title: 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 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.
authors.affiliation: Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA.
authors.forename: Siu A
authors.initials: SA
authors.lastname: Chin
authors.affiliation: Science Program, Texas A&M University at Qatar, P.O. Box 23874 Doha, Qatar.
authors.forename: Omar A
authors.initials: OA
authors.lastname: Ashour
authors.affiliation: Science Program, Texas A&M University at Qatar, P.O. Box 23874 Doha, Qatar.
authors.forename: Milivoj R
authors.initials: MR
authors.lastname: Belić