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In physical reality, the phenomena of plasma physics is actually a three-dimensional one. On the other hand, a vast majority of theoretical studies only analyze a one-dimensional prototype of the situation. So, in this communication, we tried to treat the quantum electron–ion plasma in a full 3D setup and the modulational stability of envelope soliton was studied in a quantum electron–ion plasma in three dimensions. The Krylov–Bogoliubov–Mitropolsky method was applied to the three-dimensional plasma governing equations. A generalized form of the nonlinear Schrödinger (NLS) equation was obtained, whose dispersive term had a tensorial character, which resulted in the anisotropic behavior of the wave propagation even in absence of a magnetic field. The stability condition was deduced ab initio and the stability zones were plotted as a function of plasma parameters. The modulational stability of such a three-dimensional NLS equation was then studied as a function of plasma parameters. It is interesting to note that the nonlinear excitation of soliton took place again here due to the balance of nonlinearity and dispersion. The zones of contour plots are given in detail.

We study the nanoscale magnetization evolution of a one dimensional ferromagnetic nanowire with the inclusion of next-nearest-neighbor interaction and Dzyaloshinkii-Moriya (DM) interaction. The system is modelled using Heisenberg Hamiltonian in the quasi classical approximation of spin operators. The dynamics of the system is found to be governed by generalized discrete nonlinear Schrodinger equation. Further we investigate analytically the modulational instability in ferromagnetic nanowire system, where the effect of next-nearest neighbour (NNN) interaction and DM interaction are taken into an account to show graphically the effect of interactions on Modulational Instability (MI) gain spectra.

An axon or nerve fiber is a long, slender projection of a nerve cell, or neuron, in vertebrates that typically conducts electrical impulse known as action potentials away from the nerve cell body. Axons are the cable-like protrusions of neurons which wire up the nervous system. These delicate structures frequently need to survive for a lifetime. Parallel polar, bundles of microtubules run beside axons to form their structural backbones and are acting as highways for life-sustaining transportation. In this paper, we investigate the excitations of soliton along axons that are governed by the Nerve Soliton (NS) model. The NS model of nerve axon is governed by the discrete nonlinear equation of motion derived using the finite difference method. We done modulational instability (MI) analysis for the discrete equation of motion with effect of damping force by using Rotating wave approximation (RWA). The inherent intrinsic localized wave modes were generated which is responsible for energy localization in the nerve axons.

In this study, the relation between rogue wave occurrence statistics and properties of the directional wind-wave spectrum is investigated. The study is conducted by means of the fully nonlinear phase-resolving numerical model and the high-order spectral model. Benjamin-Feir Index (BFI), Π numbers and Modulation Index, which are designed to represent instabilities in nonlinear wave systems, are used as indicators for the occurrence of rogue waves. Rogue wave statistics and indicator parameters’ performance are assessed based on numerical simulations and compared against in situ measurements. Directional Π 2 is found to be the best performing indicator. It is also observed that probability of rogue wave occurrence further peaks up at high directionality where modulational instability should be suppressed. It is argued that this condition is associated with the linear superposition of waves coming from different directions. Therefore, both directionality and instability, i.e. both linear and nonlinear effects, appear to be important physical mechanisms contributing to the extreme wave statistics. However, it is also argued that rogue wave occurrence is a transient (fast) event and cannot be accurately represented with conventional probability distribution functions.

We present a one-line closed-form expression for the three-parameter breather of the nonlinear Schrodinger equation. This provides an analytic proof of the time period doubling observed in experiments. The experimental check that some pulses generated in optical fibers are indeed such generalized breathers will be drastically simplified.

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 $\alpha$. 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 Schr\"odinger equation. We then perform the continuum limit and we obtain a fractional nonlinear Schr\"odinger-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 $\alpha < 3$. 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 $\alpha$.

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.

The modulational instability of dust acoustic (DA) waves propagating in a four-component dusty plasma system comprising negatively and positively charged dust grains as well as kappa-distributed electrons and ions has been studied in the presence of a generalized polarization force. The derivative expansion technique is used to derive a nonlinear Schrodinger equation (NLSE), which describes the nature of the modulational instability of DA waves. The conditions for the existence of the modulational instability of DA waves has been discussed. The results exhibit that the plasma system supports both a modulationally unstable domain and a stable domain, which are significantly modified by the related plasma parameters (viz., polarization force parameter $$R$$ , dust charge number ratio $$Z$$ , and superthermal parameters of electrons and ions). Moreover, the instability growth rate is found to be effectively modified by the presence of these parameters. Our results could be applicable for different space and astrophysical plasma environments.

