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\author{Emmanuel KENGNE\thanks{%
University of Dschang. Faculty of Science. Dept. of Mathematics \& Computer
Science\protect\\  P.O. Box 67 Dschang, CAMEROON\protect\\ E-Mail:
ekengne6@yahoo.fr\protect\\ Tel. +237 451074}}
\title{Envelope Modulational Instability in the Nonlinear Dissipative Transmission
Line}
\date{July, 2001}
\maketitle

\begin{abstract}
Biinductance LC circuit is considered and envelope modulation is reduced to
the Complex Nonlinear Schr\"odinger (CNLS) equation. The Benjamin-Feir (or,
as it is sometimes called, the modulational) instability for the CNLS
equation is investigated to lead the Lange and Newell's criterion.\\ \\ $%
{\bf KEYWORDS:}$ Nonlinear Dissipative Transmission Line; Complex Nonlinear
Schr\"odinger Equation; Envelope Modulational Instability.
\end{abstract}

\section{Introduction}

Our work deals with the derivation of a nonlinear wave equation for
electromagnetic wave propagation on a nonlinear dissipative biinductance
transmission line$^{1-4)}$ shown in Fig. 1. 
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\\ In this transmission line, $C(V)\approx C_1-C_NV,$ where $C_N$ is a
nonlinear capacitor such as a ''VARICAP'' or a reverse-biased $p-n$ junction
diode, the capacitance of which depends on the voltage applied across it.
One can write the set of partial differential equations for the currents and
voltages as 
\begin{eqnarray}
L_2I_{2n,t}+R_2I_{2n} &=&V_{2n-1}-V_{2n} \\
L_1I_{2n+1,t}+R_1I_{2n+1} &=&V_{2n}-V_{2n+1}
\end{eqnarray}
\begin{eqnarray}
Q_{2n,t} &=&I_{2n}-I_{2n+1}-GV_{2n} \\
Q_{2n+1,t} &=&I_{2n+1}-I_{2n+2}-GV_{2n+1},
\end{eqnarray}
where $I_{2n}$ is the current through the inductance $L_2$ and the
resistance $R_2,$ $Q_{2n}$ being the charge density stored in the $2n$ $th$
capacitor, $V_{2n}$ is the $ac$ voltage across it, subscript $t$ denote the
differentiation with respect to $t$.

The dependence of $Q(V)$ ($Q$ is a charge density, Coulomb/length) must be
specified before we proceed. The simplest choice is to expand $Q(V)$ in a
Taylor series as 
\begin{equation}
Q(V)=C_0(V-aV^2).  \label{5}
\end{equation}

>From (1)-(4), we can eliminate the currents, and using (5), we obtain the
following set of discrete equations 
\begin{eqnarray}
&&C_0\frac{\partial ^2}{\partial t^2}(V_{2n}-aV_{2n}^2)+\frac{R_2C_0}{L_2}%
\frac \partial {\partial t}(V_{2n}-aV_{2n}^2)+G\frac \partial {\partial
t}V_{2n}  \nonumber  \label{6} \\
&=&\frac 1{L_1}(V_{2n-1}-V_{2n})-\frac 1{L_2}(V_{2n}-V_{2n+1})-\frac{R_2G}{%
L_2}V_{2n}  \label{6}
\end{eqnarray}
\begin{eqnarray}
&&C_0\frac{\partial ^2}{\partial t^2}(V_{2n+1}-aV_{2n+1}^2)+\frac{R_2C_0}{L_2%
}\frac \partial {\partial t}(V_{2n+1}-aV_{2n+1}^2)+G\frac \partial {\partial
t}V_{2n+1}  \nonumber \\
&=&\frac 1{L_2}(V_{2n}-V_{2n+1})-\frac 1{L_1}(V_{2n+1}-V_{2n+2})-\frac{R_2G}{%
L_2}V_{2n+1}  \label{7}
\end{eqnarray}
with the duresses 
\begin{equation}
R_2L_1=R_1L_2,\text{ }L_1\geq L_2.  \label{8}
\end{equation}

