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

TL;DR

This paper explores a broader class of solutions to the focusing nonlinear Schrödinger equation, revealing new phenomena like recurrence, period doubling, and amplification outside the typical modulation instability band, with implications for rogue wave modeling.

Contribution

It introduces and analyzes doubly periodic solutions within a general three-parameter family, uncovering novel effects beyond traditional breather solutions.

Findings

Discovery of doubly periodic solutions and their Fourier expansions

Identification of regular and shifted recurrence phenomena

Observation of period doubling and amplification outside the conventional MI band

Abstract

Solitons on a finite a background, also called breathers, are solutions of the focusing nonlinear Schr\"odinger 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.