Growth Rate Of Modulational Instability Research Articles

Modulational instability and chirped modulated wave, chirped optical solitons for a generalized (3+1)-dimensional cubic-quintic medium with self-frequency shift and self-steepening nonlinear terms

We consider a generalized (3+1)-dimensional nonlinear Schrödinger with cubic-quintic nonlinear and self-frequency shift and self-steepening terms. We disrupt the plane wave to study the stability of the wave in this media. We examine how various factors—such as the amplitude of the plane wave, cubic and quintic nonlinear terms, self-steepening, and self-frequency shift—affect the modulational instability (MI). Our findings reveal that the amplitude of the plane wave and the quintic nonlinear term can expand the MI bands and increase the amplitude of the MI growth rate. Conversely, the self-steepening and self-frequency shift terms exert opposite effects, narrowing the MI bands and reducing the amplitude of the MI growth rate. We investigate the existence of modulated chirped rational and polynomial Jacobi elliptic function solutions and chirped optical solitons.

Modulation instability of magnetosonic waves in semiconducting quantum plasma considering Fermi degenerate pressure, exchange correlation potential, and Bohm potential

This study investigates the modulation instability of magnetosonic waves in a semiconducting quantum plasma system. Utilizing the quantum hydrodynamic model, we derive the nonlinear Schrödinger equation and its solution through the reductive perturbation technique. The growth rate of modulation instability for magnetosonic waves has been derived. This study incorporates various quantum corrections, including Fermi degenerate pressure, exchange-correlation potential, and Bohm potential. We study the bright soliton profiles of magnetosonic waves, and additionally, we conduct graphical analyses of the linear dispersion relation and the product of the dispersive coefficient (P) and nonlinear coefficient (Q) of the nonlinear Schrödinger equation. It has been found that magnetosonic waves exhibit modulation instability within specific parameter ranges of the semiconductor plasma and at certain wavelength regimes. Further, we present a comparative study between GaSb and InSb semiconducting plasma systems. The exchange-correlation potential and Fermi degenerate pressure have significantly impacted the modulation instability growth rate, whereas the effect of the Bohm potential is much lower.

Freak wave generation modulated by high wind and linear shear flow in finite water depth

In finite water depths, the effects of high winds and linear shear flow (LSF), encompassing both uniform flow and constant vorticity shear flow on freak wave generation are explored. A nonlinear Schrödinger equation, adjusted for high wind and LSF conditions, is derived using potential flow theory and the multiscale method. This equation accounts for the modulational instability (MI) of water waves and the evolution of freak wave amplitudes. MI analysis reveals that for waves to maintain MI, high tail winds (moving in the same direction as the wave) require less vorticity and deeper water, while adverse winds (moving in the opposite direction) necessitate more vorticity and shallower water depths compared to conditions without wind. Uniform up-flows (down-flows), positive (negative) vorticity, and high tail (adverse) winds, which inhibit (promote) wave propagation, increase (decrease) the MI growth rate and amplify (diminish) freak wave heights. It is through this MI that the generation of freak waves is either promoted or inhibited.

Modulation instability in nonlinear media with sine-oscillatory nonlocal response function and pure quartic diffraction.

Modulation instability of one-dimensional plane wave is demonstrated in nonlinear Kerr media with sine-oscillatory nonlocal response function and pure quartic diffraction. The growth rate of modulation instability, which depends on the degree of nonlocality, coefficient of quartic diffraction, type of the nonlinearity and the power of plane wave, is analytically obtained with linear-stability analysis. Different from other nonlocal response functions, the maximum of the growth rate in media with sine-oscillatory nonlocal response function occurs always at a particular wave number. Theoretical results of modulation instability are confirmed numerically with split-step Fourier transform. Modulation instability can be controlled flexibly by adjusting the degree of nonlocality and quartic diffraction.

