The linear and nonlinear instability of the Akhmediev breather

The Akhmediev breather (AB) and its M-breather generalisation, hereafter called AB M , are exact solutions of the focusing NLS equation periodic in space and exponentially localised 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 in nature. It is therefore important to establish the stability properties of these solutions under perturbations. Concerning perturbations of these solutions within the NLS dynamics, there is the following common belief 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), then the AB M solution is neutrally stable. In this paper we argue instead that the AB M solution is always linearly unstable, even in the saturation case M = N, and we prove it in the simplest case M = N = 1, constructing two examples of x-periodic solutions of the linearised theory growing exponentially in time. Then we sketch the proof of completeness of the basis of periodic solutions of the linearised theory. We also investigate the nonlinear instability showing that (i) a perturbed AB initial condition evolves into a recurrence of ABs; (ii) the AB solution is more unstable than the background solution, and its instability increases as T → 0, where T is the AB appearance time. Although the AB solution is linearly and nonlinearly unstable, its instability generates a recurrence of ABs, and this recurrence implies its relevance in the natural phenomena described by the NLS equation, as well as its orbital stability, using a specific definition of orbital stability present in the literature.