The standard nonlinear Schrödinger Equation (NLSE) is one of the elegant equations to find detailed information about the modulational instability criteria of dust-ion-acoustic (DIA) waves and associated DIA rogue waves (DIARWs) in a three-component dusty plasma medium with inertialess super-thermal kappa distributed electrons, and inertial warm positive ions and negative dust grains. It can be seen that the plasma system supports both fast and slow DIA modes under consideration of inertial warm ions along with inertial negatively charged dust grains. It is also found that the modulationally stable parametric regime decreases with κ. The numerical analysis has also shown that the amplitude of the first and second-order DIARWs decreases with ion temperature. These results are to be considered the cornerstone for explaining the real puzzles in space and laboratory dusty plasmas.

We consider the nonlinear Schrödinger equation modified by a rational nonlinear term. The model appears in various studies often in the context of the Ginzburg-Landau equation. We investigate modulational instability by means of a linear stability analysis and show how the nonlinear terms affect the growth rate. This analytical result is confirmed by a numerical simulation. The latter analysis shows that breather-like solitons are generated from the instability, and the effects of the nonlinear terms are again clearly seen. Moreover, by employing an auxiliary-equation method we obtain kink and anti-kink soliton as analytical solutions. Our theoretical solution is in good agreement with our numerical investigation.

The dynamics of spatiotemporal modulated damped signals in a nonlinear LC transmission network with dissipative elements are investigated analytically. The complex cubic Ginzburg–Landau (GL) equation governing slowly modulated wave propagation is presented. Considering linear wave propagating in the network, we derive in terms of the propagating frequency the spatial decreasing rate (linear dissipation parameter) and show that its must important contribution comes from the dissipative element of the shunt branch. The modulational instability (MI) criterion of modulated Stokes wave propagating in the network is investigated and the analytical expression of the MI growth rate is derived; we show that in the case of weak dissipation, there are no significant changes for the bandwidth frequency where the network may exhibit MI. Exact and approximative envelope soliton-like solutions of the derived GL equation are presented and used to investigate the dynamics of spatiotemporal modulated damped signals along the network. We show that the solution parameters can be used for managing the evolution of the envelope soliton signals along the network. Our investigation shows that the amplitude decays in both space (cell number n) and time t, while the velocity remains constant when the envelope soliton signal propagates along the dissipative network.

The multiple Exp-function method is employed for seeking the multiple soliton solutions to the generalized (3+1)-dimensional Kadomtsev-Petviashvili (gKP) equation, where contains one-wave, two-wave, and triple-wave solutions. The periodic wave including (exponential, $ \cosh $ hyperbolic, and $ \cos $ periodic), cross-kink containing (exponential, $ \sinh $ hyperbolic, and $ \sin $ periodic), and solitary containing (exponential, $ \tanh $ hyperbolic, and $ \tan $ periodic) wave solutions are obtained. In continuing, the modulation instability is engaged to discuss the stability of obtained solutions. Also, the semi-inverse variational principle is applied for the gKP equation with four major cases. The physical phenomena of these received multiple soliton solutions are analyzed and demonstrated in figures by choosing the specific parameters. By means of symbolic computation these analytical solutions and corresponding rogue waves are obtained with the help of Maple software. Via various three-dimensional, curve, and density charts, dynamical characteristics of these waves are exhibited.

The formation of the gigantic dust-acoustic rouge waves (DARWs) in an electron depleted unmagnetized opposite polarity dusty plasma system is theoretically observed for the first time. The nonlinear Schr\"{o}dinger equation (derived by utilizing the reductive perturbation method) has been analytically as well as numerically analyzed to identify the basic features (viz., height, thickness, and modulational instability, etc.) of DARWs. The results obtained form this investigation should be useful in understanding the basic properties of these rouge waves which can predict to be formed in electron depleted unmagnetized opposite polarity dusty plasma systems like mesosphere, F-rings of Saturn, and cometary atmosphere, etc., Comment: 5 pages; 4 figures

A theoretical investigation is made to study the properties of dust–acoustic (DA) waves, and corresponding dust–acoustic rogue waves (DARWs) in a magnetized electron-depleted dusty plasma that contains opposite polarity warm dust grains, external magnetic field, and nonextensive distributed ions. By using a multiscale reductive perturbation technique (RPT), the nonlinear Schrodinger (NLS) equation is derived in this model. The effects of magnetic field parameter and nonextensive ions are examined on the profiles of the modulational instability. It is found that with the increase of the magnetic field parameter, the instability growth rate and bandwidth increase (decrease) in the case of positive (negative) nonextensive distributed ions. Moreover, the characteristics (spatial wavelength, width, and amplitude) of nonlinear coherent structures involving the bright and dark solitons, the fundamental and second-order rogue waves are also presented in detail. It is observed that these characteristics are significantly modified by effects of magnetic field parameter and nonextensive ions. The results of the present investigation may be applicable to understanding the characteristics and basic nonlinear structures of magnetized plasma environments both in space and laboratory situations.

