The linear instability of the Akhmediev breather, the "missed" modes and the regular approach

The Akhmediev breathers are used in the literature as a mathematical model of anomalous (rogue) waves. For physical applications it is important to know if they are stable. Until recently, the common opinion in the literature was that these solutions are neutrally stable due to "saturation of instabilities" mechanism. We show that these solutions are unstable by explicitly constructing the "missed modes", i.e. exponentially growing in time solutions of the linearized equation. We also demonstrate how the unstable modes can be derived using a regular procedure.