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\begin{document}
\title{Modified Ginzburg-Landau Equation and Benjamin-Feir instability}
\author{E. Kengne}
\address{University of Dschang. Faculty of Science. Dpt. of Mathematics \& Computer
Sciences. P.O. Box 173, Dschang, Cameroon}
\email{ekengne6@@yahoo.fr}
\date{September 25, 2002}
\maketitle
\dedicatory{\emph{This work is dedicated to Pr. Maurice Tchuent\'e, Minister of Higher
Education of Cameroon}}

\begin{abstract}
In this paper the modulated wave train in nonlinear mono-inductance LC
circuit is studied. Using the method of multiple scales in general form, we
establish that the evolution of nonlinear excitations is governed by what we
called the Modified Ginzburg-Landau Equation (MGLE). Benjamin-Feir
instability for the MGLE is analyzed.\\\textbf{Keywords}: Modified
Ginzburg-Landau equation; Benjamin-Feir instability; Modulational
instability; Stokes wave.
\end{abstract}

\section{Introduction}

Considering nonlinear transmission line as a convenient tool to examine wave
propagations in dispersive media, various physical systems have been studied.%
$^{1-3)}\,$ Since the pioneering works of Hirota and Suzuki$^{4,5)}$ in
order to stimulate the integrable Toda lattice$^{6)}$ by electric circuits
there has been increased interest in the propagation of wave trains in
nonlinear-dispersive transmission lines, involving phenomena such as
Benjamin-Feir instability,$^{7-9)}$ the formation of stationary localized
waves, that is, the envelope solitons$^{10,11)}$ and the dark solitons.$%
^{12,13)}$

The Benjamin-Feir (or, as it is sometimes called, the modulational)
instability is widespread and plays an important role in various nonlinear
wave phenomena. Simply put, if dispersion and nonlinearity act against each
other, monochromatic wave trains do not wish to remain monochromatic. The
sidebands of the carrier wave can draw on its energy via a resonance
mechanism with the result that the envelope becomes modulated. In one space
dimension, this envelope modulation continues to grow until the soliton
shape is reached. At this point, nonlinearity and dispersion are in exact
balance and no further distortion occurs.$^{14,15)}$

It is well known that the self-modulation of one space dimension waves in
nonlinear dispersive systems can be described by the so-called
Ginzburg-Landau (GLE) equation,$^{16-18)}$%
\begin{equation}
iu_t+Pu_{xx}+Q\left| u\right| ^2u=i\gamma ,  \label{1.1}
\end{equation}
where subscripts $t$ and $x$ denote the partial differentiation with respect
to $t$ and $x$, respectively. If $PQ<0,$ a plane wave in this system is
stable for the modulation and, otherwise, is instable. Especially in the
later case there exist special families of solutions, which are called
envelope solitons and show various interesting phenomena.$^{19,20)}.$

Recently, there has been progress towards a mathematical understanding of
equation (1.1). Kirchgassner$^{21)}$ and Mielke$^{22,23,24)}$ restrict
attention to steady-state equations and view the single unbounded spatial
direction as an evolution variable.

In this paper, we give a rigorous derivation of the full time-dependent
Modified Ginzburg-Landau Equation (MGLE). The Benjamin-Feir ( modulational)
instability for the obtained MGLE is investigated.

\section{Basic equations}

In this section we derive nonlinear wave equation for electromagnetic wave
propagation on a nonlinear-dispersive transmission line shown in Figure 1.
By using the method of multiple scales, we derive a MGLE.