If we denote by $W_n(t)$ the voltage of the even cells $V_{2n}$ and by $V_n$
the voltage of the odd cells $V_{2n+1},$ system (6),(7) becomes 
\begin{eqnarray}
&&C_0\frac{\partial ^2V_n}{\partial t^2}+\left( \frac{R_2C_0}{L_2}+G\right) 
\frac{\partial V_n}{\partial t}-aC_0\left( \frac{\partial ^2V_n^2}{\partial
t^2}+\frac{R_2}{L_2}\frac{\partial V_n^2}{\partial t}\right)   \nonumber \\
&=&\frac 1{L_1}(W_{n-1}-V_n)-\frac 1{L_2}(V_n-W_n)-\frac{R_2G}{L_2}V_n
\label{9}
\end{eqnarray}
\begin{eqnarray}
&&C_0\frac{\partial ^2W_n}{\partial t^2}+\left( \frac{R_2C_0}{L_2}+G\right) 
\frac{\partial W_n}{\partial t}-aC_0\left( \frac{\partial ^2W_n^2}{\partial
t^2}+\frac{R_2}{L_2}\frac{\partial W_n^2}{\partial t}\right)   \nonumber \\
&=&\frac 1{L_2}(V_n-W_n)-\frac 1{L_1}(W_n-V_{n+1})-\frac{R_2G}{L_2}W_n
\label{10}
\end{eqnarray}
with $n=1,2,...,N,$ where $N$ is the number of cells considered.

\section{A\ complex nonlinear Schr\"odinger equation}

In this section we consider the propagation of a group of waves centered
around the wavenumber $k$ and the frequency $\omega .$ To accomplish this,
we use the semidiscrete approximation method$^{5-7)}.$ We follow Taniuti and
Yajima$^{8)}$ and seek the solution $V_n(t)$ of the odd cells in the form 
\begin{equation}
V_n(t)=\epsilon V_{11}(n,t)\exp \{i\theta \}+\epsilon ^2V_{22}(n,t)\exp
\{2i\theta \}+(^{*})  \label{11}
\end{equation}
where $\epsilon $ is a small, dimensionless parameter related to the
amplitudes, $\theta =2kn-\omega t=\theta (n,t).$ Here $(^{*})$ stands for
the complex conjugate of the preceding expression. We use the following
ansatz$^{7)}$ to decouple the two equations (9),(10): 
\begin{eqnarray}
W_n(t) &=&\sigma _1\exp \{ik\}\left( V_{11}+\epsilon b_1V_{11,x}+\epsilon ^2%
\frac{b_2}2V_{11,xx}+c_8\epsilon ^2V_{11}^{*}V_{22}\right) \exp \{i\theta
\}+(^{*})  \nonumber \\
&&\ \ \ +\sigma _2\epsilon ^2\exp \{2ik\}\left( V_{22}+\epsilon
c_1V_{22,x}+C_9V_{11}V_{11,x}\right) \exp \{2i\theta \}+(^{*}),  \label{12}
\end{eqnarray}
where $\sigma _1,$ $\sigma _2,$ $b_1,$ $b_2,$ $c_8,$ $c_1,$ and $c_9$ are
complex constants to determined; here subscript $x$ denotes the
differentiation with respect to $x$. We order the damping coefficients so
that the effects of the damping and the nonlinearity appear in the same
perturbation equations. Thus we let 
\begin{equation}
\frac{R_2C_0}{L_2}+G=\epsilon ^2\mu   \label{13}
\end{equation}
Substituting (13) into (9) and (10) we obtain 
\begin{eqnarray}
&&\ \ \ C_0V_{n,tt}+\epsilon ^2\mu V_{n,t}-aC_0\left( V_{n,tt}^2+\frac{R_2}{%
L_2}V_{n,t}^2\right)   \nonumber  \label{14} \\
\  &=&\frac 1{L_1}(W_{n-1}-V_n)-\frac 1{L_2}(V_n-W_n)-\epsilon ^2\delta V_n
\label{14}
\end{eqnarray}
\begin{eqnarray}
&&\ \ \ C_0W_{n,tt}+\epsilon ^2\mu W_{n,t}-aC_0\left( W_{n,tt}^2+\frac{R_2}{%
L_2}W_{n,t}^2\right)   \nonumber  \label{15} \\
\  &=&\frac 1{L_2}(V_n-W_n)-\frac 1{L_1}(W_n-V_{n+1})-\epsilon ^2\delta W_n.
\label{15}
\end{eqnarray}

