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.
During these last decades the behavior of nonlinear discrete systems has received considerable attention in many areas of physics. The nonlinear electrical transmission lines (NLTLs) are good examples of such systems. They are very convenient tools for studying quantitatively the fascinating properties of wave propagation in nonlinear dispersive media. Afshari and Hajimiri [
Thus far, discrete spatial solitons (nonlinear eigenstates) have been successfully demonstrated in NLTL [12-15]. Like every nonlinear system, a NLTL can exhibit an instability that leads to a self-induced modulation of input plane wave with the subsequence generation of localized pulses [16-19]. This phenomenon is known as a Benjamin-Feir modulational instability [
Dissipative phenomena in nonlinear media with complex parameters are attracting nowadays a great deal of attention. In the present work we shall address these problems with a twofold aim. From one side, we derive the discrete CGL (DCGL) equation with nearest-neighbor nonlinearities which governs the propagation of wave in the DNLTL. From the other side, we show that the derived equation can be used to explore interesting dynamical behaviors as generate nonlinear localized modes in the DNLTL. To this regard we investigate the MI as a mechanism of the generation of bright matter-waves in the DNLTL. Dissipation is one of the main forces acting against the formation of nonlinear coherent structures in extended systems. When dissipation is present in systems without additional gain mechanisms, typically all excitations decay into the regime of linear waves.
The work is organized as follows. In section 2, the analytical model based on the DNLTL is presented and we derived the DCGL equation with nearest-neighbor nonlinearities. Then, we present a qualitative analysis concerning MI and we propose the generalized LangeNewell criterion. In section 3, since the discrete breathers solutions with small amplitudes are very close to plane waves, we focuse on the generation of nonlinear excitations induced by MI. Finally, conclusions are drawn in Section 4.
Many schematic electrical lattices have already been consider in the litterature. Recently, a one dimensional biinductance lattices which act as band-pass filters has been considered [
So, here we consider a nonlinear network of N cells as illustrated in
where, C0p is a constant corresponding to the capacitance of the nonlinear diode at the dc bias-voltage Vb. The nonlinear parameters α and β are assumed to be positive constants. In Equation (1), we keep nonlinear coefficient up to the second order for the following reasons. First, the polynomial approximation of the C-V curve and corresponding fit are justified if the voltage amplitude is small enough. Second, in this voltage range, to reduce the equation of motion to an ordinary differential equation, it is sufficient to take into account these two terms, only, to balance the first-order dispersion term. Having in mind that the compactification of solitary wave results from the nonlinear dispersion of the system, we have to choose the dispersion element properly in order to assure
that the resulting network will satisfy this requirement. It has been pointed out by Comte and Marquié [
where V is the voltage across the nonlinear capacitor with the zero-voltage value C0s. So, the operating point of this capacitor corresponds to the zero-voltage value. In order to take in to account the dissipation of the network, the conductance g is connected in parallel with Cp and Lp, respectively. The conductance g accounts for the dissipation of the inductor Lp in addition to the loss of the nonlinear capacitor C. The corresponding conductance g is given by [
The linear dispersion relation of the line is a typical band pass filter:
where, , and , are the dimensionless capacitance and characteristic frequencies of the system. The corresponding linear spectrum has a gap f0 = ω0/2π and it is limited by the cut-off frequency, with due to lattice effects. The linear dispersion curve of the network is plotted in
This group velocity is represented in
Applying Kirchhoff’s laws to this system leads to the following set of differential equations governing wave propagation in the network
with, and. For this purpose, restricting moreover our study to weak amplitude and slow temporal variations of the wave envelope, we look for a solution of Equation (6) in the form
where is small parameter () and, is unknown complex envelope function, stands for complex conjugate and ω denoting frequency. Inserting this relation in Equation (6), we collect solutions of order which give a relation between the wave function at different site of the lattice. Thereafter, one can write the relation at order, using the dispersion relation [Equation (4)] and equations resulting from the above different order, one obtains the following equation:
where the complex coefficients of Equation (8) are given by
Equation (8) is the DCGL equation with nearestneighbor nonlinearities. Note that the DCGL equation has been phenomenologically proposed to describe frustrated states in a linear array of vortices [26,27]. Also, it reproduces reasonably well characteristics of the turbulent regime below the percolation threshold. Percolation has been found to be a useful concept for the description of turbulence, and the results suggest that non adiabatic effects, such as discrete nature of the system, play a role in the system. From a physical point of view, it is of interest to study the effects of including nearest-neighbor nonlinearities terms than cubic in the equation on discrete solitons. These terms appear in different physical contexts such as Bose gases with hard core interactions in the Tonks-Girardeau regime [
