This work investigates the dynamics of modulated waves in a coupled nonlinear LC transmission line. By means of a method based on the semi-discrete limit and in suitably scaled coordinates, we derive the two-dimensional NLS equation governing the propagation of slowly modulated waves in the network. The exact transverse solution is found and the analytical criteria of stability of this solution are derived. The condition for which the network can exhibit modulational instability is also determined. The exactness of this analytical analysis is confirmed by numerical simulations performed on the exact equation of the network.
Nonlinear dissipative waves have attracted considerable attention in recent years as a result of their multiple applications in many systems. These applications can be observed in different fields of the research. Among these domains, we can list biology, chemistry and physics to mention a few. In physics precisely, nonlinear electrical transmission lines (NETLs) are good examples to provide a useful way to check how the nonlinear excitations behave inside the nonlinear medium. In particular, one of their importance lies on the easy and rapid technique to investigate the behavior of nonlinear excitations throughout the waveguide. Moreover, it allows investigations of new erotic designs. Following these introductive studies longtime after, a great number of works have been done on NETLs. Thus, the first nonlinear and dissipative transmission line was built by Hirota and Suzuki [
MI leads to a self-induced modulation of an input plane wave with the subsequent generation of localized pulses; it is responsible for many interesting physical effects such as the formation of envelope solitons [
The purpose of this work is to conduct the linear stability analysis of solitary waves propagating in coupled nonlinear LC transmission lines with respect to longwavelength transverse perturbations on the basis of the nonlinear Schrödinger (NLS) equation. Therefore, our investigation of the transverse perturbations to the twodimensional NLS equation of the NETLs may be helpful in other fields of physics. The paper is organized as follows: in Section 2, we present the basic characteristics of the coupled NETL under consideration; in the semi-discrete limit, we derive the amplitude equations and the two-dimensional NLS equation governing the propagation of slowly modulated waves in the network. The solution of NLS, the stability and the condition for which our network can support the pulse solution are determined in Section 3. In Section 4, numerical experiments are done in order to verify the validity of the theoretical predictions. Finally, concluding remarks are presented in Section 5.
The standard nonlinear discrete LC line is a structure made of elementary cells which consist of an inductance L and a nonlinear capacitor C(V) [
In the network, nonlinearity is introduced by a varicap diode which admits that the capacitance varies with the applied voltage. The voltage dependence relation is assumed to have a polynomial form given by
where are constants. In the present work, we set. Applying Kirchoff’s laws to this system leads to the following set of propagation equations:
with.
Equation (2) is the differential equation governing the wave propagation in the network under consideration. As one can see, all of the lines have the same characteristic frequency. This is due to the fact that all of the lines are identical. is the coupling frequency. The properties of the network can be studied by using a solution of the form
where is the phase and “cc” stands for the complex conjugated of the preceding expression; k and q are the wave numbers respectively in the n and m direction; is the angular frequency; is a small parameter. For the semi-discrete approximation, we set
to obtain the short wavelength envelope solitons; vg and ug are the group velocities respectively in the n and m direction. Substituting Equation (3) into Equation (2), we obtain different equations as power series of.
1) The coefficient of, proportional to exp(iθ), gives the dispersion relation
This dispersion relation shows that our network is a band-pass filter.
This group velocity is represented in
2) The coefficient of, proportional to exp(2iθ) leads to the following relation:
3) From the coefficient of, proportional to exp(iθ),
we obtain the following two-dimensional nonlinear Schrödinger equation for A:
with the following definitions
The numbers P1, P2 and P3 are the dispersion coefficients, while Q is the nonlinearity coefficient of the nonlinear Schrödinger equation.
