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

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
Siu A ChinMilivoj Belić

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.

References

Jan 13, 1986·Physical Review Letters·K TaiA Tomita
Dec 14, 2007·Nature·D R SolliB Jalali
Jul 1, 1991·Optics Letters·S Trillo, S Wabnitz
Jan 17, 2012·Physical Review Letters·Miro ErkintaloGoëry Genty

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