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Solitons on a finite a background, also called breathers, are solutions ofthe focusing nonlinear Schr\"odinger equation, which play a pivotal role in thedescription of rogue waves and modulation instability. The breather familyincludes Akhmediev breathers (AB), Kuznetsov-Ma (KM), and Peregrine solitons(PS), which have been successfully exploited to describe several physicaleffects. These families of solutions are actually only particular cases of amore general three-parameter class of solutions originally derived byAkhmediev, Eleonskii and Kulagin [Theor. Math. Phys. {\bf 72}, 809--818(1987)]. Having more parameters to vary, this significantly wider family has apotential to describe many more physical effects of practical interest than itssubsets mentioned above. The complexity of this class of solutions preventedresearchers to study them deeply. In this paper, we overcome this difficultyand report several new effects that follow from more detailed analysis. Namely,we present the doubly periodic solutions and their Fourier expansions. Inparticular, we outline some striking properties of these solutions. Among thenew effects, we mention (a) regular and shifted recurrence, (b) period doublingand (c) amplification of small periodic perturbations with frequencies outsidethe conventional modulation instability gain band.

physical review. a/physical review, a

Doubly periodic solutions of the focusing nonlinear Schr\"odinger  equation: recurrence, period doubling and amplification outside the  conventional modulation instability band