The linear and nonlinear instability of the Akhmediev breather

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