Nonlinear Stage Of Modulational Instability Research Articles

Parametric instability in the pure-quartic nonlinear Schrödinger equation

We study the nonlinear stage of modulation instability (MI) in the non-intergrable pure-quartic nonlinear Schrödinger equation where the fourth-order dispersion is modulated periodically. Using the three-mode truncation, we reveal the complex recurrence of parametric resonance (PR) breathers, where each recurrence is associated with two oscillation periods (PR period and internal oscillation period). The nonlinear stage of parametric instability admits the maximum energy exchange between the spectrum sidebands and central mode occurring outside the MI gain band.

Drift waves enstrophy, zonal flow, and nonlinear evolution of the modulational instability

The interaction of the drift wave (DW) turbulence and zonal flow (ZF) is investigated with the modified Hasegawa–Mima equation taking into account the backreaction of ZF velocity on DW turbulence. It is shown that the y-averaged enstrophy of DW turbulence and the velocity of ZF are intrinsically related. By utilizing this feature, a nonlinear stage of DW modulational instability is considered within the framework of the wave kinetic equation. It is shown that in this approximation, the nonlinear stage of the modulational instability results in the collapsing solutions, accompanied by the “wave breaking” phenomenon. Numerical simulations based on the Hasegawa–Mima equation show that for a weak DW turbulence, Φ̃=(eφ̃/Te) (Ln/ρs)⪝1, the collapsing-like features on both ZF and y-averaged enstrophy of DW turbulence decay in time and then re-emerge again at different locations. For the case of a strong DW turbulence, Φ̃>1, where nonlinear interactions of DW harmonics dominate, stable spatial structures of ZF and y-averaged enstrophy of DW turbulence emerge.

Exact superregular breather solutions to the generalized nonlinear Schrödinger equation with nonhomogeneous coefficients and dissipative effects.

Superregular breathers are peculiar solutions to the integrable nonlinear Schrödinger equation that constitute the building blocks for analysis of the nonlinear stage of modulation instability developing from a localized perturbation on the nonvanishing condensate background. Here superregular breather solutions are extended to the generalized nonlinear Schrödinger equation with nonhomogeneous coefficients and in the presence of dissipation. Concrete examples are shown that may allow observation of new solutions in fiber optics where dissipation is unavoidable, nonhomogeneous spatial distribution of the amplification profile can be controlled, and current technology allows design of the longitudinal dispersion profile.

Statistical Properties of the Nonlinear Stage of Modulation Instability in Fiber Optics.

We present an optical fiber experiment in which we examine the space-time evolution of a modulationally unstable plane wave initially perturbed by a small noise. Using a recirculating fiber loop as an experimental platform, we report the single-shot observation of the noise-driven development of breather structures from the early stage to the long-term evolution of modulation instability. Performing single-point statistical analysis of optical power recorded in the experiments, we observe decaying oscillations of the second-order moment together with the exponential distribution in the long-term evolution, as predicted by Agafontsev and Zakharov [Nonlinearity 28, 2791 (2015).NONLE50951-771510.1088/0951-7715/28/8/2791]. Finally, we demonstrate experimentally and numerically that the autocorrelation of the optical power g^{(2)}(τ) exhibits some unique oscillatory features typifying the nonlinear stage of the noise-driven modulation instability and of integrable turbulence.

Quantitative approach to breather pair appearance in nonlinear modulational instability

We investigate the generation of symmetric and asymmetric pairs of breathers from localized disturbances in the nonlinear stage of modulational instability governed by the nonlinear Schrödinger equation. An asymptotic matching approach allows us to predict with great accuracy how their emergence and evolution depend on the parameters of the perturbation. The same approach turns out to be applicable to other classes of perturbations which, due to their different decays in time, do not give rise to breathers.

Nonlinear Evolution of the Locally Induced Modulational Instability in Fiber Optics.

We report an optical fiber experiment in which we study the nonlinear stage of modulational instability of a plane wave in the presence of a localized perturbation. Using a recirculating fiber loop as the experimental platform, we show that the initial perturbation evolves into an expanding nonlinear oscillatory structure exhibiting some universal characteristics that agree with theoretical predictions based on integrability properties of the focusing nonlinear Schrödinger equation. Our experimental results demonstrate the persistence of the universal evolution scenario, even in the presence of small dissipation and noise in an experimental system that is not rigorously of an integrable nature.

