Occurrence Of Modulational Instability Research Articles

Modified three-dimensional nonlinear Schrödinger equation in finite water depth for gravity waves with influence of slowly varying currents

A new modified three-dimensional (3D) nonlinear Schrödinger equation (3DMNLSE) is derived for gravity waves in the presence of wind, dissipation, and two-dimensional slowly varying currents, which include transverse and longitudinal currents in finite water depth. The effect of currents on modulational instability (MI) is investigated. The divergence (convergence) effect of longitudinal favorable (adverse) currents decreases (increases) the MI growth rate and region of instability while suppressing (enhancing) the formation of freak waves. Meanwhile, the transverse currents hardly affect the occurrence of MI. Furthermore, some results from the simulations with the space evolution of 3DMNLSE are presented. The results show the ubiquitous occurrence of freak waves in 3D wave fields under certain sets of initial conditions. We demonstrate that larger waves can be triggered when a weakly modulated wave train enters a region of adverse currents. The maximum amplitude of a freak wave depends on the ratio of the current velocity to the wave phase velocity.

Modulation effect of uniform flow on three-dimensional freak wave generation in arbitrary water depth

In arbitrary water depths, the influence of uniform flow, which includes transverse and longitudinal flows, on the generation of three-dimensional (3D) freak waves is examined. A modified Davey–Stewartson equation is derived using potential flow theory and the multiscale method. This equation describes the evolution of 3D freak wave amplitude under the influence of uniform flow. The relationship between two-dimensional (2D) modulational instability (MI) and the generation of 3D freak waves, as represented by the modified 3D Peregrine Breather solution, is explored. The characteristics of 2D MI depend on the orientation of the longitudinal and transverse perturbations. In shallow waters, the generation of freak waves by MI is challenging due to the minimal orientation difference, and longitudinal flows hardly affect the occurrence of MI. Variations in relative water depth can contribute to forming shallow-water freak waves. In finite-depth waters, oblique modulation leads to MI, whereas in deep and infinite-depth waters, longitudinal modulation gains significance. In environments of finite-depth, deep, and infinite-depth waters, the divergence (convergence) effect of longitudinal favorable (adverse) currents reduces (increases) the MI growth rate and suppresses (facilitates) freak wave generation.

Enhanced higher-order modulational instability in a parity–time-symmetric fiber Bragg grating system with modified saturable nonlinearity

With the use of cubic, quintic, and septic nonlinearities, we demonstrate the influence of modified nonlinear saturation on modulational instability (MI) in a nonlinear complex parity–time (PT)-symmetric fiber Bragg grating (FBG) structure. Using a modified coupled nonlinear Schrodinger equation and linear stability analysis, we derive a dispersion relation for instability gain spectra in a complicated PT-symmetric system. Our main aim is to examine the MI in non-Kerr nonlinearities with nonlinear saturation in three PT-symmetric regimes: below threshold point, at threshold point (breaking point), and above threshold point. The occurrence of MI is known to be problematic at the PT-symmetry threshold point in a standard FBG structure (A.K. Sharma, 2014). At the same time, MI can exist in the normal group velocity dispersion domain when the modified nonlinear saturation effect is used. With the help of a modified form of saturable nonlinearity, we discovered that MI could exist in all three regimes in a complex PT-symmetric FBG structure. In anomalous group velocity dispersion alone, we found bistability behavior in a PT-symmetric FBG structure with higher-order saturable nonlinearity. In the presence of a modified nonlinear saturation effect and higher-order non-Kerr nonlinearities, we found a novel type of dynamics in the PT-symmetric FBG structure. All alterations in the photonic device bandgap directly result from changes in the refractive index of the medium caused by the interaction of PT-symmetric potential with the cubic–quintic–septic and modified form of nonlinear saturation. As a result, we provide approaches for generating and managing the MI in a complex PT-symmetric FBG structure under the influence of the modified nonlinear saturation effect.

Higher-order dispersion and nonlinear effects of optical fibers under septic self-steepening and self-frequency shift.