The focal point here corresponds to the determination of the solution of Equation (8). Before the discovery of solitons, mathematicians thought that nonlinear differential equations could not be solved, at least not exactly. However, solitons lead to the recognition that through a combination of such diverse subjects as quantum physics and algebraic geometry, one can actually solve some nonlinear equations exactly. This innovation opens up a wide window in the world of nonlinearity [
where a(z) is the amplitude, g(z) is the phase, represents the spectral parameter of the wave and the single variable for the amplitude, depending on ve which is the velocity of the wave packet. By substituting Equation (10) into the two-dimensional NLS Equation (8), and equating real and imaginary parts to zero, the following two coupled ordinary differential equations are obtained:
where the prime stands for derivation with respect to z and. By multiplying the first equation of (11) by a(z) and integrating once, it follows that the phase g is related to the amplitude a(z) through the expression:
where k1 is the constant of integration, which can naturally be taken as k1 = 0 for all continue solution at the origin a = 0. Taken then k1 = 0, Equation (12) yields
By substituting Equation (13) into the second equation of (11), we arrive to the following differential equation satisfied by the amplitude a(z):
from which the first integral is obtained by multiplying Equation (14) by and integrating the resulting equation:
with k2, another constant of integration. Let us mention that, Equation (15) can be also derived from the auxiliary Hamiltonian and lagrangian defined as follows:
This Hamiltonian may be viewed as the energy of a particle with an effective mass m(a) = 1 moving in the effective potential
It is obvious that Equation (14) can be transformed into the following equivalent autonomous dynamic system:
where solutions are the fixed points of the system. The number of equilibrium points, and consequently the dynamic of this system depend on the sign of the quantity
In fact, when F0 > 0, the system (18) admits only the equilibrium point (0, 0) and consequently, no nonlinear localized wave (NLW) can be obtained. However, for F0 < 0, the system admits three equilibrium points: (0, 0) and (0, ±Aeq), with
From the linear stability analysis, it appears that the stability of these equilibrium points depends on the sign of the product PQ (the saddle point is obtained if
and the center point else). In factwhen PQ > 0, the equilibrium point (0, 0) is a saddle while the two others, (0, ±Aeq), are the centers. This analysis is confirmed by the phase plane plot of the system sketched in
where A0 is the maximum amplitude of the envelope wave. The condition (21) leads to the following constraint to be satisfied by the integrating constant k2 and the spectral parameter Ω
However, when PQ < 0, there is a change in the properties of the above equilibrium points; (0, 0) becomes a center while (0, ±Aeq) are the saddle points. The phase plane plot sketched in
from which the following expressions of the spectral parameter and the integration constant are obtained:
The plot of the effective potential U(a) indicates the presence of a double wells when PQ > 0 and a single well for PQ < 0 which are in agreement with results of the phase plane plots.
Now we focus our attention on the derivation of bright solution of the NLS. For this end, the integration constant k1 = 0, while k2 and the spectral parameter Ω will be taking as given in Equation (22); thus, Equation (15) can be rearranged as
with; is a parameter describing the pulse width. From Equation (13), the phase g(z) is given by
where z0 is the initial position of the wave which can be equal to zero. Hence the solution of the NLS equation can explicitly be rewritten as:
vp is the carrier velocity, with the following expression
As for the particular case of solution with stationary phase in time (vp = 0), we have:
Having found this solution, we then check its stability. The stability of the BSW is determined here by the dependence of the norm (the power) on the velocity ve. Solitons are stable if dN/dve > 0 and unstable otherwise [
Substituting the ve obtained in Equation (29) into (30), one obtains
which is an increasing function of the envelope velocity for PQ > 0 and then pulse soliton is stable.
To determine the conditions of instability of the modulated waves in the network, we use the plane wave solution given below:
By inserting Equation (32) into Equation (8), we have the following dispersion relation:
The linear stability of this continuous wave can be investigated by looking for a solution of the form
where and are small perturbations for the amplitude and for the phase respectively; they can be writen as follow:
Substituting (34) and (35) into Equation (8), one obtains a system for the perturbations. For the nontrivial solutions of this system, we then have:
It appears that the behavior of depends on the quantity. On one hand, if this quantity is negative, the plane wave solution of NLS equation is stable. On the other hand, if this quantity is positive,
could be negative under certain conditions and the consequence is that the plane wave solution of NLS equation is unstable; hence, it appears MI phenomenon in the line. This instability induces the formation of small wave packets or envelope pulse solitons train, solution of the NLS equation (8).
We present in this section the numerical experiments on the propagation of slowly modulated waves in the network, this to check the analytical calculations presented in the previous sections. The numerical experiments are carried out in Equation (2) describing the propagation of waves in the NETL of
where fm is the modulation frequency, V0 is the amplitude of the wave and is the modulation rate. We take fm = 54 kHz, V0 = 0.2 V and. A fourth-order RungeKutta algorithm has been used and a normalized integration time step is used for numerical simulations. Similarly, the number of cells N in the n direction is chosen to be equal to 3000 and we have used periodic boundary conditions so that we do not encounter the wave reflection at the end of the line. In the m direction, we have taken M = 18. The parameters of the network are the same as in