers on stationary periodic waves

We report an experimental study on the modulation instability process and associated rogue breathers for the case of stationary periodic background waves, namely dnoidal and cnoidal envelopes. Despite being well-known solutions of the nonlinear Schrödinger equation (NLSE), the stability of such background waves has remained unexplored experimentally until now, unlike the constant-amplitude plane wave. By means of two experimental setups, namely, in nonlinear optics and hydrodynamics, we observe the spontaneous modulation instability gain seeded by input random noise and the formation of rogue breathers induced by a coherent perturbation. Our observations are in excellent agreement with the NLSE dynamics.