Since the realization of Bose-Einstein condensates (BECs) in optical potentials, intensive experimental and theoretical investigations have been carried out for matter-wave solitons, coherent structures, modulational instability (MI), and nonlinear excitation of BEC matter waves, making them objects of fundamental interest in the vast realm of nonlinear physics and soft condensed-matter physics. Ubiquitous models, which are relevant to the description of diverse nonlinear media are provided by the nonlinear Schrodinger (NLS), alias Gross-Pitaevskii (GP) equations. In many settings, nontrivial solitons and coherent structures, which do not exist or are unstable in free space, can be created or stabilized by means of various management techniques, which are represented by NLS and GP equations with spatiotemporal coefficients in front of linear or nonlinear terms. Developing this direction of research in various settings, efficient schemes of the spatiotemporal modulation of coefficients in the NLS/GP equations have been designed to engineer desirable robust nonlinear modes. This direction and related ones are the main topic of the present review. A broad and important theme is the creation and control of 1D solitons in BEC by means of combination of the temporal or spatial modulation of the nonlinearity strength and a time-varying trapping potential. An essential ramification of this topic is analytical and numerical analysis of MI of continuous-wave states, and control of the nonlinear development of MI. In addition to that, the review also includes some topics that do not directly include spatiotemporal modulation but address physically important phenomena which demonstrate similar soliton dynamics. These are soliton motion in binary BEC, three-component solitons in spinor BEC, and dynamics of two-component solitons under the action of spin-orbit coupling., This paper is accepted for publication at Physics Report

In this paper, the nonlinear ion acoustic rogue waves (IARWs) are investigated in an unmagnetized, collisionless plasma which consist of positive ions, superthermal electrons and positrons, respectively. A nonlinear Schrodinger equation (NLSE) of IAW packets has been obtained by using the reductive perturbation technique. Furthermore, the effects of superthermal electrons and positrons on the nonlinear dispersion relation, the group velocity and the modulational instability of nonlinear ion acoustic wave (IAW) packets have been studied in detail. At the same time, the characteristics of first- and second-order IARW also have been discussed. It seems that there is some significant influence of superthermal electrons and positrons on the nonlinear ion acoustic waves. Meanwhile, it is also found that the first- and second-order IARW consisting of Peregrine breathers can be excited by using the modulational instability in this system.

The nonlinear propagation of dust acoustic (DA) waves (DAWs) in an unmagnetized electron depleted dusty plasma (containing opposite polarity warm dust grains and non-extensive positive ions) has been theoretically investigated. The nonlinear Schrodinger equation (NLSE) is derived by employing the reductive perturbation method. It is observed that the dusty plasma system under consideration supports two branches of modes, namely, fast and slow DA modes, and that both of these two modes can be modulationally stable or unstable depending on the sign of ratio of the dispersive and nonlinear coefficients of the NLSE. The conditions for the formation of bright and dark envelope solitons as well as the first- and second-order rogue waves have been examined. The numerical analysis has shown that the basic features (viz., stability/instability, amplitude and width of the envelope solitons and rogue structures, etc.) of the DAWs associated with the fast and slow DA modes are significantly modified by dust masses, dust charges, non-extensivity of ions, and other various plasma parameters. The implications of the results to space plasma research (viz., magnetosphere of Jupiter, upper mesosphere, Saturn’s F-ring, cometary tail, etc.) are briefly discussed.

We investigate the frequency conversion associated with the nonlinear stage of vector modulational instability in high-birefringence fibers with normal group-velocity dispersion. A complex heteroclinic structure of instability reveals all possible dynamic trajectories of frequency conversion. It is shown that different Fermi–Pasta–Ulam recurrent regimes are separated by Akhmediev breathers corresponding to a separatrix on the heteroclinic structure. We also demonstrate that the optimal frequency conversion unexpectedly occurs outside the parametric gain bandwidth when the input light is polarized close to 45° from a principal axis of the fiber, whereas the optimal conversion frequency gradually shifts into the gain region when the polarization direction of input light tilts toward the principal axis of the fiber.

It is well known that the ponderomotive force of a relatively high-frequency magnetosonic wave (HMSW) can excite the low-frequency magnetosonic wave (LMSW). By using the coupled system of dynamical equations of LMSW in the presence of a ponderomotive force of HMSW (pump wave), we have studied the modulational instability of the coupled waves between the HMSW and the LMSW. It is found that there are instabilities for the coupled waves between the HMSW and the LMSW. It is also found that the growth rate depends on several system parameters, such as the frequency of HMSW, the perturbation wave vector, and the background magnetic field.

