  Solitons on a finite a background, also called breathers, are solutions of
the focusing nonlinear Schr\"odinger 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. {\bf 72}, 809--818
(1987)]. Having more parameters to vary, this significantly wider family has a
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 new 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
new 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.