We investigate the modulational instability (MI) of a continuous wave (cw) under the combined effects of higher-order dispersions, self steepening and self-frequency shift, cubic, quintic, and septic nonlinearities. Using Maxwell's theory, an extended nonlinear Schrödinger equation is derived. The linear stability analysis of the cw solution is employed to extract an expression for the MI gain, and we point out its sensitivity to both higher-order dispersions and nonlinear terms. In particular, we insist on the balance between the sixth-order dispersion and nonlinearity, septic self-steepening, and the septic self-frequency shift terms. Additionally, the linear stability analysis of cw is confronted with the stability conditions for solitons. Different combinations of the dispersion parameters are proposed that support the stability of solitons and the occurrence of MI. This is confronted with full numerical simulations where the input cw gives rise to a broad range of behaviors, mainly related to nonlinear patterns formation. Interestingly, under the activation of MI, a suitable balance between the sixth-order dispersion and the septic self-frequency shift term is found to highly influence the propagation direction of the optical wave patterns.

Dust-acoustic envelope solitons in strongly coupled dusty plasmas

The propagation nature and the modulational instability (MI) criteria of low frequency amplitude modulated dust-acoustic envelope solitons are analyzed theoretically and numerically in an unmagnetized dusty plasma consisting of strongly coupled micron-sized inertial dust grains, Boltzmann distributed non-inertial electrons and ions. A hydrodynamical fluid model approach is used, based on [V. V. Yaroshenko, F. Verheest, H. M. Thomas and G.E. Morfill, New J. Phys. 11, 073013 (2009); V. V. Yaroshenko, V. Nosenko, and G. E. Morfill, Phys. Plasmas, 17, 103709 (2010)], where the strong coupling between dust grains for by an effective electrostatic pressure is accounted, which is in fact a dynamical function of the dust-density and the electrostatic potential. It is seen that the strong coupling between dust particles due to the effective electrostatic ``pressure” modifies the wavepacket stability and affects the occurrence of MI. A cubic nonlinear Schrodinger-type equation, which describes the evolution of the wavepacket’s envelope, is derived by adopting a multiscale technique for the amplitude dynamics. It is shown that long-wavelength dust-acoustic wavepackets are stable, while MI sets in beyond a carrier wavenumber threshold, which is evaluated in terms of intrinsic plasma parameters of relevance. On the other hand, the MI window is larger for higher ion density. The effect of the ion number density on MI is twofold, in that a higher ion density leads to a higher threshold (thus, the wave will be modulationally unstable for the larger values of the carrier wavenumber) but it also entails a wider instability window (hence, external perturbations must be of specific wavenumbers to trigger the instability). Numerical simulation confirms that the bright-type envelope soliton solution is stable beyond the carrier wavenumber threshold, i.e., in the modulationally unstable region, while the dark type envelope soliton is stable for wavenumbers below the threshold.

Low relativistic effects on the modulational instability of rogue waves in electronegative plasmas

Relativistic ion-acoustic waves are investigated in an electronegative plasma. The use of the reductive perturbation method summarizes the hydrodynamic model to a nonlinear Schrödinger equation which supports the occurrence of modulational instability (MI). From the MI criterion, we derive a critical value for the relativistic parameter alpha _{1}, below which MI may develop in the system. The MI analysis is then conducted considering the presence and absence of negative ions, coupled to effects of relativistic parameter and the electron-to-negative ion temperature ratio. Under high values of the latter, additional regions of instability are detected, and their spatial expansion is very sensitive to the change in alpha _{1} and may support the appearance of rogue waves whose behaviors are discussed. The parametric analysis of super-rogue wave amplitude is performed, where its enhancement is debated relatively to changes in alpha _{1}, in the presence and absence of negative ions.

