ИСТИНА

The Akhmediev breather (AB) and its M-soliton generalization, hereafter called $AB_M$, 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. It is therefore important to establish the stability properties of these solutions under perturbations, to understand if they appear in nature, and in which form. 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 they coincide with this set ($M=N$), the so-called ``saturation of the instability'', then the $AB_M$ solution is neutrally stable. In this paper 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 prove the linear instability, constructing two examples of $x$-periodic solutions of the linearized theory growing exponentially in time. Moreover, these solutions grow in both time directions, and using this fact one can explain why small perturbations near the peak of the breather affects the AB recurrence stronger than the same perturbations of the background (this effect was observed in numerical simulations). In the know literature these solutions were missed. We also explain how to derive these solutions using the technique developed by Krichever for the KP equation.