Raman scattering instability (RSI) is investigated in a magnetized quantum electron–ion plasma by applying the separate spin evolution-quantum hydrodynamic (SSE-QHD) model. The governing differential equations of large amplitude transverse electromagnetic waves (TEM) and electron plasma waves (EP) are derived in spin polarization effect. The nonlinear interaction of spin polarized TEM and EP waves is studied by using the standard technique of phasor matching to formulate general dispersion relation of RSI, which admits both three-wave decay and modulational instabilities. It is found that the growth rates of the obtained instabilities are a strong function of amplitude of TEM and quantum effects such as Bohm potential, electron spin effect and spin dependence Coulomb exchange interaction. Here, the growth rate of three-wave decay instability is found to increase with the increase of spin effect, whereas for oscillatory modulational instability the growth rate is observed to occur at a particular wavelength.

Wave field data are affected not only by the accuracy of instruments recording them but also by sampling variability, an uncertainty due to the limited number of observations. For stationary meteorological conditions, due to the randomness of the sea surface elevation, wave parameters derived from a temporal or spatial wave record will depend on which part of a wave record is used in an analysis as well as on the length of a wave record and/or size of the investigated ocean area. This study demonstrates, using numerical simulations, challenges that sampling variability brings in the interpretation of nonlinear wave characteristics of the surface elevation when single 20- or 30-min field wave records are used in an analysis. As examples, we use sea states in which rogue waves were observed in the North Sea and investigate them using linear, second-, and third-order numerical simulations. The third-order wave data are simulated by a numerical solver based on the higher order spectral method (HOSM) which includes the leading order nonlinear dynamical effects, accounting for the effect of modulational instability. Wave steepness, the maximum wave crest, skewness, and kurtosis are investigated in unidirectional and directional wave fields. The study shows that having a single 20- or 30-min wave record may make it difficult to determine on the degree of wave field nonlinearity and the accuracy of derived wave parameters, as well as to evaluate the validity of wave models. Both single-point temporal and stereo-video camera data are discussed. We demonstrate that numerical simulations represent important supporting tools for wave field measurements.

The modulational instability of ion-acoustic rogue waves in a three-dimensional magnetized, dissipative plasma consisting of inertial warm positive ions and nonextensive electrons has been theoreti...

Nonlinear evolution equations, correct up to fourth order in wave steepness, are derived for a pair of obliquely interacting wave packets in the presence of a thin pycnocline. These evolution equations are then employed to perform stability analysis of a pair of obliquely interacting uniform wave trains. Figures plotted here reveal that the growth rate of instability decreases with the increase in pycnocline depth and also with the increase in density difference across the pycnocline. When the angle of interaction between the two wave packets is less than certain critical value the growth rate of instability decreases with the increase in angle and beyond this critical value the result is reversed.

Modulational instability of dust-acoustic waves, which propagate in an opposite polarity dusty plasma system containing inertial warm negatively and positively charged massive dust grains as well as nonextensive q-distributed electrons and ions has been theoretically investigated. The nonlinear Schrodinger equation is derived by employing the reductive perturbation method. The nonlinear Schrodinger equation predicts the conditions of the modulational instability of dust-acoustic waves and the formation of dust-acoustic rogue waves in a nonlinear and dispersive plasma medium. It is observed that the basic features of the dust-acoustic rogue waves (viz., amplitude and width) are significantly modified by the various plasma parameters such as nonextensivity of electrons and ions, electron number density, electron temperature, ion number density, ion temperature, the mass and number density of the dust grains, etc. The application of the results in space and laboratory opposite polarity dusty plasma is briefly discussed.

The amplitude modulation of electron acoustic waves (EAWs) is investigated in an unmagnetized four-component quantum plasma containing dynamic cold electron fluid, Thomas–Fermi distributed hot electrons and positrons and immobile ions in presence of exchange-correlation potentials. For this purpose, basic set of quantum hydrodynamic equations is reduced to a nonlinear Schrodinger equation (NLSE) adopting a reductive perturbation technique. The regions of the stable and unstable wave packets are obtained precisely in terms of the intrinsic plasma parameters. It is shown that the NLSE leads to the modulational instability (MI) of EAWs, and the formation of EA rogue waves, which are due to the effects of nonlinearity and dispersion in the propagation of EAWs. The effects of relevant plasma parameters on the MI criterion are numerically investigated. It is also found that the modulated EAW packets can propagate in the form of bright envelope solitons as well as dark envelope solitons. Within the modulational instability region, the dependence of first- and second-order EA rogue wave profiles on the system parameters is discussed.

