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.

Exact solitary wavelike solutions in a nonlinear microtubule RLC transmission line.

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 Schrödinger equation. The processes of the modulational instability are studied and, motivated with a solitary wave type of solution to the nonlinear Schrödinger (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.

On the effect of discreteness in the modulation instability for the Salerno model.

A Salerno model with first-and second-neighbor couplings is derived for the nonlinear transmission lines. We revisit the problem of modulation instability in the Salerno model. We derive the expression for the modulation instability gain and use them to explore the role of discreteness. We show that discreteness has an impact on the mechanism by which wave trains of soliton type can be generated in the Salerno model. We also show that second-neighbor couplings have an effect on the signal voltage.

Dynamics of modulated waves in electrical lines with dissipative elements.

By a means of a method based on the reductive perturbation method, we show that the amplitude of waves on the nonlinear electrical transmission lines (NLTLs) is described by the cubic-quintic complex Ginzburg-Landau (CGL) equation. Then, we revisit analytically and numerically the processes of modulational instability (MI). The evolution of dissipative modulated waves through the network is also examined, and we show that solitonlike excitations can be induced by MI. Analytical results, illustrating the nature of MI of plane-wave solution, are also found to be in good agreement with numerical findings.

Discrete Lange-Newell criterion for dissipative systems.

We report on the derivation of the discrete complex Ginzburg-Landau equation with first- and second-neighbor couplings using a nonlinear electrical network. Furthermore, we discuss theoretically and numerically modulational instability of plane carrier waves launched through the line. It is pointed out that the underlying analysis not only spells out the discrete Lange-Newell criterion by the means of the linear stability analysis at which the modulational instability occurs for the generation of a train of ultrashort pulses, but also characterizes the long-time dynamical behavior of the system when the instability grows.

Modulated waves and pattern formation in coupled discrete nonlinear LC transmission lines.

The conditions for the propagation of modulated waves on a system of two coupled discrete nonlinear LC transmission lines with negative nonlinear resistance are examined, each line of the network containing a finite number of cells. Our theoretical analysis shows that the telegrapher equations of the electrical transmission line can be reduced to a set of two coupled discrete complex Ginzburg-Landau equations. Using the standard linear stability analysis, we derive the expression for the growth rate of instability as a function of the wave numbers and system parameters, then obtain regions of modulational instability. Using numerical simulations, we examine the long-time dynamics of modulated waves in the line. This leads to the generation of nonlinear modulated waves which have the shape of a soliton for the fast and low modes. The effects of dissipative elements on the propagation are also shown. The analytical results are corroborated by numerical simulations.

Modulational instability in a purely nonlinear coupled complex Ginzburg-Landau equations through a nonlinear discrete transmission line.

We study wave propagation in a nonlinear transmission line with dissipative elements. We show analytically that the telegraphers' equations of the electrical transmission line can be modeled by a pair of continuous coupled complex Ginzburg-Landau equations, coupled by purely nonlinear terms. Based on this system, we investigated both analytically and numerically the modulational instability (MI). We produce characteristics of the MI in the form of typical dependence of the instability growth rate on the wavenumbers and system parameters. Generic outcomes of the nonlinear development of the MI are investigated by dint of direct simulations of the underlying equations. We find that the initial modulated plane wave disintegrates into waves train. An apparently turbulent state takes place in the system during the propagation.