Superregular solitonic solutions: a novel scenario for the nonlinear stage of modulation instability

Abstract

We describe a general N -solitonic solution of the focusing nonlinear Schrödinger equation in the presence of a condensate by using the dressing method. We give the explicit form of one-and two-solitonic solutions and study them in detail as well as solitonic atoms and degenerate solutions. We distinguish a special class of solutions that we call regular solitonic solutions. Regular solitonic solutions do not disturb phases of the condensate at infinity by coordinate. All of them can be treated as localized perturbations of the condensate. We find a broad class of superregular solitonic solutions which are small perturbations at a certain moment of time. Superregular solitonic solutions are generated by pairs of poles located on opposite sides of the cut. They describe the nonlinear stage of the modulation instability of the condensate and play an important role in the theory of freak waves.

Superregular Breathers in Optics and Hydrodynamics: Omnipresent Modulation Instability beyond Simple Periodicity

“…[33,34] obtained within the framework of the self-focusing NLSE in dimensionless form, iψ ξ þ 1=2ψ ττ þ │ψ│ 2 ψ ¼ 0, where subscripted variables stand for partial differentiations. Here, ψ is a wave group or wave envelope which is a function of ξ (a scaled propagation distance or longitudinal variable) and τ (a comoving time, or transverse variable, moving with the wave-group velocity).…”

“…In general, the one-breather solution is a localized object with characteristic size δτ ∼ 1=2η ¼ ½ðR − 1=RÞ cos α −1 , moving on top of the continuous wave with breather group velocity V g ¼ γ=η ¼ 2sinαðR 4 þ 1Þ=½RðR 2 − 1Þ , and oscillating with period T ¼2π=2ω¼2πR 2 = ½ðR 4 −1Þcos2α [34]. Thus, R, α control the main breather properties.…”

“…The examples of complicated two-and three-pairs scenarios as well as a discussion of general N-pairs solutions can be found in Refs. [33,34]. Note that now the limit N → ∞ is under theoretical study.…”

“…Namely, Zakharov and Gelash proposed superregular breather solutions to describe a novel and general scenario of modulation instability that develops from localized perturbations of the plane wave [33,34]. The latter have more physical and practical significance than periodic perturbations in Benjamin-Feir instability that require the whole infinite space.…”

“…In addition to the development of perturbation into special pairs of breathers, the reverse process is another important scenario. Indeed, N pairs of quasi-Akhmediev breathers almost annihilate (we call this process "quasiannihilation") to a small localized perturbation as a result of their collision [33,34]. Such mathematical entities (i.e., N-pair breather solutions with their shifting parameters) are able to describe different configurations of the nonlinear stage of MI at any propagation distance, such as the amplification, annihilation, or even ghost interaction of nearly any individual or ensemble of localized perturbations (i.e., local concentration of energy) on a finite background (as revealed later by the special cases of the one-pair breather solution shown in Fig.…”