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We study the dynamic of modulated solitary wave patterns in alpha-helical proteins going beyond the standard nearest-neighbor interaction by taking into account influence of long-range dipole–dipole interactions of the power low type. In particular, self-trapping of amide-I energy quanta or exciton by the induced phonon deformation of the hydrogen-bonded lattice of peptide groups is capable of generating moving solitary waves following the Davydov soliton model. Under the adiabatic approximation, the discrete biexciton system is reduced to two coupled nonlinear Schrodinger equations where the dispersive coefficient depends on the long-range interactions parameter and the coupled term depends on the nonlinear saturation effects. We perform the linear stability analysis of continuous wave solutions of the coupled system. This analysis reveals that both nonlinear saturation and long-range interaction parameters affect the dispersion area. More new analytical multisoliton, bright and dark soliton solutions-like patterns are obtained using extended auxiliary equation method. We also observe the time behavior of modulating bright soliton-like pattern due to the nonlinear saturation and long-range interactions terms through numerical simulations.

We report on the existence and stability of Gaussian-type soliton in the nonlinear Schrodinger (NLS) equation with interplay of cubic–quintic nonlinearity, fourth-order diffraction (FOD) and novel quartic anharmonic parity-time ( $$\mathcal {PT}$$ )-symmetric Gaussian potential. We study numerically the impact of the FOD coefficient on the regions of unbroken/broken linear $$\mathcal {PT}$$ -symmetric phases. In the nonlinear domain, we derive exact soliton solutions of the one-dimensional and two-dimensional cubic–quintic NLS equation with $$\mathcal {PT}$$ -symmetric Gaussian potential and FOD coefficients. Moreover, the stability of the constructed soliton solution is investigated. The results of linear stability analysis are validated by comparison with numerical simulations. Furthermore, we also show that the relative strength of the FOD coefficient influences the direction of the power flow.

Nonlinear excitations of Ca2+ waves are investigated in a two-dimensional cell network with bidirectional paracrine signaling, both in the longitudinal and transversal directions. The semi-discrete...

The competitive effect between electric and magnetic flux couplings is used, in the context of modulational instability, to describe the collective dynamics in a modified Hindmarsh–Rose neural networks. The multiple-scale expansion is utilized to reduce the system to a nonlinear differential-difference equation, whose plane wave solutions are found to be unstable for some values of parameters. Particular interest is given to the influence of changing both the electric and magnetic coupling strengths, and confirmation of analytical results is given via numerical integration of the generic Hindmarsh–Rose model. The model presents a rich variety of spatiotemporal patterns propagating in the network, as the result of the interplay between nonlinear and dispersive effects. The electromagnetic induction appears to be responsible for regular bursting patterns and synchronous states in the network. With increasing the electric coupling, full synchronization is difficult to realize and irregular spatiotemporal patterns of action potentials are predominant.

In this paper, we study the dynamics of granular mediums. By comparing each grain of the medium with a ball, and the interaction between various balls like linear elastic interactions, we have established that the dynamics of the system of pendulum used by Fermi–Pasta–Ulam can be governed by a discrete nonlinear Schrodinger equation. Like second advanced, we also established under which conditions this system could be prone to a modulational instability. The long-time dynamics of modulated waves is examined using numerical simulations which are corroborated with analytical results leading to the generation of nonlinear modulated waves which have the shape of a soliton. At the end, we compare our results with what have been carried out previously.

In this paper, the existence and stability of the nonlinear spatial localized modes have been investigated for both self-focusing and self-defocusing in the nonlinear Schrodinger (NLS) equation with interplay of third-order dispersion (TOD), momentum term and complex parity-time $$(\mathcal {PT})$$ -symmetric Scarf potential. The impact of the TOD and momentum coefficients on the regions of unbroken/broken linear $$\mathcal {PT}$$ -symmetric phases has been studied numerically. For the nonlinear case, exact analytical expressions of the localized modes are obtained, respectively, in one- and two-dimensional nonlinear Schrodinger equation with TOD and momentum coefficients. The effects of both TOD and momentum term on the stability/instability structure of these localized modes have also been discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. It was found that the relative strength of the TOD and momentum coefficients can utterly change the direction of the power flow which may be used to control the energy exchange among gain or loss regions.

