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

TL;DR

This paper analyzes the formation and recurrence of Akhmediev breathers in the nonlinear Schrödinger equation, revealing cascading instabilities, predicting formation times, and explaining FPU recurrence through spectral and energy considerations.

Contribution

It introduces a fundamental mechanism of cascading instability in MI, analytically predicts breather formation time, and links FPU recurrence to energy conservation, extending understanding of nonlinear wave dynamics.

Findings

Higher modes are enslaved to the fundamental mode during MI.

Analytical prediction of Akhmediev breather formation time.

FPU recurrence period is twice the breather formation time.

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.