Electrostatic Rogue Waves in a Degenerate Thomas-Fermi Plasma

In this paper, we search for the modulational instability (MI) and the possibility of rogue wave generation in a degenerated Thomas-Fermi plasma, consisting of cold inertial ions and Thomas-Fermi distributed electrons and positrons. A multiscale perturbation method is used to derive a nonlinear Schrodinger equation (NLSE) for the envelope amplitude, based on which the occurrence of MI and the generation of rogue as well as super rogue waves is investigated at length. The plasma configurational parameters (namely the positron concentration and the ratio of electron Fermi temperature to positron Fermi temperature) are shown to affect the conditions for MI significantly, and in fact altering the associated threshold. In particular, the positron concentration as well as the temperature ratio leads to an increase in the critical wavenumber, suggesting that MI sets in for larger values of the wavenumber.

Modulation Instability in Two Component Bose-Einstein Condensate with Dissipation

In this paper, we consider the dynamic evolution of a binary mixture of a Bose-Einstein condensate taking into account the presence of dissipation inside the components. Using the introduction of the dissipative function, the modified Gross-Pitaevskii equations are obtained. These equations, in contrast to the usual Gross-Pitaevskii equations for two-component condensate, allow us to take into account the dissipation in the system. The influence of dissipative processes on the development of modulation instability in a spatially homogeneous two-component Bose-Einstein condensate is investigated. In contrast to the one-component Bose-Einstein condensate, in which modulation instability arises only when there are forces of attraction between atoms, in a two-component Bose-Einstein condensate nonlinear dynamics, leading to modulation instability is more complex. It essentially depends on the signs and values of the constant interaction of the components, which leads to a greater variety of possible scenarios for the development of modulation instability. The paper considers two cases. The first case is when repulsive forces act inside the components, and the second is when repulsive forces act in the first component, and in the second one - attractive forces. At the same time, the situation when there is a repulsion in the first component, and attraction between the particles in the second component differs significantly from the case of only positive interaction inside the components. The relations between the interaction constants that determine the development of the modulation instability turn out to be different. Given the relations between the interaction constants, taking into account dissipation processes, the occurrence of modulation instability in two-component Bose-Einstein condensates was studied, the maximum growth rate of oscillations was found, and the limits of the existence of modulation instability in the space of wave numbers were found. It is shown that the small effect of dissipation on the modulation instability in the Bose – Einstein condensate is explained not only by the smallness of the friction forces. For wave vectors corresponding to a mode with a maximum increment, the contribution of dissipation in the linear approximation with respect to the dissipative parameter is strictly zero. Thus, the condition for the development of the most rapidly growing mode of oscillations, which determines the beginning of the modulation instability, remains the same as in the nondissipative case.

Ion-acoustic waves in ultracold neutral plasmas: Modulational instability and dissipative rogue waves

Progress is reported on the modulational instability (MI) of ion-acoustic waves (IAWs) and dissipative rogue waves (RWs) in ultracold neutral plasmas (UNPs). The UNPs consist of inertial ions fluid and Maxwellian inertialess hot electrons, and the presence of an ion kinematic viscosity is allowed. For this purpose, a modified nonlinear Schrödinger equation (NLSE) is derived and then solved analytically to show the occurrence of MI. It is found that the (in)stability regions of the wavepacks are dependent on time due to of the existence of the dissipative term. The existing regions of the MI of the IAWs are inventoried precisely. After that, we use a suitable transformation to convert the modified NLSE into the normal NLSE whose analytical solutions for rogue waves are known. The rogue wave propagation condition and its behavior are discussed. The impact of the relevant physical parameters on the profile of the RWs is examined.

Amplitude modulation of quantum-ion-acoustic wavepackets in electron-positron-ion plasmas: Modulational instability, envelope modes, extreme waves

