Conditions For Modulational Instability Research Articles

Effect of κ-deformed Kaniadakis distribution on the modulational instability of electron-acoustic waves in a non-Maxwellian plasma

In this paper, the modulational instability (MI) of the high-frequency electron-acoustic waves (EAWs) is reported in a non-Maxwellian plasma composed of two distinct types of electrons and stationary ions. One type of electrons is treated as a cold inertial fluid, whereas the other type is considered as inertialess species following κ-deformed Kaniadakis distribution. The fluid equations to the current model are reduced via a reductive perturbation technique to a nonlinear Schrödinger equation, which is then used to compute the MI and the growth rate of the EAWs. It is instructive to note that the deformation parameter (which develops the Kaniadakis entropy) and the hot-to-cold electron density ratio (hot electron concentration) significantly affect the conditions for MI. The modulated envelope black (dark and gray) solitons are investigated. The current results are beneficial in analyzing the spectrum of the cosmic rays, which violates manifestly the Boltzmann–Gibbs statistics. Moreover, the obtained results can be used to understand the mystery of many observations in stars where the presence of non-Maxwellian particles dominates.

Modulated ion-acoustic wavepackets in a multi-component dense plasma with Thomas-Fermi distribution

Modulated ion acoustic wavepackets have been investigated in a multi-component, dense plasma containing degenerate electrons/positrons, and positive/negative ions. The electrons and positrons assumed to be followed Thomas-Fermi distribution, while the ions considered to be classic. Employing a multi-scale perturbation technique, the nonlinear Schrödinger equation is derived. The criteria for the occurrence of modulational instability of the modulated ion acoustic waves are analyzed. The influences of plasma parameters on the modulational instability and its growth rate have been discussed. The envelope solitons have been discussed as well. The results showed that both modulationally unstable domain and stable domain can be found, depending on the plasma parameters. It is also found that the width of the envelop solitons are significantly modified by related plasma parameters. The present results will be useful in understanding the conditions for modulational instability of ion acoustic waves which are relevant to both space and laboratory dense plasmas.

Modulational instability of a resonantly polariton condensate in discrete lattices

We study modulational instability of a resonantly polariton condensate in a discrete lattice. Employing a discrete gain-saturation model, we derive the dispersion relation for the modulational instability by means of the linear-stability analysis. Effects of the pumping strength, the nonlinearity, the strength of the detuning, and the coupling strength on the modulation instability are investigated. It is found that the interplay between these parameters will dramatically change the modulational instability condition. We believe that the predicted results in this work can be useful for future possible experiment of exciton-polariton condensate in lattices.

Hyperbolic (3+1)‐Dimensional Nonlinear Schrödinger Equation: Lie Symmetry Analysis and Modulation Instability

The hyperbolic nonlinear Schrödinger equation in the (3 + 1)‐dimension depicts the evolution of the elevation of the water wave surface for slowly modulated wave trains in deep water. Many researchers have studied the applicability and practicality of this model, but the analytical approach has been virtually absent from the literature. We adapted the lie symmetry analysis method to obtain a new complex solution in this work. The obtained complex solution contains bright and dark solitons. Furthermore, modulation instability is applied to this model to explain the interplay between nonlinear and dispersive effects. As a result, the modulation instability condition and the explosive rate are also discussed.

Modulational Instability of Ion-Acoustic Waves in Pair-Ion Plasma

The modulational instability (MI) of ion-acoustic waves (IAWs) is examined theoretically in a four-component plasma system containing inertialess electrons featuring a non-thermal, non-extensive distribution, iso-thermal positrons, and positively as well as negatively charged inertial ions. In this connection, a non-linear Schrödinger equation (NLSE), which dominates the conditions for MI associated with IAWs, is obtained by using the reductive perturbation method. The numerical analysis of the NLSE reveals that the increment in non-thermality leads to a more unstable state, whereas the enhancement in non-extensivity introduces a less unstable state. It also signifies the bright (dark) ion-acoustic (IA) envelope solitons mode in the unstable (stable) domain. The conditions for MI and its growth rate in the unstable regime of the IAWs are vigorously modified by the different plasma parameters (viz., non-thermal, non-extensive q-distributed electron, iso-thermal positron, the ion charge state, the mass of the ion and positron, non-thermal parameter α, the temperature of electron and positron, etc.). Our findings may supplement and add to prior research in non-thermal, non-extensive electrons and iso-thermal positrons that can co-exist with positive as well as negative inertial ions.