Nonlinear theory of the modulational instability at the ion–ion hybrid frequency and collapse of ion–ion hybrid waves in two-ion plasmas

We study the dynamics of two-dimensional (2D) nonlinear ion–ion hybrid waves propagating perpendicular to an external magnetic field in plasmas with two ion species. We derive nonlinear equations for the envelope of electrostatic potential at the ion–ion hybrid frequency to describe the interaction of ion–ion hybrid waves with low-frequency acoustic-type disturbances. The resulting nonlinear equations also take into account the contribution of the second harmonics of the ion–ion hybrid frequency. A nonlinear dispersion relation is obtained, and, for a number of particular cases, modulational instability growth rates are found. By neglecting the contribution of second harmonics, the phenomenon of the collapse of ion–ion hybrid waves is predicted. It is shown that taking into account the interaction with the second harmonic results in the existence of a stable two-dimensional soliton.

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.

Solitonic rogue and modulated wave patterns in the monoatomic chain with anharmonic potential

Modulation instability and rogue wave structures have been investigated in this work. This study is an extension of the work in Abbagari (2023), where a nonlinear Schrödinger equation with higher-order dispersion is derived to show only the development of the modulated waves bounded to bright soliton as a nonlinear exhibition of modulation instability. Here, the coupled nonlinear Schrödinger equation is derived by using the multi-scale scheme. An overview of the analytical calculations of the perturbed plane wave is carried out to show the effects of the nonlinear chain parameters on the modulation instability growth rate and bandwidths. The interest of this study lies equally in the nonlinear modes of excitation, where solitonic waves are generated under certain conditions in lower and upper frequency bands. On the other hand, relevant results have been developed to show the features of the type I and type II rogue waves of the Manakov system. Such investigations are obtained under the variation of the interaction potential parameters and the free parameter of the similarity method. Via a numerical simulation, rogue wave structures have been generated as a consequence of the long-time evolution of the perturbed plane wave. At a specific time of propagation, another localized object has been obtained to show the Akhmediev breathers and Kuznetsov-Ma solitons clusters under a strong perturbed wave number. These results have opened up new features, and many applications could follow in the future.

Influence of the discrete lattice spacing on the formation of intrinsic localized structures in the Salerno model

We report in this paper the influence of the discrete lattice spacing on the formation of intrinsic localized structures in the Salerno model. Through the linear stability analysis, appearance criterions of the modulation instability (MI) are derived. We show via the growth rate that, the discrete lattice spacing and the nonlinearity term can affect significantly the formation of instability zones. We find that for large values of the discrete lattice spacing, the MI is developed to increase the number instability zones whereas for very low values of this parameter, the MI growth rate is arisen up to enlarge the bandwidth. The theoretical findings have been numerically tested via the discrete spatial Fourier transform method and the direct simulations, and have been found to be in full agreement with the analytical prediction. Moreover, it has been revealed that a higher modulation index can dramatically modify the shape and the amplitude of the modulated waves.

Multisolitons-like patterns in a one-dimensional MARCKS protein cyclic model

In this paper, we study the nonlinear dynamics of the MARCKS protein between cytosol and cytoplasmic membrane through the modulational instability phenomenon. The reaction–diffusion generic model used here is firstly transformed into a cubic complex Ginzburg–Landau equation. Then, modulational instability (MI) is carried out in order to derive the MI criteria. We find the domains of some parameter space where nonlinear patterns are expected in the model. The analytical results on the MI growth rate predict that phosphorylation and binding rates affect MARCKS dynamics in opposite way: while the phosphorylation rate tends to support highly localized structures of MARCKS, the binding rate in turn tends to slow down such features. On the other hand, self-diffusion process always amplifies the MI phenomenon. These predictions are confirmed by numerical simulations. As a result, the cyclic transport of MARCKS protein from membrane to cytosol may be done by means of multisolitons-like patterns.

