Doubly periodic solutions of the focusing nonlinear Schrödinger equation: recurrence, period doubling and amplification outside the conventional modulation instability band

Abstract

Solitons on a finite a background, also called breathers, are solutions of the focusing nonlinear Schrödinger equation, which play a pivotal role in the description of rogue waves and modulation instability. The breather family includes Akhmediev breathers (AB), Kuznetsov-Ma (KM), and Peregrine solitons (PS), which have been successfully exploited to describe several physical effects. These families of solutions are actually only particular cases of a more general three-parameter class of solutions originally derived by Akhmediev, Eleonskii and Kulagin [Theor. Math. Phys. {\bf 72}, 809--818 (1987)]. Having more parameters to vary, this significantly wider family has a potential to describe many more physical effects of practical interest than its subsets mentioned above. The complexity of this class of solutions prevented researchers to study them deeply. In this paper, we overcome this difficulty and report several new effects that follow from more detailed analysis. Namely, we present the doubly periodic solutions and their Fourier expansions. In particular, we outline some striking properties of these solutions. Among the new effects, we mention (a) regular and shifted recurrence, (b) period doubling and (c) amplification of small periodic perturbations with frequencies outside the conventional modulation instability gain band.

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Cited by in corpus (9)

• "Extraordinary" modulation instability in optics and hydrodynamics
• Observation of modulation instability and rogue breathers on stationary periodic waves
• Non-degenerate Kuznetsov-Ma solitons of Manakov equations and their physical spectra
• The effect of a small loss or gain in the periodic NLS anomalous wave dynamics. I
• Modulation instability and non-degenerate Akhmediev breathers of Manakov equations
• Fundamental and second-order dark soliton solutions of 2- and 3-component Manakov equations in the defocusing regime
• Higher-order modulation instability and multi-Akhmediev breathers of Manakov equations: Frequency jumps over the stable gaps between the instability bands
• Optical amplification and transmission of attenuated multi-soliton based on spectral characteristics of Akhmediev breather
• Domain formation of modulation instability in spin-orbit-Rabi coupled Gross-Pitaevskii equation with cubic-quintic interactions