Modulational Instability of Ion-Acoustic Waves and Associated Envelope Solitons in a Multi-Component Plasma

A generalized plasma model with inertial warm ions, inertialess iso-thermal electrons, super-thermal electrons and positrons is considered to theoretically investigate the modulational instability (MI) of ion-acoustic waves (IAWs). A standard nonlinear Schrödinger equation is derived by applying the reductive perturbation method. It is observed that the stable domain of the IAWs decreases with ion temperature but increases with electron temperature. It is also found that the stable domain increases by increasing (decreasing) the electron (ion) number density. The present results will be useful in understanding the conditions for MI of IAWs which are relevant to both space and laboratory plasmas.

Modelling of Ocean Waves with the Alber Equation: Application to Non-Parametric Spectra and Generalisation to Crossing Seas

The Alber equation is a phase-averaged second-moment model used to study the statistics of a sea state, which has recently been attracting renewed attention. We extend it in two ways: firstly, we derive a generalized Alber system starting from a system of nonlinear Schrödinger equations, which contains the classical Alber equation as a special case but can also describe crossing seas, i.e., two wavesystems with different wavenumbers crossing. (These can be two completely independent wavenumbers, i.e., in general different directions and different moduli.) We also derive the associated two-dimensional scalar instability condition. This is the first time that a modulation instability condition applicable to crossing seas has been systematically derived for general spectra. Secondly, we use the classical Alber equation and its associated instability condition to quantify how close a given nonparametric spectrum is to being modulationally unstable. We apply this to a dataset of 100 nonparametric spectra provided by the Norwegian Meteorological Institute and find that the vast majority of realistic spectra turn out to be stable, but three extreme sea states are found to be unstable (out of 20 sea states chosen for their severity). Moreover, we introduce a novel “proximity to instability” (PTI) metric, inspired by the stability analysis. This is seen to correlate strongly with the steepness and Benjamin–Feir Index (BFI) for the sea states in our dataset (>85% Spearman rank correlation). Furthermore, upon comparing with phase-resolved broadband Monte Carlo simulations, the kurtosis and probability of rogue waves for each sea state are also seen to correlate well with the PTI (>85% Spearman rank correlation).

Dust–Acoustic Envelope Solitons in an Electron-Depleted Plasma

A theoretical investigation of the modulational instability (MI) of dust–acoustic waves (DAWs) by deriving a nonlinear Schrodinger equation in an electron-depleted opposite polarity dusty plasma system containing non-extensive positive ions is presented. The conditions for MI of DAWs and formation of envelope solitons are investigated. The sub-extensivity and super-extensivity of positive ions are seen to change the stable and unstable parametric regimes of DAWs. The addition of dust grains causes changes the width of both bright and dark envelope solitons. The findings of this study can help understanding the nonlinear features of DAWs in Martian atmosphere, cometary tails, solar system, laboratory experiments, etc.

Modulational instability of dust-ion-acoustic waves and associated first and second-order rogue waves in a super-thermal plasma

A proper theoretical research has been carried out to explore the modulational instability (MI) conditions of dust-ion-acoustic (DIA) waves (DIAWs) in a three-component dusty plasma system containing inertialess κ-distributed electrons, and inertial warm positive ions and negative dust grains. The nonlinear Schrödinger equation (NLSE) has been derived by employing the reductive perturbation method. The numerical analysis under consideration demonstrates two types of modes, namely, fast and slow DIA modes. The MI conditions of DIAWs and the configuration of the energetic rogue waves (RWs) associated with DIAWs in the modulationally unstable parametric regime have been rigorously changed by the variation of various plasma parameters, namely, charge, mass, temperature, and number density of the plasma species. The findings of our present investigation will be useful in understanding the criteria for the formation of electrostatic RWs in both astrophysical environments (viz., Jupiter’s magnetosphere, cometary tails, pulsar magneto-sphere, interstellar medium, Earth’s mesosphere, Saturn’s rings, etc.) and laboratory experiments.

Modulation instability of ion-acoustic waves and exchange interaction in magnetized quantum plasma with relative density effects of spin-up and spin-down electrons

The separate-spin evolution quantum hydrodynamic (SSE-QHD) model is used to study Ion-Acoustic Waves (IAWs) in a magnetized quantum plasma having electrons with different spin populations in the presence of the spin particle exchange interactions (Coulomb type) in spin-down electrons only. The plasma consists of inertia-less separated spin-up and spin-down electrons states, while ions are inertial and classical. Besides, a uniform magnetic field is assumed to be directed in the Z-direction i.e. . An interesting interplay between exchange interaction and density polarization ratio κ has been observed on dispersive properties, effective Debye screening length and effective ion-acoustic speed. The non-linear Schrödinger equation is derived through a reductive perturbation approach. Moreover, the modulation instability condition and resulting envelope waves are analyzed. The parametric effects of density polarization ratio κ and Coulomb exchange interaction on dispersion, effective screening length, effective ion-acoustic speed, instability growth rate, and on the solitons profile have been discussed in detail. This model can be helpful to study properties of the dilute magnetic semiconductors and degenerate 2D & 3D Fermi electron gas, where exchange interactions are significant.

