Anatomy of the Akhmediev breather: Cascading instability, first formation time, and Fermi-Pasta-Ulam recurrence
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By invoking Bogoliubov's spectrum, we show that for the nonlinear Schrödinger equation, the modulation instability (MI) of its n=1 Fourier mode on a finite background automatically triggers a further cascading instability, forcing all the higher modes to grow exponentially in locked step with the n=1 mode. This fundamental insight, the enslavement of all higher modes to the n=1 mode, explains the formation of a triangular-shaped spectrum that generates the Akhmediev breather, predicts its formation time analytically from the initial modulation amplitude, and shows that the Fermi-Pasta-Ulam (FPU) recurrence is just a matter of energy conservation with a period twice the breather's formation time. For higher-order MI with more than one initial unstable mode, while most evolutions are expected to be chaotic, we show that it is possible to have isolated cases of "super-recurrence," where the FPU period is much longer than that of a single unstable mode.
author list (cited authors)
Chin, S. A., Ashour, O. A., & Belić, M. R.