Modulation effect of linear shear flow, wind, and dissipation on freak wave generation in finite water depth

This paper studies the modulation effect of linear shear flow (LSF), comprising a uniform flow and a shear flow with constant vorticity, combined with wind and dissipation on freak wave generation in water of finite depth. A nonlinear Schrödinger equation (NLSE) modified by LSF, strong wind, and dissipation is derived. This can be reduced to consider the effects of LSF, light wind, and dissipation, and further reduced to include only LSF. The relation between modulational instability (MI) of the NLSE and freak waves represented as a modified Peregrine Breather solution is analyzed. When considering only LSF, the convergence (divergence) effect of uniform up-flow (down-flow) and positive (negative) vorticity increases (decreases) the MI growth rate and promotes (inhibits) freak wave generation. The combined effect of LSF and light wind shows that a light adverse (tail) wind can restrain (amplify) MI and bury (trigger) freak waves. Under the effect of a light tailwind, LSF has the same effect on the MI growth rate and freak wave generation as the case without any wind. The combination of LSF and strong wind enables both adverse and tail winds to amplify MI and trigger freak waves. In the presence of strong wind, LSF has the opposite effect to the case of a light tailwind.

Modulation instability and modulated wave patterns in a nonlinear electrical transmission line with the next-nearest-neighbor coupling

In this paper, we investigate modulation instability and solitary waves in a one-dimensional nonlinear electrical transmission line with second-neighbor interactions. Through the quasi-discrete multiple scale method, we derive coupled nonlinear Schrödinger equations. To investigate the modulation instability in the structure, we use the linear stability analysis, and an expression for the modulation instability spectrum is derived. From the analytical investigation, we show that the second neighbor coupling affects both the modulation instability growth rate and modulation instability bands. Furthermore, we show via single and coupled mode excitation that bright and dark solitary waves can propagate at the lower and upper cutoff frequencies. To evaluate the robustness of the analytical investigation, we use a direct numerical simulation of the continuous wave. We conclude that the second-neighbor couplings generate rogue waves in the structure during the long-time evolution of the plane wave. We also demonstrate that with a variation of the excitation wave number, the nonlinear electrical transmission line can support new objects as wave molecules for high values of the wave number. This feature is not yet observed in nonlinear electrical transmission lines and will be useful in many fields of physics.

Unstable wave patterns of information in neural network under light illumination and magnetic field

Functional neurons built from neural circuits are capable of perceiving and processing external signals such as light illumination and magnetic radiation, by converting the physical signals into modulated bioelectric signals called action potential with diverse forms and shapes. Through modulational instability (MI), modulated wave formation and pattern transition are studied in a chain memristive network of [Formula: see text] photosensitive neurons. Memristors and photocells are incorporated in a simple FitzHugh–Nagumo neuron to detect and process the external magnetic flux and light illumination. To determine regions of modulated wave formation, linear stability analysis is performed on a nonlinear envelope equation which resulted from the asymptotic expansion of the generic dynamical equations. The growth rate of MI is plotted and the distinct zones of stable/unstable MI are presented. We confirm the analytical result through numerical simulations whereby the initial plane wave solutions lead to the emergence of localized structures with traits of spiking, bursting and chaotic states. High-frequency photocurrent changes orderly localized patterns to chaotic-like patterns while high-frequency magnetic flux promotes pattern transition from bursting to 2-period spiking state and a 4-period spiking state. This could provide an adequate way to influence the behaviors of artificial neurons as well as potential mechanism of information coding in the nervous system.

Nonlocal cubic and quintic nonlinear wave patterns in pure-quartic media

In this work, pure-quartic soliton (PQS)formation is investigated in the framework of a nonlinear Schrödinger equation with competing Kerr (cubic) and non-Kerr (quintic) nonlocal nonlinearities and quartic dispersion. In the process, the modulational instability (MI) phenomenon is activated under a suitable balance between the nonlocal nonlinearities and the quartic dispersion, both for exponential and rectangular nonlocal nonlinear responses. Interestingly, the maximum MI growth rate and bandwidth are reduced or can completely be suppressed for some specific values of the cubic and quintic nonlocality parameters, depending on the type of nonlocal response. The analytical results are confirmed via direct numerical simulations, where the instability supports the signature of pure-quartic dark and bright solitons. These results may provide a better understanding of PQS structures for their potential applications in the next generation of nonlinear optical devices.

