The Akhmediev Breather is unstable

TL;DR

This paper rigorously proves that the periodic Akhmediev breather solution to the nonlinear Schrödinger equation is unstable, demonstrating that it cannot maintain stability in finite energy periodic Sobolev spaces.

Contribution

It provides a rigorous proof of the instability of the NLS periodic Akhmediev breather, extending previous ideas to a new class of solutions.

Findings

Akhmediev breather is unstable in periodic Sobolev spaces

The proof uses modifications of the Stokes wave as a global attractor

The instability holds for sufficiently large time

Abstract

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.