Periodic and solitary waves in an inhomogeneous optical waveguide with third-order dispersion and self-steepening nonlinearity

We demonstrate the formation of periodic waves and envelope solitons in dispersive optical media having a Kerr nonlinear response under the influence of third-order dispersion and self-steepening effect. The stability properties of the bright- and dark-soliton solutions are proved using the stability criterion based on the theory of nonlinear dispersive waves. Regimes for the modulation instability of a continuous wave signal propagating inside the dispersive optical medium are also investigated. The results show that the gain spectrum depends crucially on the self-steepening parameter, while the third-order dispersion has no effect on the modulation instability condition. A similarity transformation is presented to reduce the generalized extended nonlinear Schr\"odinger equation with distributed coefficients which models the pulse evolution in the presence of the inhomogeneities of media to the related constant-coefficient one. The propagation behaviors of self-similar bright and dark solitons are discussed in a periodic distributed fiber system and an exponential dispersion decreasing fiber.

Dust-acoustic envelope solitons and rogue waves in an electron depleted plasma

A theoretical investigation of the modulational instability (MI) of dust-acoustic waves (DAWs) by deriving a nonlinear Schr\"{o}dinger equation in an electron depleted opposite polarity dusty plasma system containing non-extensive positive ions has been presented. The conditions for MI of DAWs and formation of envelope solitons have been investigated. The sub-extensivity and super-extensivity of positive ions are seen to change the stable and unstable parametric regimes of DAWs. The addition of dust grains causes to change the width of both bright and dark envelope solitons. The findings of this study may be helpful to understand the nonlinear features of DAWs in Martian atmosphere, cometary tail, solar system, and in laboratory experiments, etc.

Modulation instability and rogue wave spectrum for the generalized nonlinear Schrödinger equation

We discuss modulation instability for the generalized nonlinear Schrödinger equation based on nonzero background wave frequency. First of all, we analyze the existence condition of modulation instability under different perturbed frequency. The influences of the background amplitude, background frequency and perturbed frequency on the modulation instability gain are researched, respectively. Also, we obtain the correspondences between several nonlinear excitations (Kuznetsov-Ma breather, general breather, rogue wave, bright soliton and plane wave) and modulation instability according to new parameters. Furthermore, by the Fourier transformation method, we perform spectrum analyses of the first-order and second-order rogue waves. The perturbed frequency of the rogue wave can affect the location and profile of the spectrum. And we find that the spectrum of the second-order rogue wave is jagged due to the collision of the rogue waves. These results would help us further understand the dynamics of rogue wave in complex systems.

Interplay of three-body and higher-order interactions on the modulational instability of Bose–Einstein condensate

We investigate the modulational instability (MI) of trapped Bose–Einstein condensates with three-body and higher-order interactions employing both semi-analytical and numerical methods. Using the time-dependent variational approach, we derive variational equations for the time evolution of the amplitude, phase of modulational perturbation, and effective potential of the system. By means of an effective potential, we retrieve the corresponding MI condition of the dynamical system under consideration. The interplay between three-body interaction and higher-order interaction is discussed in detail. The semi-analytical predictions are confirmed through numerical simulations.

Generation of localized patterns in two-component condensates trapped in variable shape optical lattices

Abstract In this work, we discuss the generation of modulational instability in two component Bose-Einstein condensates loaded in deformable shape external optical potentials. As theoretical analysis, we use the time-dependent variational approach which allows us to obtain ordinary differential equations for the time evolution of the amplitude and phase of modulational perturbation. We determine the modulational instability condition by means of effective potential. We show the effects of the optical lattice deformability as well as the cubic and quintic nonlinear interactions on the dynamics of the mixture. Performing direct numerical simulations of the two coupled Gross-Pitaevskii equations (GPEs), we find that our variational and numerical calculations well coincide with each other. The high degree of control on every parameter suggest that our system is relevant for future applications.