In particular when the nearest-neighbor parameter D = 0, Equation (8) becomes the well-known DCGL equation [
Modulation instability is a generic nonlinear phenomenon governing nonlinear wave propagation in dispersive media. It refers to a weak space-time dependence (modulation) of the wave amplitude, due to intrinsic medium nonlinearity, however weak. Under the effect of external perturbations (e.g., noise), the wave amplitude (the envelope) may potentially grow, eventually leading to energy localization via the formation of localized structures (envelope solitons) [
To analyze MI, which is responsible for energy localization, we seek a solution of Equation (8) in the form of plane wave disturbed as follow
where f0 is the initial complex constant amplitude, k and ω are, respectively, the wave number and the angular frequency of the carrier wave. The quantity Bn(τ) is the perturbation assumed to be small in comparison with the amplitude of the carrier wave. It would be important to ask what happens to the plane waves when the amplitude increases sufficiently so that the nonlinearity occurs. In the linear approximation an equation for Bn(τ) yields the dispersion relation for the evolution of small perturbations,
where λr, λi, χr, and χi are defined in the Appendix. The frequency Ω can be written as
Equation (11) has been established for the case. We easily get the perturbation as follow
where K, b1 and b2 are the wave number and the complex constants, respectively.
The amplitude Bn will be unbounded as if and only if: , in order to get this relation, it is necessary that λi < 0. Because, , the relation, holds and from this inequality we can easily derive the following inequality,
Relation (13) represents the amplitude threshold, for the MI versus the wave number k of the carrier wave and the K of the perturbation for the dissipative coefficients: α1 = 2.6710 × 104 Ω−1·F−1 (see
Assume that the necessary condition , is satisfied, then we can write the inequality, that is
Relation (14) represents the MI criterion associated with the DCGL equation with higher-order nonlinearities.
This result is the generalized Discrete Lange and Newell criterion for Stokes waves.
The growth rate of the perturbation is given by Ωi. This quantity has been plotted in
From this figure, one can see that our system can be really stable unstable (the two branches).
Let us check the theoretical predictions concerning the existence of MI in the system. So, to further explore MI, we compute numerical simulations. In particular, our results are based on the theory of linear stability analysis. However, we know that the linear stability analysis is limited because it can only predict the onset of instability and does not tell us anything about the long-time dynamical behavior of the system when the instability grows. When the perturbation amplitude grows large enough compared to that of the initial wave, the numerical analysis must be adopted. To further confirm that the linear instability analysis given above can correctly describe the initial stage of instability, we have performed numerical simulations of Equation (1). A fourth-order Runge-Kutta algorithm has been used. A normalized
integration time step Δt = 5 × 10−3 is used for numerical simulations. Similarly, the number of cells N is chosen to be equal to 1600 and we have used periodic boundary conditions so that we do not encounter the wave reflection at the end of the line. The parameters of the system are choosen in accordance with
where, V0 is the amplitude of the unperturbed plane wave, m0 designates the modulation rate and fm the frequency of modulation. MI has been analyzed for lattices with respect to discrete breathers. As a specific example, we use the following value V0 = 0.90 V, fp = 800 kHz, m0 = 0.01 and fm = 8 kHz. Then, we launch solution (15) in the network. As time goes on, the modulation increases and the continuous wave breaks into a periodic pulse or envelope soliton train as shown in
In this work, we have introduced the generalized discrete complex Ginzburg-Landau equation with nearest-neighbor nonlinearities in the nonlinear discrete transmission lattices. The appearence of MI has been investigated and the generalized discrete Lange-Newell proposed. The theoretical findings have been numerically tested through direct simulations and solitonic excitations of the pulse train have been generated. The theory of “bushes” of nonlinear normal modes has been also point out.
The MI is the first step in the generation of soliton like excitations in physical systems. Therefore the study of
the conditions in which this phenomenon takes place is of special importance. This result is very useful for either the investigation of nonlinear transmission lines or of there similar physical problems, such as nonlinearity, in the plasma, dusty plasma, Bose-Einstein condensates, etc.
Finally, it is important to mention that, in recent years, the development in NLTL has demonstrated its capacity to work as signal processing tools. To cite only very few examples, it has been demonstrated that the nonlinear uniform electrical line can be used for extremely wide band signal shaping applications [
A.M is very grateful for the hospitality of the CMSPS of the Abdus Salam ICTP of Trieste-Italy.