Modulational stability of solitary states in a lossy nonlinear electrical line.(Report)

1. Introduction

Distributed electrical transmission lines consisting of a large number of identical sections have been used for theoretical and experimental study of nonlinear propagation of signals. About a decade ago, Marquie et al. [1] and Bilbault et al. [2] considered a nonlinear network composed of a set of cells containing two linear inductances in series and one in parallel together with a nonlinear capacitance diode in the shunt branch. It has been shown that the system of equations governing the physics of this network can be reduced to a cubic nonlinear Schrodinger equation or a pair of coupled nonlinear Schrodinger equations. These equations admit the formation of envelope solitons, which have been observed experimentally [1, 2]. By assuming that the nonlinear capacitance C is of the form C = [C.sub.0]/[1 + (V/[V.sub.0])] where V is the voltage in the transmission line and [C.sub.0] and [V.sub.0] are constants, Kengne et al. [3-5] presented a model for wave propagation in the lossy nonlinear transmission line shown in Fig. 1 (here G = 1/R). This model is based on the modified complex Ginzburg-Landau (MCGL) equation, derived in the small amplitude and long-wavelength limit using a standard semidiscrete approximation method [6-8] of the governing nonlinear equations. The modulational instability of the Stokes wave solution for the MCGL equation are presented in ref. 3.

In this paper, we study the pulse propagation in a single lossy nonlinear transmission line described by a single complex quintic nonlinear Schrodinger equation with gradient terms, also called the generalized complex Ginzburg-Landau equation. In a reference frame traveling with the given group velocity, this equation, in the case of the network of Fig. 1, coincides with the MCGL equation of the form [3]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [??](x, t) is the electrical-field envelope, the coefficients P = Pr + [iP.sub.i], [Q.sub.0] = [Q.sub.0r] + [iQ.sub.0i], [gamma] = [[gamma].sub.r] + i[[gamma].sub.i], [Q.sub.1] = [Q.sub.1r] + [iQ.sub.1i], [Q.sub.2] = [Q.sub.2r] + [Q.sub.2i], and [Q.sub.3] = [Q.sub.3r] + [iQ.sub.3i] are expressed in terms of the line parameters. Here and in the rest of the paper, the asterisk * denotes the complex conjugate. In the case of a lossless nonlinear transmission line (R = 0), (1) becomes a derivative nonlinear Schrodinger equation with quintic nonlinearity. The modulational instability criterion of the Stokes wave solution of (1) is given by eq. (4.12) of ref. 3.

[FIGURE 1 OMITTED]

We note that the Galilean transformation

[xi] = x - vt, [tau] = t, w([xi], t)= [??](x, t)exp [-i v/2p (x - v/2 t)], v = const (2)

transforms (1) into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

which implicitly contains the parameter v. This means that a stationary wave is transformed into a traveling wave by transformation (2).

Generally, in the physical literature, (3) is called either the quintic derivative Ginzburg-Landau equation [9-11], or the quintic complex Ginzburg-Landau equation with nonlinear gradient terms [12]. The last two terms, [u.sup.2][partial derivative][u.sup.*]/[partial derivative][xi] and [[absolute value of u].sup.2][partial derivative]u/[partial derivative][xi], are the nonlinear gradient terms appearing in the asymptotic derivation. Deissler and Brand [13] showed numerically that these two terms can significantly slow down the pulse propagation speed and also cause nonsymmetric pulses [14]. For fixed-shape solutions, these terms cause the solution to become asymmetric and to move with a nonzero velocity. We note that (3) is the most general equation to the lowest consistent order in an envelope expansion near the onset of perturbation [15] and is thus generic in nature.

The most comprehensive mathematical treatment of exact solutions of the special case of (3) when [Q.sub.1] = [Q.sub.2] = 0 (in this case, (3) is also called the cubic-quintic complex Ginzburg-Landau equation) using Painlelve analysis and symbolic computations is given in ref. 16. The general approach used in ref. 16 is to reduce the differential equation to a purely algebraic problem, which allows one to obtain the formulas for solitary wave-like solutions with parameters that are expressed implicitly through the coefficients of the equation but some work is still needed to compute the pulse shapes numerically. In the special case when (3) coincides with the derivative nonlinear Schrodinger equation ([P.sub.i] = [Q.sub.i] = [[gamma].sub.i] = [Q.sub.3i] = [Q.sub.1r] = [Q.sub.2r] = 0), we have obtained in our recent works [17] the higher order soliton solutions using Hirota's method, and some exact nonlinear wave solutions with periodic amplitude [18] using the theory of the Jacobian elliptic functions.

In general, (3) is not integrable, and only a few exact solutions can be obtained under …