The modulational instability of ion-acoustic waves (IAWs) in an unmagnetized multicomponent plasma is investigated, including a hot positrons, hot isothermal electrons and cold ions. Employing the reductive perturbation technique, the nonlinear Schr $$\ddot{o}$$ dinger equation (NLSE) is derived. The effects of positron concentration and temperature ratio of electron to positron significantly modify the modulational instability and its growth rate. Results show that increasing the strength of these parameters leads to localization of IAWs. Further, the exact traveling wave solutions are studied using a modified extended tanh function method. The relevance of theoretical results may be beneficent in understanding the localized electrostatic disturbances in space and astrophysical situations where electron–positron–ion plasmas are present.

Modulational collapse of a wave pulse in a complex nonlinear plasma described by a complex nonlinear Schrodinger equation (CNSE) including the effects of viscous heating and nonlinear damping, and the adjusting by an external potential are investigated. Theoretical and numerical results reveal that the original pulse first suffers modulational instability, the growth rate of the modulational instability increases with the increase of viscous heating, but it is not sensitive to the nonlinear damping. The instability induces the self-similar collapsing, then the fields rapidly transform into a turbulent state with short-wavelength modes. The results are explained in terms of the evolution of the total energy, which can become invariant during the evolution even though the system is nonconservative. The external potential can delay the self-defocusing process of the pulse. Those results can apply to the phenomena described by the CNSE.

A realistic multicomponent collisionless complex magnetized dusty plasma system consisting of warm ions, $\kappa $ -distributed positrons and electrons, and immobile dust grains is considered to study the modulational instability (MI) (stable/unstable frequency regimes) of dust-ion-acoustic waves (DIAWs). By employing the reductive perturbation method, a 3-D nonlinear Schrodinger equation (NLSE), which leads to the MI of DIAWs, is derived. Furthermore, the NLSE gives a special type of 3-D rogue wave (RW) solution. It is observed that the spectral index $\kappa $ plays a crucial role on the behavior of DIAWs. The basic properties like amplitude and width of the first- and second-order dust-ion-acoustic RWs are affected by the key complex magneto-plasma configuration parameters, such as spectral index $\kappa $ , temperature ratio, number densities of magneto-plasma species, and ion cyclotron frequency $(\omega _{ci})$ . The implication of our results for space and laboratory plasmas is briefly discussed.

This work deals with the stimulation of cardiac spiral waves in a two-dimensional FitzHugh–Nagumo model through modulational instability phenomenon in the presence of intracellular magnetic flux. The nonlinear generic model is firstly transformed into a two-dimensional complex Ginzburg–Landau equation with a small real damping term. Then, the latter is explored to perform the linear stability analysis and some modulational instability properties are derived and found to be modified with the change of either the magnetic coupling strength or the diffusion coefficient which are set as the control parameters. The bifurcation theory analysis is numerically performed, and the range of values of the stimulation parameter for which firing patterns emerge in cardiac media is well estimated. The recording of membrane potential as well as the magnetic flux as functions of time or spatial coordinates allows to picture some features such as time series, phase portraits and spatial patterns of membrane potential. As a result, diamond-shaped, squared-shaped and spiral-shaped waves are obtained. An elimination process of multi-spiral waves is proposed by exposing such patterns to an external magnetic flux. Thus, for suited values of parameters, the cardiac spiral waves are transmuted into target waves which obviously protect heart against harmful attacks such as ventricular tachycardia and fibrillation. Our results suggest that uncontrolled electromagnetic induction within cardiac tissue may be considered as the principal source of many heart injuries which are the first cause of mortality in the industrial world.

A central question in sea-state modeling is the role that various physical effects have on the evolution of the statistical properties of random sea states. This becomes a critical issue when one is concerned with the likelihood of rare events such as rogue, or freak, waves which can have significant destructive potential on deep sea ships and other offshore structures. In this paper then, using a recently derived higher-order model of deep water nonlinear waves, we examine the impact of constant vorticity currents on the statistical properties of nonlinearly evolving random sea states. As we show, these currents can both decrease and increase the kurtosis of the affiliated distributions of the sea states, thereby diminishing or enhancing the likelihood of rare events. We likewise numerically study the relationship between the kurtosis and a non-dimensional parameter, the Benjamin–Feir Index, which has proven to be a useful measure of when rare events are likely in oceanographic application.

The Ritz variational method has been applied to the nonlinear partial differential equation to construct a model for travelling wave solution. The spatially periodic trial function was chosen in the form of combination of Jacobian Elliptic functions, with the dependence of its parameters.