Chaos and complexity in the dynamics of nonlinear Alfvén waves in a magnetoplasma.

The nonlinear dynamics of circularly polarized dispersive Alfvén wave (AW) envelopes coupled to the driven ion-sound waves of plasma slow response is studied in a uniform magnetoplasma. By restricting the wave dynamics to a few number of harmonic modes, a low-dimensional dynamical model is proposed to describe the nonlinear wave-wave interactions. It is found that two subintervals of the wave number of modulation k of AW envelope exist, namely, (3/4)kc<k<kc and 0<k<(3/4)kc, where kc is the critical value of k below which the modulational instability (MI) occurs. In the former, where the MI growth rate is low, the periodic and/or quasi-periodic states are shown to occur, whereas the latter, where the MI growth is high, brings about the chaotic states. The existence of these states is established by the analyses of Lyapunov exponent spectra together with the bifurcation diagram and phase-space portraits of dynamical variables. Furthermore, the complexities of chaotic phase spaces in the nonlinear motion are measured by the estimations of the correlation dimension as well as the approximate entropy and compared with those for the known Hénon map and the Lorenz system in which a good qualitative agreement is noted. The chaotic motion, thus, predicted in a low-dimensional model can be a prerequisite for the onset of Alfvénic wave turbulence to be observed in a higher dimensional model that is relevant in the Earth's ionosphere and magnetosphere.

Modulation instability gain and wave patterns in birefringent fibers induced by coupled nonlinear Schrödinger equation

In this work, we studied the behavior of the modulation instability (MI) growth rate and modulated wave (MW) bright and dark-soliton in nonlinear optical fibers (OFs) where both birefringence and four-wave mixing (4WM) terms are employed. We have shown the effects of the 4WM and the group velocity dispersion (GVD) on the MI gain and MI bands as well as on the MWs patterns. For certain conditions on the coupled nonlinear Schrödinger equation (CNLSE) parameters, we exhibited stable and unstable zones of the MI. More precisely, in a normal dispersion regime, we shown how the variation of 4WM can generate additional sides lobes and consequently the formation of new MI bands. Furthermore, it is observed that when the coupling coefficient (CC) increases, its effects could reduce the stability zones by increasing strongly the number of sides lobes in a normal dispersion regime. To seek the behavior of the MWs patterns, we used numerical investigation (NI). From these results, it has been observed the propagation of the MW bright-soliton and dark-soliton with the effects of the 4WM. We noticed that the MWs patterns can preserve high energy on the peak and keep constant its shapes in birefringent OFs. These results show that the 4WM and the GVD behave as being energy sources for MI and MWs. Our finding can help to manufacture OF tools and could open the way for the use of the 4WM and cross-phase modulation (XPM) to generate MI bands and long-live temporal MWs patterns.

Modulational instability in nonlinear saturable media with competing nonlocal nonlinearity.

The modulational instability (MI) phenomenon is addressed in a nonlocal medium under controllable saturation. The linear stability analysis of a plane-wave solution is used to derive an expression for the growth rate of MI that is exploited to parametrically discuss the possibility for the plane wave to disintegrate into nonlinear localized light patterns. The influence of the nonlocal parameter, the saturation coefficient, and the saturation index are mainly explored in the context of a Gaussian nonlocal response. It is pointed out that the instability spectrum, which tends to be quenched by the high nonlocality parameter, gets amplified under the right choices of the saturation parameters, especially the saturation index. Via direct numerical simulations, confirmations of analytical predictions are given, where competing nonlocal and saturable nonlinearities enable the emergence of trains of patterns as manifestations of MI. The comprehensive parametric analysis carried out throughout the numerical experiment reveals the robustness of the obtained rogue waves of A- and B-type Akhmediev breathers, as the nonlinear signature of MI, providing the saturation index as a suitable tool to manipulate nonlinear waves in nonlocal media.