Electron acoustic envelope solitons in non-Maxwellian plasmas

The NASA Van Allen Probes spacecraft has recently observed broadband electrostatic turbulence in the inner magnetosphere which has been conjectured to be produced by large amplitude electrostatic solitary waves of generally two types [D.M. Malaspina, J.R. Wygant, R.E. Ergun, G.D. Reeves, R.M. Skoug, B.A. Larsen, J. Geophys. Res.: Space Phys. 120, 4246 (2015); C.S. Dillard, I.Y. Vasko, F.S. Mozer, O.V. Agapitov, J.W. Bonnell, Phys. Plasmas 25, 022905, (2018)]. The solitary waves with highly asymmetric bipolar parallel electric field have been recently shown to correspond to the electron-acoustic plasma mode (existing due to two-temperature electron population) [C.S. Dillard, I.Y. Vasko, F.S. Mozer, O.V. Agapitov, J.W. Bonnell, Phys. Plasmas 25, 022905, (2018)]. These findings along with the observations of non-Maxwellian electrons in terrestrial magnetosheath and planetary magnetospheres [W. Masood, S.J. Schwartz, M. Maksimovic, A.N. Fazakerley, Ann. Geophys. 24, 1725 (2006); W. Masood, S.J. Schwartz, J. Geophys. Res. 113, A01216 (2008); M.N.S. Qureshi, W. Nasir, W. Masood, P.H. Yoon, H.A. Shah, S.J. Schwartz, J. Geophys. Res.: Space Phys. 119, 10059 (2014); M.N.S. Qureshi, W. Nasir, R. Bruno, W. Masood, MNRAS 488, 954 (2019)] have prompted us to theoretically investigate the problem of amplitude modulation of the electron acoustic waves (EAWs) in plasmas whose ingredients are stationary ions, inertial cold electrons and warm (r,q) distributed inertialess electrons. The nonlinear Schrodinger equation (NLSE) that governs the modulational instability (MI) of the EAWs has been derived using the standard reductive perturbation technique (RPT). The presence of the warm (r,q) distributed electrons has been shown to influence the existence conditions of MI of the EAWs and both the indices have been found to cause a decrease in the growth rate of MI. A detailed comparison has also been drawn between double spectral index (r,q), kappa and Maxwellian distribution functions and the differences are highlighted. Additionally, the nondimensional parameter α=nc0/ne0 which is the equilibrium density ratio of the cold to hot electron component, has been shown to play an important role in the formation of envelope solitary excitations.

Modulational instability of dust-ion acoustic waves in the presence of generalized (r, q) distributed electrons

We have investigated the modulational instability (MI) of dust-ion acoustic waves in a dusty plasma model containing warm ions fluid, generalized (r, q) distributed electrons, and immobile dust grains. Using the Krylov-Bogoliubov-Mitropolsky method, the amplitude modulation of a monochromatic plane wave is analyzed through derivation of a nonlinear Schrödinger equation. The MI condition is calculated. Moreover, the effects of the plasma parameters such as the ion-to-electron temperature ratio, the wavenumber, and the unperturbed dust-to-electron number density ratio as well as the electron distribution dual parameters (r, q) are discussed. Our investigations have shown that the unstable region is bounded by two stable regions, i.e., two critical wavenumbers [kcl(lower) and kch(higher)] are found. The applications of the present theoretical study are briefly discussed in space physics whether positive or negative dust grains are enclosed.

Dust-acoustic rogue waves in an electron depleted plasma

A rigorous theoretical investigation is made to study the characteristics of dust-acoustic (DA) waves (DAWs) in an electron depleted unmagnetized opposite polarity dusty plasma system that contains super-thermal ($\kappa$-distributed) ions, mobile positively and negatively charged dust grains for the first time. The reductive perturbation method is employed to obtain the NLSE to explore the modulational instability (MI) conditions for DAWs as well as the formation and characteristics of gigantic rogue waves. The nonlinear and dispersion properties of the dusty plasma medium are the prime reasons behind the formation of rogue waves. The height and thickness of the DARWs associated with DAWs as well as the MI conditions of DAWs are numerically analyzed by changing different dusty plasma parameters, such as dust charges, dust and ion number densities, and ion-temperature, etc. The implications of the results for various space dusty plasma systems (viz., mesosphere, F-rings of Saturn, and cometary atmosphere, etc.) as well as laboratory dusty plasma produced by laser-matter interaction are briefly mentioned.

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 induced by intermodal cross-phase modulation in step-index multimode fiber.

Based on linear stability analysis, we theoretically investigate the onset condition and gain spectrum of intermodal cross-phase modulation-induced noise-seeded modulation instability taking place in the normal dispersion regime of step-index multimode fiber, specially designed to support four modes (LP01, LP11, LP21, LP02) at 1064nm. For the case of total power being distributed equally in certain bimodal combinations, the peak sideband frequency scales as the square root of total power, and peak gain increases linearly with total power. For the case of total power being distributed unequally, there exists an optimal power ratio to obtain peak gain, which does not depend on total power significantly. When the mode group velocity mismatch is considered, the range and peak values of gain spectra are notably increased.