Auto-modulation versus breathers in the nonlinear stage of modulational instability.

The nonlinear stage of modulational instability in optical fibers induced by a wide and easily accessible class of localized perturbations is studied using the nonlinear Schrödinger equation. It is shown that the development of associated spatio-temporal patterns is strongly affected by the shape and the parameters of the perturbation. Different scenarios are presented that involve an auto-modulation developing in a characteristic wedge, possibly coexisting with breathers which lie inside or outside the wedge.

The finite gap method and the analytic description of the exact rogue wave recurrence in the periodic NLS Cauchy problem. 1

The focusing nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media; MI is considered the main physical mechanism for the appearance of rogue (anomalous) waves (RWs) in nature. In this paper we study, using the finite gap method, the NLS Cauchy problem for periodic initial perturbations of the unstable background solution of NLS exciting just one of the unstable modes. We distinguish two cases. In the case in which only the corresponding unstable gap is theoretically open, the solution describes an exact deterministic alternate recurrence of linear and nonlinear stages of MI, and the nonlinear RW stages are described by the 1-breather Akhmediev solution, whose parameters, different at each RW appearence, are always given in terms of the initial data through elementary functions. If the number of unstable modes is >1, this uniform in t dynamics is sensibly affected by perturbations due to numerics and/or real experiments, provoking corrections to the result. In the second case in which more than one unstable gap is open, a detailed investigation of all these gaps is necessary to get a uniform in t dynamics, and this study is postponed to a subsequent paper. It is however possible to obtain the elementary description of the first nonlinear stage of MI, given again by the Akhmediev 1-breather solution, and how perturbations due to numerics and/or real experiments can affect this result. Since the solution of the Cauchy problem is given in terms of different elementary functions in different time intervals, obviously matching in the corresponding overlapping regions, an alternative approach, based on matched asymptotic expansions, is suggested and presented in a separate paper.

Nonlinear Stage of Modulation Instability for a Fifth-Order Nonlinear Schrödinger Equation

Abstract In this article, a fifth-order nonlinear Schrödinger equation, which can be used to characterise the solitons in the optical fibre and inhomogeneous Heisenberg ferromagnetic spin system, has been investigated. Akhmediev breather, Kuzentsov soliton, and generalised soliton have all been attained via the Darbox transformation. Propagation and interaction for three-type breathers have been studied: the types of breather are determined by the module and complex angle of parameter ξ; interaction between Akhmediev breather and generalised soliton displays a phase shift, whereas the others do not. Modulation instability of the generalised solitons have been analysed: a small perturbation can develop into a rogue wave, which is consistent with the results of rogue wave solutions.

Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects.

We study the higher-order generalized nonlinear Schrödinger (NLS) equation describing the propagation of ultrashort optical pulse in optical fibres. By using Darboux transformation, we derive the superregular breather solution that develops from a small localized perturbation. This type of solution can be used to characterize the nonlinear stage of the modulation instability (MI) of the condensate. In particular, we show some novel characteristics of the nonlinear stage of MI arising from higher-order effects: (i) coexistence of a quasi-Akhmediev breather and a multipeak soliton; (ii) two multipeak solitons propagation in opposite directions; (iii) a beating pattern followed by two multipeak solitons in the same direction. It is found that these patterns generated from a small localized perturbation do not have the analogues in the standard NLS equation. Our results enrich Zakharov's theory of superregular breathers and could provide helpful insight on the nonlinear stage of MI in presence of the higher-order effects.

Mode-locking via dissipative Faraday instability.

Emergence of coherent structures and patterns at the nonlinear stage of modulation instability of a uniform state is an inherent feature of many biological, physical and engineering systems. There are several well-studied classical modulation instabilities, such as Benjamin–Feir, Turing and Faraday instability, which play a critical role in the self-organization of energy and matter in non-equilibrium physical, chemical and biological systems. Here we experimentally demonstrate the dissipative Faraday instability induced by spatially periodic zig-zag modulation of a dissipative parameter of the system—spectrally dependent losses—achieving generation of temporal patterns and high-harmonic mode-locking in a fibre laser. We demonstrate features of this instability that distinguish it from both the Benjamin–Feir and the purely dispersive Faraday instability. Our results open the possibilities for new designs of mode-locked lasers and can be extended to other fields of physics and engineering.

