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.

It is well known that Akhmediev breathers of the nonlinear cubic Schrödinger equation can be superposed nonlinearly via the Darboux transformation to yield breathers of higher order. Surprisingly, we find that the peak height of each Akhmediev breather only adds linearly to form the peak height of the final breather. Using this new peak-height formula, we show that at any given periodicity, there exist a unique high-order breather of maximal intensity. Moreover, these high-order breathers form a continuous hierarchy, growing in intensity with increasing periodicity. For any such higherorder breather, a simple initial wave function can be extracted from the Darboux transformation to dynamically generate that breather from the nonlinear Schrödinger equation.

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