Oscillation structure of localized perturbations in modulationally unstable media

Biondini, Gino; Li, Sitai; Mantzavinos, Dionyssios

doi:10.1103/physreve.94.060201

Cited by 49 publications

(66 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In this paper, we make a systematic attempt to fill this gap by developing the IST for studying pulses that asymptotically approach plane waves with equal amplitudes and frequencies in both backward and forward times, i.e., for MBEs with nonzero background (NZBG).The direct and inverse scattering components of our study closely parallel those in our work on the focusing nonlinear Schrödinger (NLS) equation with nonzero boundary conditions (NZBC) [69]. More generally, the study of nonlinear systems with NZBG has also received considerable interest thanks in part to its connections to the theory of modulational instability, rogue waves and integrable turbulence [70][71][72][73][74][75][76][77][78][79]. The continuous spectra for the scattering problems of both the NLS and MBEs comprise the real axis plus a symmetric interval along the imaginary axis with length twice that the amplitude of the CW background.…”

supporting

confidence: 65%

Inverse scattering transform for two-level systems with nonzero background

Biondini

Gabitov

Kovačič

et al. 2019

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

We formulate the inverse scattering transform for the scalar Maxwell-Bloch system of equations describing the resonant interaction of light and active optical media in the case when the light intensity does not vanish at infinity. We show that pure background states in general do not exist with a nonzero background field. We then use the formalism to compute explicitly the soliton solutions of this system. We discuss the initial population of atoms and show that the pure soliton solutions do not correspond to a pure state initially. We obtain a representation for the soliton solutions in determinant form, and explicitly write down the one-soliton solutions. We next derive periodic solutions and rational solutions from the one-soliton solutions. We then analyze the properties of these solutions, including discussion of the sharp-line and small-amplitude limits, and thereafter show that the two limits do not commute. Finally, we investigate the behavior of general solutions, showing that solutions are stable (i.e., the radiative parts of solutions decay) only when initially atoms in the ground state dominant, i.e., initial population inversion is negative. arXiv:1907.06231v1 [nlin.SI] 14 Jul 2019 their parameters and inputs, the two-level MBEs exhibit a rich variety of dynamical regimes. While some regimes of laser operation are chaotic [29][30][31][32][33], and can be described by the Lorenz equations [34,35], or are even turbulent [36][37][38][39], and have a finite-dimensional attractor [40][41][42][43] or a slow manifold [44], other regimes of light propagating through rarefied gases may be approximated by completely-integrable, soliton-type equations [4,9,10,17,[45][46][47][48][49][50][51][52].The integrable MBEs are derived under the above-mentioned approximations, and have addressed phenomena including self-induced transparency [9, 10, 45], area theorem [46], photon echo [48], amplification [47,49,50], and superfluorescence [17,51,52]. Typically, these phenomena are addressed from the viewpoint of the initial-boundary-value or signaling problem for a finitely or infinitely long narrow tube containing the active medium, into which a narrow optical pulse is injected at one end, and whose initial state is given far in the past. The MBEs can be represented in terms of a Lax pair [45,52], thus, the initial-value problem can be solved using the inverse-scattering transform (IST) [45,[52][53][54]. The temporal piece of the Lax pair is in the AKNS form [55,56], so just as for the nonlinear Schrödinger equation appearing in [57,58], which stretches from the infinite past to the infinite future, and pulses are evolved along the spatial variable, describing its propagation from the location, where the pulse is launched into the medium. The evolution operator in the Lax pair of the MBEs has an unusual form containing a Hilbert transform, which complicates the derivation and form of the equations describing the evolution of the spectral data, as well as reflects the fact that the phenomena described by the MBEs are irrev...

show abstract

supporting

confidence: 65%

Inverse scattering transform for two-level systems with nonzero background

Biondini

Gabitov

Kovačič

et al. 2019

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Concerning the NLS Cauchy problems in which the initial condition consists of a perturbation of the exact background (2), what we call the Cauchy problem of the AWs, if such a perturbation is localized, then slowly modulated periodic oscillations described by the elliptic solution of (1) play a relevant role in the longtime regime [16,17]. The relevance of the Kuznetsov -Kawata -Inoue -Ma solitons and of the superregular solitons (constructed by Zakharov and Gelash [93], see also [94], [95]) in this problem was investigated in [37].…”

Section: Introductionmentioning

confidence: 99%

Effect of a small loss or gain in the periodic nonlinear Schrödinger anomalous wave dynamics

2020

View full text Add to dashboard Cite

The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. Using the finite gap method, two of us (PGG and PMS) have recently solved, to leading order and in terms of elementary functions of the initial data, the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of NLS (what we call the Cauchy problem of the AWs), in the case of a finite number of unstable modes. In this paper, concentrating on the simplest case of a single unstable mode, we study the periodic Cauchy problem of the AWs for the NLS equation perturbed by a linear loss or gain term. Using the finite gap method and the theory of perturbations of soliton PDEs, we construct the proper analytic model describing quantitatively how the solution evolves, after a suitable transient, into slowly varying lower dimensional patterns (attractors) in the (x, t) plane, characterized by ∆X = L/2 in the case of loss, and by ∆X = 0 in the case of gain, where ∆X is the x-shift of the position of the AW during the recurrence, and L is the period. This process is described, to leading order, in terms of elementary functions of the initial data. Since dissipation can hardly be avoided in all natural phenomena involving AWs, and since a small dissipation induces O(1) effects on the periodic AW dynamics, generating the slowly varying loss/gain attractors analytically described in this paper, we expect that these attractors, together with their generalizations corresponding to more unstable modes, will play a basic role in the theory of periodic AWs in nature.

show abstract

“…On the other hand, localized perturbations of a constant plane wave background produce a very different evolution pattern that depends on the soliton content of the localized perturbation. For localized "solitonless" initial perturbations, the nonlinear dynamics of MI is characterized by the development of a universal nonlinear oscillatory structure that expands in time with finite speed [29][30][31][32]. Localized perturbations having pure solitonic content may induce the generation of pairs of breathers with opposite velocities, termed super-regular breathers in ref.…”

Section: Introductionmentioning

confidence: 99%

From modulational instability to focusing dam breaks in water waves

et al. 2020

View full text Add to dashboard Cite

We report water wave experiments performed in a long tank where we consider the evolution of nonlinear deep-water surface gravity waves with the envelope in the form of a large-scale rectangular barrier. Our experiments reveal that, for a range of initial parameters, the nonlinear wave packet is not disintegrated by the Benjamin-Feir instability but exhibits a specific, strongly nonlinear modulation, which propagates from the edges of the wavepacket towards the center with finite speed. Using numerical tools of nonlinear spectral analysis of experimental data we identify the observed envelope wave structures with focusing dispersive dam break flows, a peculiar type of dispersive shock waves recently described in the framework of the semi-classical limit of the 1D focusing nonlinear Schrödinger equation (1D-NLSE). Our experimental results are shown to be in a good quantitative agreement with the predictions of the semi-classical 1D-NLSE theory. This is the first observation of the persisting dispersive shock wave dynamics in a modulationally unstable water wave system.

show abstract

Oscillation structure of localized perturbations in modulationally unstable media

Cited by 49 publications

References 40 publications

Inverse scattering transform for two-level systems with nonzero background

Inverse scattering transform for two-level systems with nonzero background

Effect of a small loss or gain in the periodic nonlinear Schrödinger anomalous wave dynamics

From modulational instability to focusing dam breaks in water waves

Contact Info

Product

Resources

About