A semirelativistic fluid model is employed to describe the nonlinear amplitude modulation of low-frequency (ionic scale) electrostatic waves in an unmagnetized electron-positron-ion plasma. Electrons and positrons are assumed to be degenerated and inertialess, whereas ions are warm and classical. A multiscale perturbation method is used to derive a nonlinear Schrödinger equation for the envelope amplitude, based on which the occurrence of modulational instability is investigated in detail. Various types of localized ion acoustic excitations are shown to exist, in the form of either bright type envelope solitons (envelope pulses) or dark-type envelope solitons (voids, holes). The plasma configurational parameters (namely, the relativistic degeneracy parameter, the positron concentration, and the ionic temperature) are shown to affect the conditions for modulational instability significantly, in fact modifying the associated threshold as well as the instability growth rate. In particular, the relativistic degeneracy parameter leads to an enhancement of the modulational instability mechanism. Furthermore, the effect of different relevant plasma parameters on the characteristics (amplitude, width) of these envelope solitary structures is also presented in detail. Finally, the occurrence of extreme amplitude excitation (rogue waves) is also discussed briefly. Our results aim at elucidating the formation and dynamics of nonlinear electrostatic excitations in superdense astrophysical regimes.

Modulation Instability of an Intense Laser Beam in an Unmagnetized Plasma

The modulation instability of an intense elliptically polarized laser beam propagating in an unmagnetized plasma is investigated. Adopting a generalized Karpman method, a three-dimensional nonlinear equation is shown to govern the laser fleld. Conditions indicating the occurrence of modulation instability and its temporal growth rate, taking into account the longitudinal and the transverse perturbations, are obtained analytically, and the modulated growth rate is shown to be maximal for any flxed wave number when the initial laser fleld is coplanar with the perturbations. In the weakly relativistic region, the growth rate increases with increasing laser pump intensity and grows signiflcantly when the laser approaches the critical surface.

Electrostatic solitary waves in the presence of excess superthermal electrons: modulational instability and envelope soliton modes

The nonlinear dynamics of electrostatic solitary waves in the form of localized modulated wavepackets is investigated from first principles. Electron-acoustic (EA) excitations are considered in a two-electron plasma, via a fluid formulation. The plasma, assumed to be collisionless and uniform (unmagnetized), is composed of two types of electrons (inertial cold electrons and inertialess kappa-distributed superthermal electrons) and stationary ions. By making use of a multiscale perturbation technique, a nonlinear Schrödinger equation is derived for the modulated envelope, relying on which the occurrence of modulational instability (MI) is investigated in detail. Stationary profile localized EA excitations may exist, in the form of bright solitons (envelope pulses) or dark envelopes (voids). The presence of superthermal electrons modifies the conditions for MI to occur, as well as the associated threshold and growth rate. The concentration of superthermal electrons (i.e., the deviation from a Maxwellian electron distribution) may control or even suppress MI. Furthermore, superthermality affects the characteristics of solitary envelope structures, both qualitatively (supporting one or the other type, for different κ) and quantitatively, changing their characteristics (width, amplitude). The stability of bright and dark-type nonlinear structures is confirmed by numerical simulations.

Modulation instability of plane electromagnetic beams in centro-symmetric crystals due to two-photon photorefractive effects

This paper presents an investigation of the modulation instability (MI) of a broad optical beam in biased centro-symmetric two-photon photorefractive media under undepleted pump approximation. From the evolution equation of the perturbation field, we obtain the expression for growth rate of the sidebands. The occurrence of MI has been found to be constrained by the system parameters. The MI gain function, its maximum value, critical frequency and bandwidth have been evaluated in the unstable region.

Modulation instability in nonlinear oppositely directed coupler with a negative-index metamaterial channel

The nonlinear optical coupler with a negative-index metamaterial (NIM) channel is different from a conventional one, mainly in that the former has an effective feedback mechanism due to the opposite directionality of the phase velocities in the two channels. In this paper, we study modulation instability (MI) in this new kind of couplers, trying to identify the different MI properties from those in conventional couplers. It is found that MI in the nonlinear optical coupler with a NIM channel is significantly influenced by the ratio of the forward- to backward-propagating wave's power and the nonlinear parameters. And the threshold condition for the ratio and input power can exist only in the normal dispersion regime where the ratio is positive. In addition, it is shown that MI occurs only for finite values of input power when the NIM channel is self-defocusing nonlinear material and another channel is self-focusing nonlinear material in the anomalous dispersion regime. Further, increasing input power may suppress the MI, quite different from the MI in the conventional couplers. For the conventional one, the increase of input power generally promotes the occurrence of MI. These findings suggest that the controllability of NIMs provides more ways to manipulate MI and soliton in couplers with NIM channels.

