Anatomy of the Akhmediev breather: cascading instability, first formation time and Fermi-Pasta-Ulam recurrence

Abstract: By invoking Bogoliubov's spectrum, we show that for the nonlinear Schrodinger 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 which 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 modes, 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.