In the small amplitude limit, we use the reductive perturbation method and the continuum limit approximation to derive a coupled nonlinear Schro dinger (CNLS) equation describing the dynamics of two interacting signal packets in a discrete nonlinear electrical transmission line (NLTL) with linear dispersion. With the help of the derived CNLS equations, we present and analyze explicit expressions for the instability growth rate of a purely growing modulational instability (MI). We establish that the phenomenon of the MI can be observed only for “small” nonzero modulation wavenumbers. Also, we point out the effects of the linear dispersive element, as well as of the frequencies of the signal packets, on the instability growth rate. It is shown that the linear dispersion and the frequencies of signal packets can be well used to control the instability domain. Through the CNLS equations, we analytically investigate the propagation of solitary waves in the network. Our analytical studies show four types of interaction of signal packets propagating in the network: bright–bright, dark–dark, bright–dark and dark–bright soliton interactions.

David J. Benney (1930–2015) was an applied mathematician and fluid dynamicist whose highly original work has shaped our understanding of nonlinear wave and instability processes in fluid flows. This article discusses the new paradigm he pioneered in the study of nonlinear phenomena, which transcends fluid mechanics, and it highlights the common threads of his research contributions, namely, resonant nonlinear wave interactions; the derivation of nonlinear evolution equations, including the celebrated nonlinear Schrödinger equation for modulated wave trains; and the significance of three-dimensional disturbances in shear flow instability and transition.

The Hamiltonian describing the interaction between the two neighbouring filaments of neuronal microtubules (nMTs) and in addition of u and ф model, which may be represented as longitudinal and radial displacements. From this Hamiltonian, we get discrete Nonlinear Schrodinger (DNLS) like equations by using the Rotating wave approximation (RWA). In order to analyze the instability nature of the system, we perform modulational instability (MI) analysis for the DNLS equations of motion which attribute for the energy localization phenomenon and it paves way for bio-energy transport in intracellular communications.

In this paper, we are interested in studying the modulational dynamics of interfacial waves rising buoyantly along a conduit of a viscous liquid. Formally, the behavior of modulated periodic waves on large space and time scales may be described through the use of Whitham modulation theory. The application of Whitham theory, however, is based on formal asymptotic (WKB) methods, thus removing a layer of rigor that would otherwise support their predictions. In this study, we aim at rigorously verifying the predictions of the Whitham theory, as it pertains to the modulational stability of periodic waves, in the context of the so-called conduit equation, a nonlinear dispersive PDE governing the evolution of the circular interface separating a light, viscous fluid rising buoyantly through a heavy, more viscous, miscible fluid at small Reynolds numbers. In particular, using rigorous spectral perturbation theory, we connect the predictions of Whitham theory to the rigorous spectral (in particular, modulational) stability of the underlying wave trains. This makes rigorous recent formal results on the conduit equation obtained by Maiden and Hoefer., 31 pages, 2 figures. Minor typos corrected

In this paper, we fnd new soliton solutions of the Sasa-Satsuma equation according to the Sardar sub-equation scheme. Diferent forms of soliton solutions, including the optical dark, optical bright, and optical singular function solutions are formally extracted. To better realize the physical phenomena of the gained solutions, some physical explanations of the exhibited solutions under suitable parameters for the physical values via the contour and 3D simulations are given. The confrontation between the dispersion and nonlinear terms has permitted survey the treatment of the Modulation Instability gain spectra.

This chapter discusses the nonlinearity of silica-based fiber which is quite small. Several nonlinear optical effects manifest themselves conspicuously owing to the fact that the density is very high because light is confined in a small cross-sectional area. The interaction length between the light wave and fiber material is quite long due to the low-loss property of fibers, and coherent interaction is possible since the modal field distribution and polarization are well prescribed and maintained over the long length. The chapter explains various nonlinear optical effects in fibers, such as optical solitons, stimulated Raman scattering, Brillouin scattering, and second-harmonic generation. The interesting and important nonlinear effects in optical fibers utilizing optical Kerr effect are optical solitons, optical pulse compression, and modulational instabilities. Modulational instability is a factor observed in a nonlinear dispersive medium in which the side-band component of the amplitude-modulated light grows exponentially when a certain condition is satisfied. When the perturbation is the amplitude modulation, then the phenomenon implies that the modulation depth grows exponentially. Therefore such a phenomenon is called the modulational instability.

The radial modes of the coronal loops are used to interpret the observed microwave and hard x-ray pulsations during flares. The properties of radial oscillations are well studied in the approximation of linear magnetohydrodynamics, and their further study requires a nonlinear approach. Obviously, the first step in this path is the weakly nonlinear approximation, which is used in this work. The study uses cylindrical geometry with the previously obtained nonlinear Schrodinger equation for the amplitude of radial oscillations. The conditions under which modulation instability of radial modes takes place, leading to the appearance of quasiperiodic oscillations, are studied.

