Alidou Mohamadou

Modulation instability in nonlinear positive–negative index couplers with saturable nonlinearity

We have investigated the modulation instability (MI) in a nonlinear optical coupler using a generalized model describing the pulse propagation of a waveguiding structure composed of two adjacent waveguides. The model consists of a nonlinear tunnel-coupled structure consisting of right- and left-handed media. The optical coupler considered here includes a local saturable nonlinear refractive index. In particular, we discuss the impact of the saturable nonlinearity for the MI of plane waves and formation of spatial solitons. The results show that MI can exist not only in the normal group velocity dispersion (GVD) regime but also in the normal GVD regime in the nonlinear positive-negative index coupler in the presence of saturable nonlinearity. The saturable nonlinearity can increase/decrease the number of sidebands or shift the existing sidebands. The maximum value of the MI gain, as well as its bandwidth, has been also affected by the saturable nonlinearity. Moreover, the saturable nonlinearity may influence considerably the number of wave trains induced by MI.

Modulated pressure waves in large elastic tubes.

Modulational instability is the direct way for the emergence of wave patterns and localized structures in nonlinear systems. We show in this work that it can be explored in the framework of blood flow models. The whole modified Navier-Stokes equations are reduced to a difference-differential amplitude equation. The modulational instability criterion is therefore derived from the latter, and unstable patterns occurrence is discussed on the basis of the nonlinear parameter model of the vessel. It is found that the critical amplitude is an increasing function of α, whereas the region of instability expands. The subsequent modulated pressure waves are obtained through numerical simulations, in agreement with our analytical expectations. Different classes of modulated pressure waves are obtained, and their close relationship with Mayer waves is discussed.

A variational approach to the modulational instability of a Bose–Einstein condensate in a parabolic trap

By means of the time-dependent variational approach, we study the modulational instability of Bose–Einstein condensates (BECs), with both two- and three-body interatomic interactions, trapped in an external parabolic potential. Within this framework, we derive and analyse ordinary differential equations for the explicit time evolution of the amplitude and phase of modulational perturbation. The effect of the trapping coefficients as well as the quintic nonlinear interaction on the dynamics of the BEC are examined. Numerical simulations are carried out in order to support our theoretical findings.

Soliton excitation in the DNA double helix

We study the nonlinear dynamics of the DNA double-helical chain using the Peyrard–Bishop–Dauxois (PBD) model. By using the Fourier series approach, we have found that the DNA dynamics in this case is governed by the modified discrete nonlinear Schrodinger (MDNLS) equation. Through the Jacobian elliptic function method, we investigate a set of exact solutions of this model. These solutions include the Jacobian periodic solution as well as bubble solitons. The stability of these solutions is also studied.

Formation of localized structures in the Peyrard–Bishop–Dauxois model

We explore in detail the properties of modulational instability (MI) and the generation of soliton-like excitations in DNA nucleotides. Based on the Peyrard–Bishop–Dauxois (PBD) model of DNA dynamics, which takes into account the interaction with neighbors in the structure, we derive through the semidiscrete approximation a modified discrete nonlinear Schrodinger (MDNLS) equation. From this equation, we predict the condition for the propagation of modulated waves through the system. To verify the validity of these results we have carried out numerical simulations of the PBD model and the initial conditions in the form of planar waves whose modulated amplitudes are given by the examples studied in the MDNLS equation. In the simulations we have found that a train of pulses are generated when the lattice is subjected to MI, in agreement with the analytical results obtained in an MDNLS equation. Also, the effects of the harmonic longitudinal and helicoidal constants on the dynamics of the system are notably pointed out. The process of energy localization from a nonsoliton initial condition is also explored.

Nonlinear charge transport in the helicoidal DNA molecule

Charge transport in the twist-opening model of DNA is explored via the modulational instability of a plane wave. The dynamics of charge is shown to be governed, in the adiabatic approximation, by a modified discrete nonlinear Schrödinger equation with next-nearest neighbor interactions. The linear stability analysis is performed on the latter and manifestations of the modulational instability are discussed according to the value of the parameter α, which measures hopping interaction correction. In so doing, increasing α leads to a reduction of the instability domain and, therefore, increases our chances of choosing appropriate values of parameters that could give rise to pattern formation in the twist-opening model. Our analytical predictions are verified numerically, where the generic equations for the radial and torsional dynamics are directly integrated. The impact of charge migration on the above degrees of freedom is discussed for different values of α. Soliton-like and localized structures are observed and thus confirm our analytical predictions. We also find that polaronic structures, as known in DNA charge transport, are generated through modulational instability, and hence reinforces the robustness of polaron in the model we study.

Intramolecular vibrations and noise effects on pattern formation in a molecular helix.

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 Schrödinger 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.

Wave instability of intercellular Ca2+ oscillations

Modulational instability is exclusively addressed in a minimal model for calcium oscillations in cells. The cells are considered to be coupled through paracrine signaling. The endoplasmic recticulum and cytosolic equations are reduced to a single differential-difference amplitude equation. The linear stability analysis of a plane wave is performed on the latter and the paracrine coupling parameter is shown to deeply influence the instability features. Our analytical expectations are confirmed by numerical simulations, as instability regions give rise to unstable wave patterns. We also discuss the possibility of perfect intercellular communication via the activation of modulational instability.

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.

Modulated Wave Packets in DNA and Impact of Viscosity

We study the nonlinear dynamics of a DNA molecular system at physiological temperature in a viscous media by using the Peyrard–Bishop model. The nonlinear dynamics of the above system is shown to be governed by the discrete complex Ginzburg–Landau equation. In the non-viscous limit, the equation reduces to the nonlinear Schrodinger equation. Modulational instability criteria are derived for both the cases. On the basis of these criteria, numerical simulations are made, which confirm the analytical predictions. The planar wave solution used as the initial condition makes localized oscillations of base pairs and causes energy localization. The results also show that the viscosity of the solvent in the surrounding damps out the amplitude of wave patterns.