This paper explores the higher-order nonlinear Schrodinger equation with dual power-law nonlinearities and external potential that can depend on space and time coordinates. We constructed exact sol...

In this paper, the existence and stability of solitons in parity-time ( PT )-symmetric optical media characterized by a generic complex hyperbolic refractive index distribution with fourth-order diffraction (FOD) coefficient and higher-order nonlinearities have been investigated. For the linear case, we have demonstrated numerically that, the FOD parameter can alter the PT -breaking points. Exact analytical expressions of the localized modes are obtained respectively, in one and two dimensional nonlinear Schrodinger (NLS) equation with both self-focusing and self-defocusing Kerr, and higher-order nonlinearities for nonlinear case. The effects of both FOD and higher-order nonlinearities on the stability/instability structure of these localized modes have also been discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. Some stable and unstable solutions have been given and, it has been seen how higher-order self-focusing and self-defocusing nonlinearities can influence the stability/instability of the system.

A theoretical investigation on the impact of high-order nonlinear dispersions of the modulation instability (MI) spectra in the three core triangular oppositely directed coupler with negative index material channel is carried out. A keen attention was paid to find out the influence of self-steepening (SS), intrapulse Raman scattering ( T R ), second-order nonlinear dispersion (SOND) as well as saturable nonlinearity (SNL) on the MI spectra. Gain spectra has been investigated for both anomalous group velocity dispersion (GVD) and normal GVD regime. Particularly, our results show that in the normal GVD regime, the instability gain exists even if the perturbation frequency ( Ω ) is zero; and the instability gain at Ω = 0 is nil when the dispersion is anomalous. We also find that the magnitude and sign of SOND exert strong influences on MI sideband. Furthermore, by adjusting various parameters such as SS, intrapulse Raman scattering, and SOND, we observe the occurrence of new instability regions. The MI sidelobes, as well as its bandwidth, have also been affected by the SNL. Finally, the SNL may influence considerably the number of wave trains induced by MI. These results can help to understand the generation of soliton-like excitation in nonlinear three-core oppositely directed couplers with a particular attention on a negative-index material channel.

Two electrically coupled Hindmarsh–Rose neural networks are considered, each including power-law long-range dispersive interactions. The whole dynamics of the system is reduced to a set of two coupled complex Ginzburg–Landau equations. The linear stability analysis of the plane wave solutions brings about the existence of two dynamical regimes that predict direct and indirect synchronization of the two networks, under the activation of modulational instability. The conditions for the latter to develop are discussed and used to observe numerically the synchronized longtime dynamics of action potentials, under the effect of both long-range intra-coupling and electrical inter-coupling parameters. Mainly, the synchronization criterion depends on the plane wave amplitudes and for some of their values, perfect and partial inter-network synchronization phenomena are observed. It is also found that indirect synchronization between adjacent networks requires local synchronization among neurons of the same fiber. This is discussed based on some further formulation of the synchronization error, additionally to the time series of action potentials. Some spatiotemporal behaviors of the corresponding bursts of spikes are also discussed using coupling parameters.

A nonlinear model of blood flow in large vessels is addressed. The influence of radiations, viscosity and uniform magnetic fields on velocity and temperature distribution waveforms is studied. Exact solutions for the studied model are investigated through the F − expansion method. Based on the choice of parameter values, single-, multi-soliton and Jacobi elliptic function solutions are obtained. Viscosity and permanent magnetic field bring about wave spreading and reduce the velocity of blood, while radiations have reversed effects with strong impact on the waveform frequency of both the velocity and temperature distribution.

The contribution of both longitudinal and transversal nonlinear oscillations to energy localization is investigated in a zigzag molecular chain, which include simultaneously nearest- and next-nearest neighbor interactions. Coupled amplitude equations are found in the form of discrete nonlinear Schrodinger equations, whose plane wave solutions are found to be subjected to some instabilities. They are shown to be very sensitive to transverse and longitudinal couplings, which is confirmed via direct numerical simulations. The two available modes are found to be alternatively responsible for energy localization and transport. Thermal fluctuations effects bring about highly localized modes, along with narrow structures for efficient energy transport.