In order to determine $V_{11}$ and $V_{22}$, and constants $\sigma _1,$ $%
\sigma _2,$ $b_1,$ $b_2,$ $c_8,$ $c_1,$ and $c_9$ we insert $V_n(t)$ from
(10) and $W_n(t)$ from (11) and their derivatives into (14) and (15), and
impose to the resulting equations to be equivalent. These operations yield
many equations distinguished by the powers of $\epsilon $ and the powers of $%
\exp \{i\theta \}.$

Equations of $(\epsilon ,\exp \{i\theta \})$ give 
\begin{equation}
\left\{ 
\begin{array}{c}
\left( -C_0\omega ^2+\frac 1L+\frac{R_2G}{L_2}\right) V_{11}=\sigma _1\left( 
\frac{\exp \{-ik\}}{L_1}+\frac{\exp \{ik\}}{L_2}\right) V_{11} \\ 
\left( \frac{\exp \{ik\}}{L_1}+\frac{\exp \{-ik\}}{L_2}\right) V_{11}=\sigma
_1\left( -C_0\omega ^2+\frac 1L+\frac{R_2G}{L_2}\right) V_{11},
\end{array}
\right.  \label{16}
\end{equation}
where $\frac 1L=\frac 1{L_1}+\frac 1{L_2}.$ From (16) one derives the linear
dispersion relation 
\begin{equation}
\omega ^2=\frac 1{C_0}\left[ \frac 1L+\frac{R_2G}{L_2}\pm \sqrt{\frac
1{L^2}-\frac 4{L_1L_2}\sin ^2k}\right]  \label{17}
\end{equation}
which is illustrated in Fig. 2 and Fig. 3 for the line parameters $L_1=28\mu
H;$ $L_2=14\mu H;$ $C_0=540pF;$ $R_2=10\Omega ;$ and $G=38.6\times
10^{-6}\Omega ^{-1},$ when $0\leq k\leq \pi /2.$

The dispersion relation (17) provides two types of frequency modes: the Low
Frequency (LF) mode in the region $\omega _1\leq \omega \leq \omega _2$
(Fig. 2) and the High Frequency (HF) mode for $\omega _3\leq \omega \leq
\omega _4$ (Fig. 3)$,$ where the cut-off frequencies $\omega _1,$ $\omega _2,
$ $\omega _3,$ and $\omega _4$ are defined by 
\[
\omega _1=\sqrt{\frac{R_2G}{C_0L_2}};\text{ }\omega _2=\sqrt{\frac
1{C_0}\left( \frac 2{L_1}+\frac{R_2G}{L_2}\right) };\text{ }\omega _3=\sqrt{%
\frac{2+R_2G}{C_0L_2}};\text{ }\omega _4=\sqrt{\frac 1{C_0}\left( \frac 2L+%
\frac{R_2G}{L_2}\right) }.
\]
>From (16) we deduce 
\[
\sigma _1=\frac 1{H_0}\left[ \frac{\cos k}L+i\frac{\sin k}{L_0}\right] ,
\]
where 
\[
H_0=-C_0\omega ^2+\frac 1L+\frac{R_2G}{L_2},\text{ and }\frac 1{L_0}=\frac
1{L_1}-\frac 1{L_2}.\text{ From (8) we have }\frac 1{L_0}\leq 0.
\]
\begin{eqnarray*}
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&&\ Fig.2.\text{ Dispersion curve for the linearized version of the
transmission line shown } \\
&&\ \text{in Fig. 1 for the Low Frequency mode.}
\end{eqnarray*}
\\ From equations of $(\epsilon ^2,\exp \{i\theta \})$ we have

