Doubly periodic solutions of the focusing nonlinear Schrödinger equation: Recurrence, period doubling, and amplification outside the conventional modulation-instability band

Solitons on a finite background, also called breathers, are solutions of the focusing nonlinear Schrödinger equation, which play a pivotal role in the description of rogue waves and modulation instability. The breather family includes Akhmediev breathers (AB), Kuznetsov-Ma (KM), and Peregrine solitons (PS), which have been successfully exploited to describe several physical effects. These families of solutions are actually only particular cases of a more general three-parameter class of solutions originally derived by Akhmediev, Eleonskii, and Kulagin [Theor. Math. Phys. 72, 809 (1987)TMPHAH0040-577910.1007/BF01017105]. Having more parameters to vary, this significantly wider family has the potential to describe many more physical effects of practical interest than its subsets mentioned above. The complexity of this class of solutions prevented researchers to study them deeply. In this paper, we overcome this difficulty and report several effects that follow from more detailed analysis. Namely, we present the doubly periodic solutions and their Fourier expansions. In particular, we outline some striking properties of these solutions. Among the effects, we mention (a) regular and shifted recurrence, (b) period doubling, and (c) amplification of small periodic perturbations with frequencies outside the conventional modulation-instability gain band.