Heteroclinic Structure of Parametric Resonance in the Nonlinear Schrödinger Equation.

We show that the nonlinear stage of modulational instability induced by parametric driving in the defocusing nonlinear Schrödinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearized Floquet analysis.

Universal Nature of the Nonlinear Stage of Modulational Instability.

We characterize the nonlinear stage of modulational instability (MI) by studying the longtime asymptotics of the focusing nonlinear Schrödinger (NLS) equation on the infinite line with initial conditions tending to constant values at infinity. Asymptotically in time, the spatial domain divides into three regions: a far left and a far right field, in which the solution is approximately equal to its initial value, and a central region in which the solution has oscillatory behavior described by slow modulations of the periodic traveling wave solutions of the focusing NLS equation. These results demonstrate that the asymptotic stage of MI is universal since the behavior of a large class of perturbations characterized by a continuous spectrum is described by the same asymptotic state.

Optimal frequency conversion in the nonlinear stage of modulation instability.

We investigate multi-wave mixing associated with the strongly pump depleted regime of induced modulation instability (MI) in optical fibers. For a complete transfer of pump power into the sideband modes, we theoretically and experimentally demonstrate that it is necessary to use a much lower seeding modulation frequency than the peak MI gain value. Our experiment shows that, at such optimal modulation frequency, a record 95 % of the output pump power is frequency converted into the comb of sidebands, in good quantitative agreement with analytical predictions based on the simplest exact breather solution of the nonlinear Schrodinger ¨ equation.

Optical turbulence in fiber lasers

We analyze the nonlinear stage of modulation instability in passively mode-locked fiber lasers leading to chaotic or noise-like emission. We present the phase-transition diagram among different regimes of chaotic emission in terms of the key cavity parameters: amplitude or phase turbulence, and spatio-temporal intermittency.

Nonlinear Stage of Modulation Instability

We study the nonlinear stage of the modulation instability of a condensate in the framework of the focusing nonlinear Schrödinger equation (NLSE). We find a general N-solitonic solution of the focusing NLSE in the presence of a condensate by using the dressing method. We separate a special designated class of "regular solitonic solutions" that do not disturb phases of the condensate at infinity by coordinate. All regular solitonic solutions can be treated as localized perturbations of the condensate. We find an important class of "superregular solitonic solutions" which are small perturbations at a certain moment of time. They describe the nonlinear stage of the modulation instability of the condensate.

Dynamics of vortex structure formation during the evolution of modulation instability of dark solitons

The nonlinear stage of modulation instability of dark solitons is studied analytically and numerically. We propose an asymptotic description of the dynamics of these solitons in terms of their local velocity and the curvature of the lines at which solitons are concentrated. The features of the destruction of dark solitons (in particular, the formation of vortex structures from them) are analyzed.

On stochastization of monochromatic wave propagation in an optical single-mode fiber

The inhomogeneities in an optical fiber is one of the most significant problems of wavelength multiplexing in optical communications. We investigate the effect of such inhomogeneities giving rise to stochastization of the transmitted signal in a canonical formulation. We consider the nonlinear stage of modulational instability of a pump wave in a three-wave approximation. By employing equations for wave dynamics in the Hamiltonian form we show analytically that a “stochastic layer”, i.e. the region in which the phase trajectories behave chaotically, is formed in the phase space of a dynamical system even at rather weak inhomogeneity of the fiber. The estimates obtained indicate that the considered phenomenon is very likely to be observed in experiments.

The Enhancement of the Interaction between Transverse and Longitudinal Waves by Turbulent Plasmon Condensation

It is shown that three-wave interaction processes can be enhanced and change sign in the presence of strong Langmuir turbulence, which is treated as a nonlinear stage of modulational instability of plasmon condensation. In particular we point out the possibility of enhanced emissions at the double plasma frequency (ωt ≈ 2ωp). We demonstrate that depression occurs when the intensity of the transverse wave is less than a certain threshold value but that the wave is enhanced when it is larger than this value.