\[
\left\{ 
\begin{array}{c}
-2iC_0\omega V_{11,t}=\sigma _1\left[ \left( \frac{\exp \{-ik\}}{L_1}+\frac{%
\exp [-ik\}}{L_2}\right) b_1-\frac{2\exp \{-ik\}}{L_1}\right] V_{11,x} \\ 
-2iC_0\omega V_{11,t}=\left( \frac{2\sigma _1^{*}\exp \{ik\}}{L_1}%
-H_0\right) V_{11,x},
\end{array}
\right. 
\]
which gives 
\[
b_1=\frac 4{L_1H_0^2}\left( \frac 1{L_1}+\frac{\cos (2k)}{L_2}\right) -1. 
\]
\begin{eqnarray*}
&&\ \text{\FRAME{itbpxFX}{455.8125pt}{270.625pt}{0pt}{}{}{Figure }{\special{%
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&&\ \text{Fig. 3. Dispersion curve for the linearized version of the
transmission line shown } \\
&&\ \text{in Fig. 1 for the High Frequency mode.}
\end{eqnarray*}
\\Equations of $(\epsilon ^2,\exp \{2i\theta \})$ yield

\[
\left\{ 
\begin{array}{c}
\left[ H_0-3C_0\omega ^2-\left( \frac{\exp \{2ik\}}{L_2}+\frac{\exp \{-2ik\}%
}{L_1}\right) \sigma _2\right] V_{22}=aC_0\left( -4\omega ^2-2i\omega \frac{%
R_2}{L_2}\right) V_{11}^2 \\ 
\left[ \left( H_0-3C_0\omega ^2\right) \sigma _2-\left( \frac{\exp \{-2ik\}}{%
L_2}+\frac{\exp \{2ik\}}{L_1}\right) \right] V_{22}=aC_0\left( -4\omega
^2-2i\omega \frac{R_2}{L_2}\sigma _1\right) V_{11}^2,
\end{array}
\right. 
\]
from where we obtain 
\[
\sigma _2=\frac{\left( H_0-3C_0\omega ^2\right) \left( 2\omega ^2\sigma
_1+i\sigma _1^2\frac{R_2}{L_2}\right) +\left( 2\omega ^2+i\frac{R_2}{L_2}%
\right) \left( \frac{\exp \{-2k\}}{L_2}+\frac{\exp \{2ik\}}{L_1}\right) }{%
\left( 2\omega ^2+i\frac{R_2}{L_2}\right) \left( H_0-3C_0\omega ^2\right)
+\left( 2\omega ^2\sigma _1+i\sigma _1^2\frac{R_2}{L_2}\right) \left( \frac{%
\exp \{2k\}}{L_2}+\frac{\exp \{-2ik\}}{L_1}\right) } 
\]
and 
\[
V_{22}=\frac{-2aC_0\omega \left( 2\omega +i\frac{R_2}{L_2}\right) }{%
H_0-3C_0\omega ^2-\left( \frac{\exp \{2ik\}}{L_2}+\frac{\exp \{-2ik\}}{L_1}%
\right) \sigma _2}V_{11}^2. 
\]