Optical fiber communication system is one of the core supporting systems of the modern internet age. In this paper, under investigation is a variable-coefficient fourth-order nonlinear Schrodinger system, which describes the simultaneous propagation of the optical pulses in an inhomogeneous optical fiber. The first-order breathers are constructed. Breathers are converted into several types of the nonlinear waves, i.e., multi-peak solitons, antidark solitons, periodic waves and W-shaped solitons, under the transition condition. With the decrease of | λ r + a 2 | , the peak number of a multi-peak soliton increases, where λr represents the real part of the spectral parameter and a is a real constant. Peak number reaches the maximum since the multi-peak soliton is transformed into a set of the periodic waves with λ r + a 2 = 0 . Widths and velocities of the multi-peak solitons are related to the group velocity dispersion and fourth-order dispersion coefficients. We find that the condition of baseband modulational instability coincides with the transition condition when we use the rogue-wave eigenvalues, i.e., the transition between the rogue waves and multi-peak solitons can occur in the baseband modulational instability.

Domain wall motion is in the core of many information technologies ranging from storage [Beach et al., J. Magn. Magn. Mater. 320, 1272–1281 (2008)], processing [Tatara et al., Phys. Rep. 468, 213–301 (2008)], and sensing [Ralph and Stiles, J. Magn. Magn. Mater. 320, 1190–1216 (2008)] up to novel racetrack memory architectures [Parkin et al., Science 320, 190–194 (2008)]. The finding of magnetism in two-dimensional (2D) van der Waals (vdW) materials [Huang et al., Nature 546, 270 (2017); Gong et al., Nature 546, 265–269 (2017); Guguchia et al., Sci. Adv. 4, eaat3672 (2018); Klein et al., Science 360, 1218–1222 (2018)] has offered a new frontier for the exploration and understanding of domain walls at the limit of few atom-thick layers. However, to use 2D vdW magnets for building spintronics nanodevices such as domain-wall based logic [Allwood et al., Science 309, 1688–1692 (2005); Luo et al., Nature 579, 214–218 (2020); Xu et al., Nat. Nanotechnol. 3, 97–100 (2008)], it is required to gain control of their domain wall dynamics by external driving forces such as spin-polarized currents or magnetic fields, which have so far been elusive. Here, we show that electric currents as well as magnetic fields can efficiently move domain walls in the recently discovered 2D vdW magnets CrI3 and CrBr3 at low temperatures and robust down to monolayer. We realize field- and current-driven domain wall motion with velocities up to 1020 m s−1, which are comparable to the state-of-the-art materials for domain-wall based applications [Yang et al., Nat. Nanotechnol. 10, 221–226 (2015); Woo et al., Nat. Mater. 15, 501–506 (2016); Vélez et al., Nat. Commun. 10, 4750 (2019); Siddiqui et al., Phys. Rev. Lett. 121, 057701 (2018); Ryu et al., Nat. Nanotechnol. 8, 527–533 (2013)]. Domain walls keep their coherence driven by the spin-transfer torque induced by the current and magnetic fields up to large values of about [Formula: see text] A cm−2 and 5 T, respectively. For larger magnitudes of current or field, a transition to a hydrodynamic spin-liquid regime is observed with the emission of a periodic train of spin-wave solitons with modulational instability [Rabinovich and Trubetskov, Oscillations and Waves: In Linear and Nonlinear Systems, Mathematics and its Applications (Springer Netherlands, 2011)]. The emitted waveform achieves terahertz (THz) frequency in a wide range of fields and current densities, which opens up perspectives for reconfigurable magnonic devices. Moreover, we found that these spin-waves can transport spin angular momentum through the layers over distances as long as 10 μm without losses for the transport of spin information. Our results push the boundary of what is currently known about the dynamics of domain walls in 2D vdW ferromagnets and unveil strategies to design ultrathin, high-speed, and high-frequency spintronic devices.

We introduce a continuous one-dimensional non-Hermitian matrix gauge potential and study its effect on dynamics of a two-component field. The model is emulated by a system of evanescently coupled nonlinear waveguides with distributed gain and losses. The considered gauge fields lead to a variety of unusual physical phenomena in both linear and nonlinear regimes. In the linear regime, the field may undergo superexponential convective amplification. A total power of an input Gaussian beam may exhibit a finite-distance blowup, which manifests itself in absolute delocalization of the beam at a finite propagation distance, where the amplitude of the field remains finite. The defocusing Kerr nonlinearity initially enhances superexponential amplification, while at larger distances it suppresses the growth of the total power. The focusing nonlinearity at small distances slows down the power growth and eventually leads to the development of the modulational instability. Complex periodic gauge fields lead to the formation of families of stable fundamental and dipole solitons., Comment: accepted for Phys. Rev. A (Letters)