Long-range diffusive effects are included in a discrete Hindmarsh–Rose neural network. Their impact on the emergence of nonlinear patterns is investigated via the modulational instability. The whole system is first shown to fully reduce to a single nonlinear differential-difference equation, which has plane wave solutions. The stability of such solutions is investigated and regions of instability are found to be importantly influenced by long-range parameters. The analytical results are confirmed through direct numerical simulations, where scattered and chaotic patterns illustrate the long-range effect. Synchronized states are described by quasi-periodic patterns for nearest-neighbor coupling. The external stimulus is also shown to efficiently control strong long-range effects via more regular spatiotemporal patterns.

The dynamics of coupled nonlinear waves is addressed in the framework of the angular model of microtubules. The semi-discrete approximation is used to write the dynamics of the lower and upper cutoff modes in the form of coupled nonlinear Schrodinger equations. The linear stability analysis of modulational instability is used to confirm the existence of soliton solutions, and the growth-rate of instability is shown to be importantly influenced by the dipolar energy. Single mode solutions are found as breathers and resonant kink, while the coupled mode introduces a kink envelope solution, whose characteristics are discussed with respect to the dipolar energy. The found solution is shown to be robust, which is important for energy transport in the Polymerization/depolymerization mechanism of protofilaments.

We theoretically investigate modulation instability (MI) in a two-core nonlinear oppositely directed coupler with a negative-index material channel in the presence of a non-kerr-like nonlinear refractive-index change. Using linear stability analysis, we obtain an expression for the instability gain. It is found that the MI in the nonlinear two-core oppositely directed coupler is significantly influenced by the ratio of the forward- to backward-propagating wave power, nonlinearity, power and saturation. In the normal dispersion regime, the septic nonlinearity is found to suppress the MI by reducing both the gain and width of the instability nonlinearity. But in anomalous dispersion regime, the situation is different because septic nonlinearity enhances the MI by increasing both the gain and width of MI. The increase of septic nonlinearity both in normal and anomalous dispersion regimes increases the gain of the instability nonlinearity. Thus, we report new ways to generate and manipulate the MI and solitons in two-core oppositely directed couplers with a particular emphasis on a negative-index material (NIM) channel in presence of septic nonlinearity.

Analytically, we study the dynamics of ionic waves in a microtubule modeled by a nonlinear resistor, inductor, and capacitor (RLC) transmission line. We show through the application of a reductive perturbation technique that the network can be reduced in the continuum limit to the dissipative nonlinear Schrodinger equation. The processes of the modulational instability are studied and, motivated with a solitary wave type of solution to the nonlinear Schrodinger (NLS) equation, we use the direct method and the Weierstrass's elliptic function method to present classes of solitary wavelike solutions to the dissipative NLS equation of the network. The results suggest that microtubules are the biological structures where short-duration nonlinear waves called electrical envelope solitons can be created and propagated. This work presents a good analytical approach of investigating the propagation of solitary waves through a microtubule modeled by a nonlinear RLC transmission line.

In the present study, the dynamics of nanobeam resting on fractional order softening nonlinear viscoelastic pasternack foundations is studied. The Hamilton principle is used to derive the nonlinear equation of the motion. Approximate analytical solution is obtained by applying the standard averaging method. The Melnikov method is used to investigate the chaotic behaviors of device, the critical curve separating the chaotic and non-chaotic regions are found. It is shown that the distance between chaotic region and non-chaotic region in this kind of structure depends strongly on the fractional order parameter.

The nonlinear localization phenomena in ferromagnetic spin lattices have attracted a steadily growing interest and their existence has been predicted in a wide range of physical settings. We investigate the onset of modulational instability of a plane wave in a discrete ferromagnetic spin chain with physically significant higher order dispersive octupole-dipole and dipole-dipole interactions. We derive the discrete nonlinear equation of motion with the aid of Holstein-Primakoff (H-P) transformation combined with Glauber's coherent state representation. We show that the discrete ferromagnetic spin dynamics is governed by an entirely new discrete NLS model with complex coefficients not reported so far. We report the study of modulational instability (MI) of the ferromagnetic chain with long range dispersive interactions both analytically in the frame work of linear stability analysis and numerically by means of molecular dynamics (MD) simulations. Our numerical simulations explore that the analytical predictions correctly describe the onset of instability. It is found that the presence of the various exchange and dispersive higher order interactions systematically favors the local gathering of excitations and thus supports the growth of high amplitude, long-lived discrete breather (DB) excitations. We analytically compute the strongly localized odd and even modes. Further, we employ the Jacobi elliptic function method to solve the nonlinear evolution equation and an exact propagating bubble-soliton solution is explored. (C) 2015 Elsevier B.V. All rights reserved.

