The Akhmediev Breather is unstable

In this note, we give a rigorous proof that the NLS periodic Akhmediev breather is unstable. The proof follows the ideas in [17], in the sense that a suitable modification of the Stokes wave is the global attractor of the local Akhmediev dynamics for sufficiently large time, and therefore the latter cannot be stable in any suitable finite energy periodic Sobolev space.