  The Akhmediev breather (AB) and its M-soliton generalization $AB_M$ are exact
solutions of the focusing NLS equation periodic in space and exponentially
localized in time over the constant unstable background; they describe the
appearance of $M$ unstable nonlinear modes and their interaction, and they are
expected to play a relevant role in the theory of periodic anomalous (rogue)
waves (AWs) in nature. It is rather well established that they are unstable
with respect to small perturbations of the NLS equation. Concerning
perturbations of these solutions within the NLS dynamics, there is the
following common believe in the literature. Let the NLS background be unstable
with respect to the first $N$ modes; then i) if the $M$ unstable modes of the
$AB_M$ solution are strictly contained in this set ($M<N$), then the $AB_M$ is
unstable; ii) if $M=N$, the so-called "saturation of the instability", then the
$AB_M$ solution is neutrally stable. We argue instead that the $AB_M$ solution
is always unstable, even in the saturation case $M=N$, and we prove it in the
simplest case $M=N=1$. We first prove the linear instability, constructing two
examples of $x$-periodic solutions of the linearized theory growing
exponentially in time. Then we investigate the nonlinear instability using our
previous results showing that i) a perturbed AB initial condition evolves into
an exact Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence of ABs described in terms
of elementary functions of the initial data, to leading order; ii) the AB
solution is more unstable than the background solution, and its instability
increases as $T\to 0$, where $T$ is the AB appearance parameter. Although the
AB solution is linearly and nonlinearly unstable, it is relevant in nature,
since its instability generates a FPUT recurrence of ABs. These results
suitably generalize to the case $M=N>1$.