We consider a nonlinear oppositely direct coupler with a negative-index metamaterial channel and a modified Kerr-type saturable nonlinearity (MSN). We presented an analytical expression for MI gain. Special attention is paid to investigation of the influence of system parameters such as the forward to backward-propagating wave's, the power, nonlinear parameters and nonlinear saturable parameter on the MI. The MSN behaves in a perceptible manner such that the gain and the unstable region inherently decrease. In the normal and anomalous dispersion region, the gain of the instability spectrum increases (decreases) monotonously with increases (decreases) of saturable nonlinearity. In other term, when the saturable parameter or the input power increases, the MI peak gain and the gain band-width may increase and then decrease.

We consider a nonlinear left-handed transmission line that incorporates an array of Josephson junctions in its periodic lattice structure. We show that the system dynamics is described by a discrete sine-Gordon-like equation, where the left-handedness of the lattice manifests in the form of a non-standard second-time- derivative term. Since this modified discrete sine-Gordon equation has not yet been extensively studied in the literature, this paper opens up the possibility of additional mathematical analysis. It is also intriguing that by means of a semi-discrete approximation we can derive a nonlinear Schrodinger equation and thus show that the system supports both bright and dark envelope soliton solutions depending on the choice of carrier frequency. The left-handedness of the network is explicitly confirmed in numerical simulations which demonstrate the backward propagation of the bright and dark soliton, in good agreement with analytical predictions.

With the help of the Boussinesq perturbation expansion, a new basic equation describing the long, small-amplitude, unidirectional wave motion in shallow water with surface tension is derived to fourth order, namely a higher-order Korteweg–de Vries (KdV) equation. The procedure for deriving this equation assumes that the relation between the small parameter $$\alpha $$ , which measures the ratio of wave amplitude to undisturbed fluid depth, and the small parameter $$\beta $$ , which measures the square of the ratio of fluid depth to wave length, is taken in the form $$\beta = 0(\alpha ) = \varepsilon $$ , where $$\varepsilon $$ is a small, dimensionless parameter which is the order of the amplitude of the motion. Hirota’s bilinear method is used to investigate one- and two-soliton solutions for this new higher-order KdV equation.

In the present study, the nonlinear vibration of a nanobeam resting on the fractional order viscoelastic Winkler&ndash, Pasternak foundation is studied using nonlocal elasticity theory. The D&rsquo, Alembert principle is used to derive the governing equation and the associated boundary conditions. The approximate analytical solution is obtained by applying the multiple scales method. A detailed parametric study is conducted, and the effects of the variation of different parameters belonging to the application problems on the system are calculated numerically and depicted. We remark that the order and the coefficient of the fractional derivative have a significant effect on the natural frequency and the amplitude of vibrations.

We investigate the existence and stability of solitons in parity-time ($\mathcal{PT}$)-symmetric optical media characterized by a generic complex hyperbolic refractive index distribution and fourth-order diffraction (FOD). For the linear case, we demonstrate numerically that the FOD parameter can alter the $\mathcal{PT}$-breaking points. For nonlinear cases, the exact analytical expressions of the localized modes are obtained both in one- and two-dimensional nonlinear Schr\"odinger equations with self-focusing and self-defocusing Kerr nonlinearity. The effect of FOD on the stability structure of these localized modes is discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. Examples of stable and unstable solutions are given. The transverse power flow density associated with these localized modes is also discussed. It is found that the relative strength of the FOD coefficient can utterly change the direction of the power flow, which may be used to control the energy exchange among gain or loss regions.

We report the modulational instability (MI) analysis for the modulation equations governing the propagation of modulated waves in a practical left-handed nonlinear transmission lines with series of nonlinear capacitance. Considering the voltage in the spectral domain and the Taylor series around a certain modulation frequency, we show in the continuum limit, that the dynamics of localized signals is described by a nonlinear Schrodinger equation with a cubic–quintic nonlinear terms. The MI process is then examined and we derive the gain spectra of MI for the generation of solitonlike-object in the transmission line metamaterials. We emphasize on the effect of losses on the MI gain spectra. An exact kink-darklike solutions is derived through the auxiliary equation method. It comes out that the width of the darklike solution decreases as the attenuation constant increases. Our theoretical solution is in good agreement with our numerical observation.