Equations of $(\epsilon ^3,\exp \{i\theta \})$ give 
\begin{equation}
\left\{ 
\begin{array}{c}
C_0\left( V_{11,tt}-2i\omega V_{11,t}\right) -i\omega \mu V_{11}+\left[
2aC_0\omega \left( \omega +i\frac{R_2}{L_2}\right) -H_0c_8\right]
V_{11}^{*}V_{22} \\ 
+\left[ \sigma _1\frac{\exp \{-ik\}}{L_1}(2b_1-1)-\frac{H_0}2b_2\right]
V_{11,xx}=0 \\ 
C_0\left( V_{11,tt}-2i\omega V_{11,t}\right) -i\omega \mu V_{11}+\left[
aC_0\omega ^2\sigma _1^{*^2}\sigma _2+\left( H_0+aC_0i\omega \frac{R_2}{L_2}%
\right) c_8\right] \times \\ 
\times V_{11}^{*}V_{22}+\left[ \left( H_0+aC_0i\omega \right) \frac{b_2}2-%
\frac{\sigma _1^{*}\exp \{ik\}}{L_1}\right] V_{11,xx}-C_0b_1\left[ 2i\omega
++a\frac{R_2}{L_2}\right] V_{11,xt}=0
\end{array}
\right.  \label{18}
\end{equation}
The compatibility condition for the nonlinear term $V_{11}^{*}V_{22}$ gives 
\[
c_8=\frac{aC_0\left[ 2\omega \left( \omega +i\frac{R_2}{L_2}\right) -\omega
^2\frac{\sigma _2}{\sigma _1^2}\right] }{2H_0+aC_0i\omega \frac{R_2}{L_2}} 
\]
and the compatibility condition for $V_{11,xx}$ gives 
\[
b_2=\frac{2\left[ \sigma _1\left( 2b_1-1\right) \exp \{-ik\}+\sigma
_1^{*}\exp \{ik\}\right] }{L_1\left[ 2H_0+aC_0i\omega \right] }. 
\]

Using Galilea transformation 
\[
\xi =x-\frac{d\omega }{dk}t,\text{ }\tau =t 
\]
in the second equation of (18) and by going into the reference frame moving
with the group velocity $V_g=\frac{d\omega }{dt}$, we derive the resulting
equation for $V_{11}$ that describes the evolution of the wave packet 
\begin{equation}
iV_{11,\tau }+PV_{11,\xi \xi }+Q\left| V_{11}\right| ^2V_{11}+i\Gamma
V_{11}=0  \label{19}
\end{equation}
where 
\begin{eqnarray*}
P &=&P(k)=\frac 1{2C_0\omega }\left[ \frac{\exp \{ik\}}{\sigma _1L_1}-\left(
H_0+aC_0i\omega \right) \frac{b_2}2-C_0\left( \frac{d\omega }{dk}\right)
^2\right. \\
&&\left. -\frac{d\omega }{dk}\left( 2C_0b_1i\omega +aC_0b_1\frac{R_2}{L_2}%
\right) \right] ;
\end{eqnarray*}
\[
Q=Q(k)=\frac{a\left( 2\omega +i\frac{R_2}{L_2}\right) \left[ aC_0\omega ^2%
\frac{\sigma _2}{\sigma _1^2}+\left( H_0+aC_0i\omega \frac{R_2}{L_2}\right)
c_8\right] }{H_0-3C_0\omega ^2-\left( \frac{\exp \{2ik\}}{L_2}+\frac{\exp
\{-2ik\}}{L_1}\right) \sigma _2}; 
\]
\[
\Gamma =\frac \mu {2C_0}. 
\]
Equation (19 is a complex nonlinear Schr\"odinger equation. If we denote by $%
P_r,$ $Q_r,$ and $\Gamma _r$ the real parts of $P,$ $Q,$ and $\Gamma ,$
respectively and by $P_i,$ $Q_i,$ and $\Gamma _i$ the imaginary parts of $P,$
$Q,$ and $\Gamma ,$ respectively, then we can write 
\[
P=P_r+iP_i;\text{ }Q=Q_r+iQ_i;\text{ }\Gamma =\Gamma _r+i\Gamma _i. 
\]

\section{Envelope modulational instability for the CNLS equation}

In this section we consider the modulation of the envelope of the unstable
periodic solution of Eq. (19) by considering a first perturbation of the
amplitude of a plane wave$^{7,9-10)}.$

First we investigate periodic solution of Eq. (19). Thus we seek a solution
in the form 
\begin{equation}
V_{11}=A\exp \left\{ i\left( \tilde k\xi -\tilde \omega \tau \right) \right\}
\label{20}
\end{equation}
where $A$ is a complex constant, $\tilde \omega $ and $\tilde k$ are real
constants. Substituting (20) into (19), we obtain 
\begin{equation}
\tilde \omega -\tilde k^2P+\left| A\right| ^2Q+\Gamma =0.  \label{21}
\end{equation}