We review spectral theory of soliton gases in integrable dispersive hydrodynamic systems. We first present a phenomenological approach based on the consideration of phase shifts in pairwise soliton collisions and leading to the kinetic equation for a non-equilibrium soliton gas. Then a more detailed theory is presented in which soliton gas dynamics are modelled by a thermodynamic type limit of modulated finite-gap spectral solutions of the Korteweg-de Vries and the focusing nonlinear Schr\"odinger equations. For the focusing nonlinear Schr\"odinger equation the notions of soliton condensate and breather gas are introduced that are related to the phenomena of spontaneous modulational instability and the rogue wave formation. Integrability properties of the kinetic equation for soliton gas are discussed and some physically relevant solutions are presented and compared with direct numerical simulations of dispersive hydrodynamic systems., Comment: 59 pages, 18 figures

Stable light bullets and clusters of them are presented in the monostable regime using the mean-field Lugiato-Lefever equation [Gopalakrishnan, Panajotov, Taki, and Tlidi, Phys. Rev. Lett. 126, 153902 (2021)]. It is shown that three-dimensional (3D) dissipative structures occur in a strongly nonlinear regime where modulational instability is subcritical. We provide a detailed analysis on the formation of optical 3D crys-tals in both the super-and sub-critical modulational instability regimes, and we highlight their link to the formation of light bullets in diffractive and dispersive Kerr resonators. We construct bifurcation diagrams associated with the formation of optical crystals in both monostable and bistable regimes. An analytical study has predicted the predominance of body-centered-cubic (bcc) crystals in the intracavity field over a large variety of other 3D solutions with less symmetry. These results have been obtained using a weakly nonlinear analysis but have never been checked numerically. We shownumerically that indeed the most robust structures over other self-organized crystals are the bcc crystals. Finally, we show that light-bullets and clusters of them can occur also in a bistable regime. (C) 2021 Elsevier Ltd. All rights reserved.

We present an investigation of the fundamental physical processes involved in deep-water gravity wave breaking. Our motivation is to identify the underlying reason causing the deficiency of the eddy viscosity breaking model (EVBM) in predicting surface elevation for strongly nonlinear waves. Owing to the limitation of experimental methods in the provision of high-resolution flow information, we propose a numerical methodology by developing an EVBM enclosed standalone fully nonlinear quasi-potential (FNP) flow model and a coupled FNP plus Navier–Stokes flow model. The numerical models were firstly verified with a wave train subject to modulational instability, then used to simulate a series of broad-banded focusing wave trains under non-, moderate- and strong-breaking conditions. A systematic analysis was carried out to investigate the discrepancies of numerical solutions produced by the two models in surface elevation and other important physical properties. It is found that EVBM predicts accurately the energy dissipated by breaking and the amplitude spectrum of free waves in terms of magnitude, but fails to capture accurately breaking induced phase shifting. The shift of phase grows with breaking intensity and is especially strong for high-wavenumber components. This is identified as a cause of the upshift of the wave dispersion relation, which increases the frequencies of large-wavenumber components. Such a variation drives large-wavenumber components to propagate at nearly the same speed, which is significantly higher than the linear dispersion levels. This suppresses the instant dispersive spreading of harmonics after the focal point, prolonging the lifespan of focused waves and expanding their propagation space.

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 Schr\"odinger 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.

Separatrix crossing stands for the continuation of the modulation instability into the strong depleted regime and is responsible for the symmetry breaking nature of the recurrence phenomenon. The near-separatrix dynamic for recurrent modulational instability is of great importance because it is closely associated with the formation of rogue breather structures. Here, we have developed the underlying phase-space structure of nonlinear modulation instability by treating the three-mode truncate model as a simple oscillator. This allowed us to reveal the high sensitivity of the switching dynamics characterized by inner or outer trajectories to the initial condition around a separatrix. Our results present a major step forward towards the complete understanding of recurrent modulational instability in a truly conservative setting and can provide a guidance to experimentally measure the involved recursive behaviors.

We employ numerical methods to investigate the dissipative freak waves (FWs) and dissipative breathers (Bs) in collisional electronegative complex plasmas. The plasmas under investigation possess inertialess Maxwellian thermal electron and light negative ion in addition to stationary negatively charged dust grains. To achieve this goal, we reduce the fluid equation of the plasma species to the linear damped nonlinear Schrodinger equation (NLSE) via using the derivative expansion technique. The modulational instability (MI) is investigated and the (un)stable regions are determined precisely in order to study the dynamic of mechanism of the dissipative modulated structures. To study the effect of related plasma parameters on the behavior of the dissipative FWs and Bs, the linear damped NLSE is solved numerically using exponential time differencing method. In addition, the relative error is estimated to prove the accuracy of the proposed method. The impact of the plasma configuration parameters on the regions of the (un)stable envelope structures is discussed. Moreover, the dependence of the dissipative FWs and dissipative Bs profiles on the plasma configuration parameters is examined and reported.