The goal of this work is to investigate, analytically and numerically, the dynamics of gravity water waves with the effects of the small surface tension and the bottom topography taken into account. Using a third-order perturbative approach of the Boussinesq equation, we obtain a new third-order perturbed Korteweg-de Vries (KdV) equation which includes nonlinear, dispersive, nonlocal and mixed nonlinear-dispersive terms, describing shallow water waves with a non-flat bottom and the surface tension. We show by numerical simulations, for various bottom shapes, that this new third-order perturbed KdV equation can support the propagation of solitary waves, whose profiles strongly depend on the surface tension. In particular, we show that the instability observed in the numerical simulation can be suppressed by the inclusion of small surface tension.

We consider a twisted Peyrard-Bishop-Dauxois (PBD) model and construct the exact analytical solutions, which can describe the propagation of solitary waves by invoking a discrete Jacobian elliptic function method. These solutions include the Jacobian periodic solution as well as bubble solitons. Through the Fourier series approach, we have found that the DNA dynamics is governed by a modified discrete nonlinear Schrodinger (MDNLS) equation. A detailed analysis of the role of the twisted angle in the process of bio energy localization is presented in the form of coherent localized breather modes in a PBD model. A linear stability analysis is performed and we obtain that the stability of the solutions also depends on the twisted angle.

We are motivated by recent studies in medium formed by two tunnel-coupled waveguides. One of the waveguides is manufactured from an ordinary dielectric, while the second has negative refraction. We present an investigation of the gain spectrum permitting modulation instability in the nonlinear optical coupler with a negative-index metamaterial channel whose non-linear response includes third- and fifth-order terms. The principal motivation for our analysis stems from the impact of the inevitable presence of the effective cubic–quintic nonlinearity. We emphasize the influence of higher order nonlinear terms, over the MI phenomena, and the outcome of its development achieved by using linear stability analysis. Gain spectrum investigation has been carried out for both anomalous and normal dispersion regime in the focusing and defocusing cases of nonlinearity and near-zero dispersion regime where higher order linear dispersive effects emerge. Our results show that the MI gain spectra consist of multiple spect...

Modulational instability (MI) in negative refractive metamaterials with saturable nonlinearity, fourth-order dispersion (FOD), and second-order nonlinear dispersion (SOND) is investigated by using standard linear stability analysis and the Drude electromagnetic model. The expression for the MI gain spectrum is obtained, which clearly reveals the influence of the saturation of the nonlinearity, FOD, and SOND parameters on the temporal MI. The evolution of the MI in negative refractive metamaterials is numerically investigated. Special attention is paid to study the effects of the higher-order dispersion terms on the formation and evolution of the solitons induced by MI. It is shown that as the third-order dispersion term increases, the solitons travel toward the right. Moreover, the magnitude of the FOD term influences considerably the number of wave trains induced by MI.

A generalized dissipative discrete complex Ginzburg-Landau equation that governs the wave propagation in dissipative discrete nonlinear electrical transmission line with negative nonlinear resistance is derived. This equation presents arbitrarily nearest-neighbor nonlinearities. We analyze the properties of such model both in connection to their modulational stability, as well as in regard to the generation of intrinsic localized modes. We present a generalized discrete Lange-Newell criterion. Numerical simulations are performed and we show that discrete breathers are generated through modulational instability.

The parametric modulational instability for a discrete nonlinear Schrodinger equation with a cubic–quintic nonlinearity is analyzed. This model describes the dynamics of BECs, with both two- and three-body interatomic interactions trapped in an optical lattice. We identify and discuss the salient features of the three-body interaction in the parametric modulational instability. It is shown that the three-body interaction term can both, shift as well as narrow the window of parametric instability, and also change the behavior of a modulationally stable and parametrically unstable BEC with attractive two-body interaction. We explore this instability through the multiple-scale analysis and identify it numerically. The effect of the three body losses have also been investigated.

We investigate exact soliton solutions for the discrete nonlinear electrical transmission line by performing the simplest equation method from a trivial seed solution. Starting from the nonlinear propagation of signals in electrical transmission lines, we derive exact traveling kink and antikink solitary wave solutions. It is shown that under a safe range of parameter, the shape of kink soliton can be controlled well by adjusting the parameter of the line. The analytical solutions for the kink and antikink solitary waves are tested in direct simulations.