Next we perturb $V_{11}.$ That is, we let 
\begin{equation}
V_{11}=\left[ 1+b\right] A\exp \left\{ i\left( \tilde k\xi -\tilde \omega
\tau \right) \right\} +(^{*})  \label{22}
\end{equation}
where $b=b(\xi ,\tau )$ is assumed to be infinitesimal. Substituting (22)
into (19), using (21) and keeping only linear terms in the perturbation
quantity, we obtain 
\begin{equation}
ib_\tau +Pb_{\xi \xi }+2i\tilde kPb_\xi +\left| A\right| ^2Q\left(
b+b^{*}\right) .  \label{23}
\end{equation}

Since (23) has constant coefficients, one can represent its solutions in the
form 
\begin{equation}
b(\xi ,\tau )=b_1\exp \left\{ i\left( K\xi +\Omega \tau \right) \right\}
+b_2^{*}\exp \left\{ -i\left( K\xi +\Omega ^{*}\tau \right) \right\}
\label{24}
\end{equation}
where $b_1,$ $b_2,$ $K,$ and $\Omega $ are constants. Substituting (24) into
(23) yield 
\begin{equation}
\left\{ 
\begin{array}{c}
\left( \Omega +K^2P+2\tilde kKP-\left| A\right| ^2Q\right) b_1-\left|
A\right| ^2Qb_2=0 \\ 
\left| A\right| ^2Q^{*}b_1+\left( \Omega -K^2P^{*}+2\tilde kKP^{*}+\left|
A\right| ^2Q^{*}\right) b_2=0.
\end{array}
\right.  \label{25}
\end{equation}
For a nontrivial solution the determinant of the coefficient matrix (25)
must vanish. That is, 
\begin{equation}
\left. \left[ \Omega +\left( 2\tilde kKP_r+i\left( K^2P_i-\left| A\right|
^2Q_i\right) \right) \right] ^2=X+iY,\right.  \label{26}
\end{equation}
where 
\begin{eqnarray*}
X &=&K^4P_r^2-4\tilde k^2K^2P_i^2-\left| A\right| ^4Q_i^2+2K^2\left|
A\right| ^2P_iQ_i-2K^2\left| A\right| ^2\left( P_rQ_r+P_iQ_i\right) ; \\
Y &=&4\tilde kKP_i\left( K^2P_r-\left| A\right| ^2Q_r\right) .
\end{eqnarray*}
If we introduce the notations 
\begin{equation}
H_1=\pm \sqrt{\frac 12\left( X+\sqrt{X^2+Y^2}\right) }\text{ and }H_2=\pm 
\sqrt{\frac 12\left( -X+\sqrt{X^2+Y^2}\right) }  \label{27}
\end{equation}
then from (26) we have 
\begin{equation}
\Omega =\left( -2\tilde kKP_r\pm H_1\right) +i\left( \left| A\right|
^2Q_i-K^2P_i\pm H_2\right) .  \label{28}
\end{equation}

We deduce from (24), (27), and (28) that for the boundedness of $b(\xi ,\tau
)$ (as $\tau \rightarrow +\infty )$ it is necessary and sufficient that at
least one of the following conditions occurs

a) $K$ is a solution of equation 
\begin{equation}
\left| A\right| ^2Q_i-K^2P_i\pm H_2=0  \label{29}
\end{equation}

b) 
\begin{equation}
\left| A\right| ^2Q_i-K^2P_i\pm H_2>0.  \label{30}
\end{equation}
We should note that condition (29) means that $\Omega $ is real and
condition (30) means that the imaginary part of $\Omega $ is positive.