Generation of Bragg solitons through modulation instability in a Bragg grating structure

In this article, we consider the continuous wave (cw) propagation through the nonlinear periodic structure that consists of alternating layers of both positive and negative Kerr coefficients along the propagation direction. We investigate the modulational instability (MI) conditions required for the generation of ultrashort pulses for the nonlinearity management system. We study the occurrence of MI at the top and bottom edges of the photonic band gap (PBG) where the forward and backward propagating waves are strongly coupled because of the presence of the grating structure. We also study the MI when cw is detuned from the edges of the PBG into the anomalous and normal dispersion regimes. In addition, we discuss the existence of gap solitons for the nonlinearity management system in the upper and lower branches of the dispersion curve through the MI gain spectra. We observe the generation of higher order solitons in the nonlinear periodic structure when the input power is increased beyond a certain critical level. Finally, we discuss the generation of higher order Bragg grating solitons through the intensity evolution of the forward and backward propagating fields.

Modulational instability in a fibre and a fibre Bragg grating

In this review article, we study the influence of cross-phase modulation, higher order nonlinear effects such as self-steepening, self-induced Raman scattering and higher order dispersion effects such as third and fourth order dispersion on cross-phase modulational instability for a highly elliptical birefringent optical fibre, and obtain the conditions for the occurrence of cross-phase modulational instability in the normal dispersion regime. In addition, we also consider the pulse propagation through a fibre Bragg grating structure where we investigate the occurrence of modulational instability at the two edges of the photonic bandgap as well as on the upper and lower branches of the dispersion curves.

Modulation instability of the stimulated Brillouin scattering in fibers in a presence of acoustic diffraction

Summary An influence of acoustic wave diffraction on the modulation instability of the backward stimulated Brillouin scattering in fibers is numerically investigated, when a small feedback for the Stokes wave is present. The cases of acoustic waveguide and anti-waveguide fibers are considered. It is shown that a presence of acoustic diffraction may expand the region of occurrence of modulation instability and may lead to 2–5 times increasing the peak amplitude of output Stokes pulses. In this case, a stable train of short powerful Stokes pulses of nanosecond duration is produced at the output.

Soliton and strong Langmuir turbulence in solar flare processes

The occurrence of modulational instability in the current sheet is investigated. Particular attention is drawn to the plasma micro-instability in this current sheet (i.e., the diffusion region) and its relation to the flare process. It is found that the solitons or strong Langmuir turbulence is likely to occur in the diffusion region under solar flare conditions in which the electric resistivity could be greatly enhanced by several orders of magnitude in this diffusion region. The result is a significant heating and stochastic acceleration of particles. Physically, the occurrence of soliton and strong Langmuir turbulence can be identified with a sudden eruption of an electric current leading to a local vacuum in which an electric potential is formed and results in the release of a huge amount of free energy. A numerical example is used to demonstrate the transition of the magnetic field, velocity, and plasma density from the outer MHD region into the diffusive (resistive) region and, then, back out again with the completion of the energy conversion process. This is all made possible by an increase of resistivity by 4–5 orders of magnitude over the classical value.

Nonlinear theory of two stream instability in the presence of magnetic field

The method of perturbation has been applied to study the nonlinear development of two stream instability of a magnetized hot plasma in the presence of two finite temperature electron beams with the help of macroscopic equations. A nonlinear Schrodinger equation governing the finite amplitude wave has been derived. It has been found that the nature of axisymmetric potential is completely changed due to the presence of the magnetic field. The values of omega and k for the marginal instability increase with the increasing magnetic field. The range of the wave numbers for the instability of higher harmonics decrease due to the magnetic field. All the disturbances grow up until overtaking occurs, the growth rate being enhanced by nonlinear effects. The occurrence of modulational instability has also been predicted.