Modulation instability gain and nonlinear modes generation in discrete cubic-quintic nonlinear Schrödinger equation

In this paper, we examined the effects of the nonlinear parameters and driven amplitude on the training envelope in discrete cubic-quintic nonlinear Schrödinger equation with arbitrarily high-order nonlinearities. From the numerical simulation, we exhibited the generation of the train of pulses and dark soliton when the driven amplitude is above the threshold. For a specific time of propagation, we illustrate how the variation of the driven amplitude and the nonlinear cubic-quintic parameters can generate instability in the forbidden gap. It emerges that the model of the discrete nonlinear Schrödinger equation with arbitrarily high-order nonlinearities can be used to generate the train of waves despite the fact that it is not integrable in the continuum limit approximation. We also displayed the effects of the cubic-quintic terms on the modulation instability growth rate. The obtained results will open new features to the train of pulses and dark soliton in the nonlinear fibers optics.

Fractional dynamics and modulational instability in long-range Heisenberg chains

We study the effective dynamics of ferromagnetic spin chains in presence of long-range interactions. We consider the Heisenberg Hamiltonian in one dimension for which the spins are coupled through power-law long-range exchange interactions with exponent α. We add to the Hamiltonian an anisotropy in the z-direction. In the framework of a semiclassical approach, we use the Holstein–Primakoff transformation to derive an effective long-range discrete nonlinear Schrödinger equation. We then perform the continuum limit and we obtain a fractional nonlinear Schrödinger-like equation. Finally, we study the modulational instability of plane-waves in the continuum limit and we prove that, at variance with the short-range case, plane waves are modulationally unstable for α<3. We also study the dependence of the modulation instability growth rate and critical wave-number on the parameters of the Hamiltonian and on the exponent α.

Frequency modes of unstable spiral waves in two-dimensional Rosenzweig–MacArthur ecological networks

The existence of two frequency regimes in a two-dimensional (2D) Rosenzweig–MacArthur ecological network is debated. The semi-discrete approximation differentiates the two regimes, each described by a 2D complex Ginzburg–Landau equation. Using the standard theory of the linear stability analysis, a generalized expression for the modulational instability growth rate is derived for each frequency mode. The parametric study of the growth rate of modulational instability reveals its sensitivity to the changes in the recruitment rate of the resources. Moreover, direct numerical simulations are carried out to confirm our analytical results. Over the prolonged evolution of the perturbed plane wave solution, the high-frequency mode entertains spiral wave patterns. In contrast, the appearance of target waves manifests the low-frequency regime. In that context, we further explore the impact of the recruitment rate of resources and give the qualitative meaning of the obtained dynamical behaviors and their ecological implications. This work may additionally provide more insight into the mechanism leading to spiral and target waves in environmental systems.

Nonlinear wave transmission in a two-dimensional nonlinear electric transmission network with dissipative elements

A two-dimensional nonlinear discrete electric transmission network made of several well-known anharmonic modified Noguchi lines coupled transversely to one another with a linear inductor is considered and the dynamics of modulated waves are investigated. The linear analysis shows that increasing either the dispersive elements of the network, the coupling elements of the network, or the transverse wavenumber increases the propagating frequencies. Employing the reductive perturbation method, we establish that the dynamics of modulated waves in the network system are governed by a two-dimensional dissipative nonlinear Schrödinger (NLS) equation having, depending on both the longitudinal and transverse wavenumbers as well as on the network parameters, either the hyperbolic or the elliptic character. Through the derived dissipative NLS equation, the phenomenon of the baseband modulational instability (MI) of the electric system under consideration is investigated and the analytical expression for the baseband MI growth rate is presented. We show that the dispersive elements of the network system softens the baseband MI and enhances the baseband MI domain. Under the condition of the baseband MI, we find approximate modulated wave solutions of the governing equations which are then used for analytical investigation of the transmission of super rogue wave voltage, Peregrine soliton voltage, and bright solitary wave signals through the network system under consideration. The effects of the network and solution parameters such as the dispersive, coupling parameters, the carrier wavenumber, and the wavenumbers in both the longitudinal and transverse directions on the dissipative wave signals are presented.