The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes

Abstract

The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, the main physical mechanism for the generation of rogue (anomalous) waves (RWs) in Nature. In this paper we investigate the $x$-periodic Cauchy problem for NLS for a generic periodic initial perturbation of the unstable constant background solution, in the case of $N=1,2$ unstable modes. We use matched asymptotic expansion techniques to show that the solution of this problem describes an exact deterministic alternate recurrence of linear and nonlinear stages of MI, and that the nonlinear RW stages are described by the N-breather solution of Akhmediev type, whose parameters, different at each RW appearence, are always given in terms of the initial data through elementary functions. This paper is motivated by a preceeding work of the authors in which a different approach, the finite gap method, was used to investigate periodic Cauchy problems giving rise to RW recurrence.

arXiv comment20 pages. arXiv admin note: text overlap with arXiv:1708.00762 and substantial text overlap with arXiv:1707.05659

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• Integrable turbulence and formation of rogue waves
• Spatiotemporal pulses in a liquid crystal optical oscillator
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Cited by in corpus (8)

• Doubly periodic solutions of the focusing nonlinear Schrödinger equation: recurrence, period doubling and amplification outside the conventional modulation instability band
• The effect of a small loss or gain in the periodic NLS anomalous wave dynamics. I
• The finite gap method and the periodic NLS Cauchy problem of the anomalous waves, for a finite number of unstable modes
• Integrability and linear stability of nonlinear waves
• Chess-Board-Like Spatio-Temporal Interference Patterns and Their Excitation
• Observation of the noise-driven thermalization of the Fermi-Pasta-Ulam-Tsingou recurrence in optical fibers
• Light polarization oscillations induced by photon-photon scattering
• The linear and nonlinear instability of the Akhmediev breather