We present a systematic analysis of the effects, of higher-order dispersion, noninstantaneous nonlinear response, as well as stochastic coefficients in optical fiber. This study is motivated by recent experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber. Analytical expression of pulse amplitude is deduced with the second-order gain nonuniformity and the stimulated-Brillouin scattering-induced third-order as well as fourth-order dispersion effects involved. The influence of stochasticity, as well as the delayed Raman response in the nonconventional sidebands obtained due to the fourth-order dispersion, is considered. We note that the shape of the spectrum, and in particular the relative intensities of the higher order harmonics, is highly sensitive to the initial presence of classical noise, and can therefore be taken as a signature that the MI is seeded by vacuum fluctuations. Some direct simulation...

We investigate analytically and numerically the modulational instability (MI) in optical fiber, where the effect of noninstantaneous nonlinear response as well as stochastic coefficients are taken into account. Applying the linear stability analysis and stochastic calculus, we show that the MI gain spectrum reads as the maximal eigenvalue of a constant matrix. In the limiting cases of small fluctuations, we give explicit expressions for the MI gain spectra. In the general configurations, we derive an explicit form of the effective matrix and compute numerically the maximal eigenvalue. The moment MI peak is enhanced and the delayed Raman response reduces the maximum MI gain caused by stochasticity both in anomalous and normal dispersion regimes. Numerical simulations of the full stochastic nonlinear Schrodinger equation show that, the phenomenon of MI gives rise to periodic pulse arrays of waves train, as well as to a chain of peaks with continuously growing amplitudes.

We consider the inhomogeneous (1+1)-dimensional coupled nonlinear Schrodinger equations from the integrability point of view. This system has potential applications in inhomogeneous birefringent fiber media or in optical communication links with variable parameters. The exact analytical multisoliton solutions are obtained by employing the simple, straightforward Darboux transformation based on the obtained 3 × 3 Lax pair. Some main propagation and interaction properties of the one- and two-soliton solutions are discussed simultaneously. Also, the interaction between two neighboring combined solitary waves is numerically discussed. Excellent agreement with analytical results are found in the case of no collision. Moreover, the stability of these solutions are discussed in detail numerically with respect to a finite perturbation.

We study the nonlinear dynamics of DNA which takes into account the twist-opening interactions due to the helicoidal molecular geometry. The small amplitude dynamics of the model is shown to be governed by a solution of a set of coupled nonlinear Schrodinger equations. We analyze the modulational instability and solitary wave solution in the case. On the basis of this system, we present the condition for modulation instability occurrence and attention is paid to the impact of the backbone elastic constant K. It is shown that high values of K extend the instability region. Through the Jacobian elliptic function method, we derive a set of exact solutions of the twist-opening model of DNA. These solutions include, Jacobian periodic solution as well as kink and kink-bubble solitons.

Through the multiple scale expansion, we show that the dynamics of modulated waves in a two-component helicoidal one-dimensional model is governed by the nonlinear Schrodinger (NLS) equation. We study the occurrence of patterns formation in such a system. The impact of the helicoidal coupling is also studied which reveals that high values of the helicoidal coupling constant contribute to regulate information and charge transport throughout the two-chain model. This is made possible due to the superposition of optical and acoustic modes. In addition, the model is used to investigate internal dynamics of DNA. It is shown that the opening of the double helix of the DNA itself is controlled by the resonance mode, where the unzipping of the double helix has been illustrated by numerical simulations. Furthermore, the shape of the eyelike shape previously observed in experiments of thermal denaturation of DNA is numerically obtained.

The dynamics of nonlinear pulse propagation in an average dispersion-managed soliton system is governed by a constant coefficient nonlinear Schrodinger (NLS) equation. For a special set of parameters the constant coefficient NLS equation is completely integrable. The same constant coefficient NLS equation is also applicable to optical fiber systems with phase modulation or pulse compression. We also investigate MI arising in the cubic–quintic nonlinear Schrodinger equation for ultrashort pulse propagation. Within this framework, we derive ordinary differential equations (ODE’s) for the time evolution of the amplitude and phase of modulation perturbations. Analyzing the ensuing ODE’s, we derive the classical modulational instability criterion and identify it numerically. We show that the quintic nonlinearity can be essential for the stability of solutions. The evolutions of modulational instability are numerically investigated and the effects of the quintic nonlinearity on the evolutions are examined. Numerical simulations demonstrate the validity of the analytical predictions.

