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

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

On the chaotic pole of attraction for Hindmarsh-Rose neuron dynamics with external current input.

Since the neurologists Hindmarsh and Rose improved the Hodgkin-Huxley model to provide a better understanding on the diversity of neural response, features like pole of attraction unfolding complex bifurcation for the membrane potential was still a mystery. This work explores the possible existence of chaotic poles of attraction in the dynamics of Hindmarsh-Rose neurons with an external current input. Combining with fractional differentiation, the model is generalized with the introduction of an additional parameter, the non-integer order of the derivative σ, and solved numerically thanks to the Haar Wavelets. Numerical simulations of the membrane potential dynamics show that in the standard case where the control parameter σ=1, the nerve cell's behavior seems irregular with a pole of attraction generating a limit cycle. This irregularity accentuates as σ decreases (σ=0.9 and σ=0.85) with the pole of attraction becoming chaotic.

Base pairs opening and bubble transport in damped DNA dynamics with transport memory effects.

Transport memory effects on nonlinear wave propagation are addressed in a damped Peyrard-Bishop-Dauxois model of DNA dynamics. Under the continuum and overdamped limits, the multiple-scale expansion method is employed to show that an open-state configuration of the DNA molecule is described by a complex nonlinear Schrödinger equation. For the latter, solutions are proposed as bright solitons, which suitably represent the open-state configuration that takes place along the DNA molecule in the form of bubbles. A good agreement between numerical experiments and analytical predictions on the impact of memory effects on the angular frequency, velocity, width, and amplitude of the moving bubble is obtained. It also appears that memory effects can modify qualitatively and quantitatively the nonlinear dynamics of DNA, including the energy brought by enzymes for the initiation of the processes of replication and transcription.

Energy patterns in twist-opening models of DNA with solvent interactions.

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 Schrödinger 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.

Interplay between spin-orbit couplings and residual interatomic interactions in the modulational instability of two-component Bose-Einstein condensates.

The nonlinear dynamics induced by the modulation instability (MI) of a binary mixture in an atomic Bose-Einstein condensate (BEC) is investigated theoretically under the joint effects of higher-order residual nonlinearities and helicoidal spin-orbit (SO) coupling in a regime of unbalanced chemical potential. The analysis relies on a system of modified coupled Gross-Pitaevskii equations on which the linear stability analysis of plane-wave solutions is performed, from which an expression of the MI gain is obtained. A parametric analysis of regions of instability is carried out, where effects originating from the higher-order interactions and the helicoidal spin-orbit coupling are confronted under different combinations of the signs of the intra- and intercomponent interaction strengths. Direct numerical calculations on the generic model support our analytical predictions and show that the higher-order interspecies interaction and the SO coupling can balance each other suitably for stability to take place. Mainly, it is found that the residual nonlinearity preserves and reinforces the stability of miscible pairs of condensates with SO coupling. Additionally, when a miscible binary mixture of condensates with SO coupling is modulationally unstable, the presence of residual nonlinearity may help soften such instability. Our results finally suggest that MI-induced formation of stable solitons in mixtures of BECs with two-body attraction may be preserved by the residual nonlinearity even though the latter enhances the instability.

Modulational instability in nonlinear saturable media with competing nonlocal nonlinearity.

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

Modulation instability in helicoidal spin-orbit coupled open Bose-Einstein condensates.

We introduce a vector form of the cubic complex Ginzburg-Landau equation describing the dynamics of dissipative solitons in the two-component helicoidal spin-orbit coupled open Bose-Einstein condensates (BECs), where the addition of dissipative interactions is done through coupled rate equations. Furthermore, the standard linear stability analysis is used to investigate theoretically the stability of continuous-wave (cw) solutions and to obtain an expression for the modulational instability gain spectrum. Using direct simulations of the Fourier space, we numerically investigate the dynamics of the modulational instability in the presence of helicoidal spin-orbit coupling. Our numerical simulations confirm the theoretical predictions of the linear theory as well as the threshold for amplitude perturbations.

Modulation instability in nonlinear metamaterials modeled by a cubic-quintic complex Ginzburg-Landau equation beyond the slowly varying envelope approximation.