It follows from (28) that for $\Omega $ to be complex so that $b(\xi ,\tau )$
is unbounded (as $\tau \rightarrow +\infty )$ it is necessary and sufficient
that the wavenumber of the perturbation $K$ should be solution of system 
\begin{equation}
\left\{ 
\begin{array}{c}
\sqrt{\frac 12\left( -X+\sqrt{X^2+Y^2}\right) }+\left| A\right|
^2Q_i-K^2P_i<0 \\ 
-\sqrt{\frac 12\left( -X+\sqrt{X^2+Y^2}\right) }+\left| A\right|
^2Q_i-K^2P_i<0.
\end{array}
\right.  \label{31}
\end{equation}
A necessary condition for (31) to have a solution is that 
\begin{equation}
\left| A\right| ^2Q_i-K^2P_i<0.  \label{32}
\end{equation}
It follows from (32) that 
\begin{equation}
K^2>\left| A\right| ^2Q_iP_i^{-1},\text{ if }P_i>0  \label{33}
\end{equation}
and 
\begin{equation}
K^2\acute <\left| A\right| ^2Q_iP_i^{-1},\text{ if }P_i<0.  \label{34}
\end{equation}
>From (33) we have $K>\left| A\right| \sqrt{Q_iP_i^{-1}},$ if $P_iQ_i>0$ and $%
K>0$ if P$_iQ_i<0,$ and from (34) we have $0<K<\left| A\right| \sqrt{Q_i/P_i}
$ if $P_iQ_i>0.$ We have the following conclusions:

i) If $P_iQ_i>0$ and either $P_i>0$ and $K>\left| A\right| \sqrt{Q_iP_i^{-1}}
$ or $P_i<0$ and $0<K<\left| A\right| \sqrt{Q_1P_i^{-1}}$ then for the
modulational instability for the plane wave in the nonlinear dissipative
transmission line, it is necessary that 
\begin{equation}
P_rQ_r+P_iQ_i>\frac{K^4(2P_i^2+P_r^2)+\left| A\right| ^4Q_i^2}{2K^2\left|
A\right| ^2}>0  \label{35}
\end{equation}

ii) If $P_i>0,$ $P_iQ_i<0$ and $K>0,$ then for the modulational instability
for the plane wave in the nonlinear dissipative transmission line, it is
necessary that 
\begin{equation}
P_rQ_r+P_iQ_i>\frac{K^4(2P_i^2+P_r^2)+\left| A\right| ^4Q_i^2-2K^2\left|
A\right| ^2P_iQ_i}{2K^2\left| A\right| ^2}>0.  \label{36}
\end{equation}

>From (35) and (36) we obtain 
\[
P_rQ_r+P_iQ_i>0 
\]
which is the well known Lange and Newell's criterion$^{10)}$ of the
modulational instability.

\begin{thebibliography}{99}
\bibitem{b1}  A. Scott, ''Active and nonlinear wave propagation in
electronics'', (Wiley-Interscience, New York, 1970).

\bibitem{B2}  K.E. Lonngren and A. Scott, ''Solitons in action'', (Academic
Press, New York, 1978).

\bibitem{B3}  L.A. Ostrovsky, K.A. Gorshkov and V.V. Papko, Phys. Scripta 
{\bf 20}, 357, (1979).

\bibitem{B4}  H. Nagashima and Y. Amagishi, J. Phys. Soc. Jpn {\bf 45}, 680,
(1978).

\bibitem{B5}  A. Tsurui, Progr. Theor. Phys. {\bf 48}, 1196 (1972) and J.
Phys. Soc. Jpn 34, 1462 (1973).

\bibitem{B6}  T. Kawahara, J. Phys. Soc. Jpn {\bf 35}, 1537 (1973).

\bibitem{B7}  F.B. Pelap and Kofan\'e, Phys. Scripta {\bf 57}, 410 (1998).

\bibitem{B8}  T. Tanuiti and N. Yajima, J. Math. Phys. {\bf 10}, 1369 (1969).

\bibitem{B9}  A.C. Newell, Lect. Appl. Math. {\bf 15}, 157 (1974).

\bibitem{B10}  C.G. Lange and A.C. Newell, SIAM J. Appl. Math. {\bf 27}, 441
(1974).
\end{thebibliography}

\end{document}