The study of solitary wave solutions is of prime significance for the nonlinear Schrodinger equation with higher order dispersion and/or higher degree nonlinearities in nonlinear physical systems. We derive the discrete cubic–quintic nonlinear Schrodinger equation from a Hamiltonian using different Poisson brackets. By using the extended Jacobian elliptic function approach, we investigate the abundant exact stationary solitons and periodic waves solution of this equation. These solutions include, Jacobian periodic solutions, alternating phase Jacobi periodic solution, kink and bubble soliton solutions, alternating phase kink soliton solution and alternating phase bubble soliton solution, provided that coefficients are bound by special relation. And then with the aid of symbolic computation, we present in explicit form these solutions. The stability of bubble and kink soliton as well as alternating kink and alternating bubble soliton are also investigated.

We investigate analytically and numerically the modulational instability (MI) in a nonlinear optical fiber. We use a generalized model describing the pulse propagation of waveguiding structure composed of two adjacent waveguides, where the effect of nonlocal nonlinear response as well as stochastic coefficients are taken into account. Applying the linear stability analysis and stochastic calculus, we show that the MI gain spectra reads as the maximal eigenvalue of a constant matrix. The generic properties of the MI gain spectra are then demonstrated for the rectangular response function. We observe that random inhomogeneities extend the domain of the homogeneous MI gain spectra over the whole spectrum of modulation, and the nonlocality parameter reduces drastically the growth rate and bandwidth of instability caused by stochasticity both in anomalous and normal dispersion regimes. We observe also that MI does not appears for all values of the nonlocal parameter. Numerical simulations of the full stochastic system of nonlinear Schrodinger equations describing the dynamics of the waves are carried out and lead to the generation of a train of pulses.

We exclusively analyze the condition for modulated waves to emerge in two ephaptically coupled nerve fibers. Through the multiple scale expansion, it is shown that a set of coupled cable-like Hodgkin-Huxley equations can be reduced to a single differential-difference nonlinear equation. The standard approach of linear stability analysis of a plane wave is used to predict regions of parameters where nonlinear structures can be observed. Instability features are shown to be importantly controlled not only by the ephaptic coupling parameter, but also by the discreteness parameter. Numerical simulations, to verify our analytical predictions, are performed, and we explore the longtime dynamics of slightly perturbed plane waves in the coupled nerve fibers. On initially exciting only one fiber, quasi-perfect interneuronal communication is discussed along with the possibility of recruiting damaged or non-myelinated nerve fibers, by myelinated ones, into conduction.

We report in this paper a detailed analysis of modulational instability of a plane wave solution of discrete dissipative structures. A generalized Ablowitz–Ladik (AL) equation with a cubic and quintic nonlinearity is then considered, and analysed via the full linear stability analysis of the nonlinear plane wave solution. We derive analytical expressions for the domain of existence as well as the gain of modulational instability of moving plane waves. In particular, we find that discreteness drastically modifies the stability condition as well as the quintic nonlinearity. The analytical and numerical stability analyses are fully supported by an extensive series of numerical simulations of the full model. The generation of pulse trains with high repetition rate predicted by analytical study is then exhibited.

We studied the modulational instability in negative index materials with higher-order effects. An expression for the instability gain is derived through the linear stability analysis. We identify and discuss the salient features of the effect, like fourth-order dispersion, self-steepening and saturable nonlinearity on the modulational instability. The results show that the instability gain depends on the sign of the higher-order dispersion and, for anomalous dispersion, the instability regime is constituted of two parts: the conventional gain spectrum and the secondary gain spectrum induced by the fourth-order dispersion. For large values of the fourth-order dispersion, the secondary instability region spectrum vanishes and merges with the ordinary gain. In the normal dispersion regime, an efficient control of the modulation instability is successfully demonstrated by adjusting the fourth-order dispersion, depending on the presence of the self-steepening parameter or the normalized input power.