Considering the theory of electromagnetic waves from the Maxwell's equations, we introduce a (3+1)-dimensionsal cubic-quintic complex Ginzburg-Landau equation describing the dynamics of dissipative light bullets in nonlinear metamaterials. The model equation, which is derived beyond the slowly varying envelope approximation, includes the effects of diffraction, dispersion, loss, gain, cubic, and quintic nonlinearities, as well as cubic and quintic self-steepening effects. The modulational instability of the plane waves is studied both theoretically, using the linear stability analysis, and numerically, using direct simulations of the Fourier space of the proposed nonlinear wave equation, based on the Drude model. The linear theory predicts instability for any amplitude of the primary wave. Also, in the linear stability analysis, self-steepening effects of different orders are confronted and one discusses their effects on the behavior of the gain spectrum under both normal and anomalous group-velocity dispersion regimes. Analytical results are equally confronted to direct numerical simulations and fully agree with the predictions from the gain spectra. Modulational instability is manifested by clusters of solitons and multihump and dromion-like structures, whose emergence and features depend not only on system parameters, such as the cubic and quintic self-steepening coefficients, but also on the propagation distance under a suitable balance between nonlinear and dispersive effects.

Discrete instability in the DNA double helix.

Modulational instability (MI) is explored in the framework of the base-rotor model of DNA dynamics. We show, in fact, that the helicoidal coupling introduced in the spin model of DNA reduces the system to a modified discrete sine-Gordon (sG) equation. The MI criterion is thus modified and displays interesting features because of the helicoidal coupling. In the simulations, we have found that a train of pulses is generated when the lattice is subjected to MI, in agreement with analytical results obtained in a modified discrete sG equation. Also, the competitive effects of the harmonic longitudinal and helicoidal constants on the dynamics of the system are notably pointed out. In the same way, it is shown that MI can lead to energy localization which becomes high for some values of the helicoidal coupling constant.

Energy localization in an anharmonic twist-opening model of DNA dynamics.

Energy localization is investigated in the framework of the anharmonic twist-opening model proposed by Cocco and Monasson. This model includes the coupling between opening and twist that result from the helicoidal geometry of B-DNA. I first reduce the corresponding two-component model to its amplitude equations, which have the form of coupled discrete nonlinear Schrödinger (DNLS) equations, and I perform the linear stability analysis of the plane waves, solutions of the coupled DNLS equations. It is shown that the stability criterion deeply depends on the stiffness of the molecule. Numerical simulations are carried out in order to verify analytical predictions. It results that increasing the value of the molecule stiffness makes the energy patterns long-lived and highly localized. This can be used to explain the way enzymes concentrate energy on specific sequences of DNA for the base-pairs to be broken. The role of those enzymes could therefore be to increase the stiffness of closed regions of DNA at the boundaries of an open state.

Modulational instability of charge transport in the Peyrard-Bishop-Holstein model.

We report on modulational instability (MI) on a DNA charge transfer model known as the Peyrard-Bishop-Holstein (PBH) model. In the continuum approximation, the system reduces to a modified Klein-Gordon-Schrödinger (mKGS) system through which linear stability analysis is performed. This model shows some possibilities for the MI region and the study is carried out for some values of the nearest-neighbor transfer integral. Numerical simulations are then performed, which confirm analytical predictions and give rise to localized structure formation. We show how the spreading of charge deeply depends on the value of the charge-lattice-vibrational coupling.

Soliton-like excitation in a nonlinear model of DNA dynamics with viscosity.

The study of solitary wave solutions is of prime significance for nonlinear physical systems. The Peyrard-Bishop model for DNA dynamics is generalized specifically to include the difference among bases pairs and viscosity. The small amplitude dynamics of the model is studied analytically and reduced to a discrete complex Ginzburg-Landau (DCGL) equation. Exact solutions of the obtained wave equation are obtained by the mean of the extended Jacobian elliptic function approach. These amplitude solutions are made of bubble solitons. The propagation of a soliton-like excitation in a DNA is then investigated through numerical integration of the motion equations. We show that discreteness can drastically change the soliton shape. The impact of viscosity as well as elasticity on DNA dynamic is also presented. The profile of solitary wave structures as well as the energy which is initially evenly distributed over the lattice are displayed for some fixed parameters.