We present both theoretical description and experimental observation of the modulation instability process and related rogue breathers in the case of stationary periodic background waves, namely cnoidal and dnoidal envelopes. Despite being well-known
solutions of the nonlinear Schrodinger equation, the stability of such background waves has remained unexplored experimentally until now, unlike the fundamental plane wave. By means of two experimental setups, namely, in nonlinear optics and hydrodynamics, we report on quantitative measurements of spontaneous modulation instability gain seeded by input random noise, as well as the formation of rogue breather solutions induced by a coherent perturbation. Our results confirm the generalization of modulation instability when more complex background waves are involved.
The double-periodic solutions of the focusing nonlinear Schrodinger equation have been previously obtained by the method of separation of variables. We construct these solutions by using an algebraic method with two eigenvalues. Furthermore, we chara
cterize the Lax spectrum for the double-periodic solutions and analyze rogue waves arising on their background. Magnification of the rogue waves is studied numerically.