We consider the stability of a continuous wave solution in line of vortices. The weakly nonlinear dynamics are governed by a one-dimensional (1D) discrete complex cubic Ginzburg–Landau (DCCGL) equation. In particular, within the framework of linear stability analysis, we show how the Lange and Newell criterion for the Benjamin–Feir instability is modified by the lattice effects as well as the second neighbour coupling. A noteworthy feature of the present study is that the initial condition is not disintegrated into trains of unstable patterns which have the shape of solitons in the absence of second neighbour couplings and quintic term. The simulations also show that the main observed physical signatures of this instability are a variety of propagative patterns for short-time scales and chaotic states for long-time scales where the linear perturbation analysis fails.

Energy localization, via modulation instability, is addressed in a modified twist-opening model of DNA with solvent interactions. The Fourier expansion method is used to reduce the complex roto-torsional equations of the system to a set of discrete coupled nonlinear Schrodinger equations, which are used to perform the analytical investigation of modulation instability. We find that the instability criterion is highly influenced by the solvent parameters. Direct numerical simulations, performed on the generic model, further confirm our analytical predictions, as solvent interactions bring about highly localized energy patterns. These patterns are also shown to be robust under thermal fluctuations.

We investigate the dynamical instability of Bose-Einstein condensates (BECs) with higher-order interactions immersed in an optical lattice with weak driving harmonic potential. For this, we compute both analytically and numerically a modified Gross-Pitaevskii equation with higher-order nonlinearity and external potentials generated by magnetic and optical fields. Using the time-dependent variational approach, we derive the ordinary differential equations for the time evolution of the amplitude and phase of modulational perturbation. Through an effective potential, we obtain the modulational instability condition of BECs and discuss the effect of the higher-order interaction in the dynamics of the condensates in presence of optical potential. We perform direct numerical simulations to support our analytical results, and good agreement is found.

Realistic models of α-helix proteins are composed of three covalently-bonded strands, each of which is made of hydrogen-bonded peptide units. The modulational instability analysis of such complex molecular systems is carried out in this work. We show that the exciton-vibron coupling parameter contributes to the explosion and expansion of instability regions. The right choice of the modulational instability parameters leads to the formation of excitonic modulated pulse-like structures. It is argued that covalent bonds are compressed during the process of energy transport, while hydrogen bond oscillations display regular trains of breather-like objects. We also argue that the probable way of energy transport, from modulational instability, is through hydrogen bonds.

Modulational instability in a biexciton molecular chain is addressed. We show that the model can be reduced to a set of three coupled equations: two nonlinear Schrodinger equations and a Boussinesq equation. The linear stability analysis of continuous wave solutions of the coupled systems is performed and the growth rate of instability is found numerically. Simulations of the full discrete systems reveal some behaviors of modulational instability, since wave patterns are observed for the excitons and the phonon spectrum. We also take the effect of thermal fluctuations into account and we numerically study both the stability and the instability of the plane waves under 300 K. The plane wave is found to be stable under modulation, but displays a gradual increase of the wave amplitudes. Under modulation, the same behaviors are observed and wave patterns are found to resist thermal fluctuations, which is in agreement with earlier research on localized structure stability under thermal noise.

Considering the higher-order nonlinearities in a material can significantly change its behavior. We suggest the extended nonlinear Schrödinger equation to describe the propagation of ultrashort optical pulses through a dispersive medium with higher-order nonlinearities. Soliton trains are generated through the modulational instability and we point out the influence of the septic nonlinearity in the modulational instability gain. Experimental values are used for the numerical simulations and the input plane wave leads to the development of pulse trains, depending upon the sign of the septic nonlinearity.

We consider a higher-order complex Ginzburg-Landau equation, with the fourth-order dispersion and cubic-quintic nonlinear terms, which can describe the propagation of an ultrashort subpicosecond or femtosecond optical pulse in an optical fiber system. We investigate the modulational instability (MI) of continuous wave solution of this equation. Several types of modulational instability gains are shown to exist in both the anomalous and normal dispersion regimes. We find that depending on the sign of the fourth-order dispersion coefficient, the MI appears for normal and anomalous dispersion regime. Simulations of the full system demonstrate that the development of the MI leads to establishment of a regular or chaotic array of pulses, a chain of well-separated peaks with continuously growing or decaying amplitudes depending on the sign of the loss/gain coefficient and higher-order dispersions terms. Comparison of the calculations with reported numerical results shows a satisfactory agreement.