\subsection{The model equations}

\begin{center}
In the considered transmission line of figure 1, $C_N$ 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. 
\begin{equation*}
\left. 
\begin{array}{c}
\FRAME{itbpxFX}{269.8125pt}{126.0625pt}{0pt}{}{}{Figure }{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
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0pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";tempfilename
'C:/ARTICLES/H2ZZRTQW.wmf';tempfile-properties "XP";}} \\ 
Fig.\text{ }1.\text{\emph{A section for a distributed nonlinear-dispersive
transmission line.}}
\end{array}
\right.
\end{equation*}
\end{center}

By applying the Kirchhoff's voltage theorem and the current theorem we
obtain 
\begin{equation}
\frac{\partial I}{\partial x}+\frac{\partial \rho (V)}{\partial t}=0;\qquad 
\frac{\partial V}{\partial x}+L\frac{\partial I_1}{\partial t}=0;\qquad 
\frac{\partial ^2V}{\partial x\partial t}+\frac 1{C_s}\left( I-I_1\right) =0
\label{2.1}
\end{equation}

where the current through the nonlinear capacitor is given by $\partial \rho
(V)/\partial t.$

>From equations (2.1) we can eliminate $I$ and $I_1$ and write

System (2.1) becomes 
\begin{equation}
C_s\frac{\partial ^4V}{\partial x^2\partial t^2}+\frac 1L\frac{\partial ^2V}{%
\partial x^2}-\frac{\partial ^2\rho }{\partial t^2}=0.  \label{2.2}
\end{equation}

With no loss of generality, we may regard $\rho (0)=0$ and expand $\rho (V)$
to obtain $\rho (V)\approx \rho ^{\prime }(0)V+\frac{\rho ^{\prime \prime
}(0)}2V^2.$ For bounded solutions, we must have $\rho ^{\prime }(0)\geq 0.$
Hence we have

\begin{equation}
\rho (V)\approx C_0\left( V-\beta ^{\prime }V^2\right) =C_0V-C_NV^2.
\label{2.3}
\end{equation}

Substituting (2.3) into (2.2), we obtain the following partial differential
equation for the voltages,

\begin{equation}
C_0\frac{\partial ^2V}{\partial t^2}-\frac 1L\frac{\partial ^2V}{\partial x^2%
}-C_S\frac{\partial ^4V}{\partial x^2\partial t^2}-C_N\frac{\partial ^2V^2}{%
\partial t^2}=0.  \label{2.4}
\end{equation}

\subsection{Derivation of the Generalized Complex Ginzburg-Landau Equation}

If we introduce the notation 
\begin{equation*}
\alpha =-1/L,\text{ }\beta =-C_N,\text{ }\lambda =C_S,
\end{equation*}
equation (2.4) takes the form

\begin{equation}
C_0\frac{\partial ^2V}{\partial t^2}+\alpha \frac{\partial ^2V}{\partial x^2}%
-C_s\frac{\partial ^4V}{\partial x^2\partial t^2}+\beta \frac{\partial ^2V^2%
}{\partial t^2}=0.  \label{2.5}
\end{equation}

We follow Taniuti and Yajima$^{25,26)}$ and seek a first-order uniform
expansion by using the method of multiple scales in the form 
\begin{equation}
\left. 
\begin{array}{l}
V=\varepsilon ^{1/2}v_{11}\exp \left[ i(kX_0-\omega T_0)\right] +\varepsilon
V_{22}\exp \left[ 2i(kX_0-\omega T_0)\right] \\ 
\qquad +\varepsilon ^{3/2}v_{33}\exp \left[ 3i(kX_0-\omega T_0)\right]
+\varepsilon ^2\left[ v_{42}\exp \left[ 2i(kX_0-\omega T_0)\right] \right.
\\ 
\qquad \left. +v_{44}\exp \left[ 4i(kX_0-\omega T_0)\right] \right] +cc+...,
\end{array}
\right.  \label{2.6}
\end{equation}
where $cc$ stands for the complex conjugate, $\varepsilon $ is a small,
dimensionless parameter related to the amplitudes ($0<\varepsilon \ll
1),v_{ij}=v_{ij}(X_1,T_1,T_2),$ $T_n=\varepsilon ^nt,$ and $X_n=\varepsilon
^nx.$

Substituting (2.6) into (2.5) and equating coefficients of like powers of $%
\varepsilon $ and $\exp \left[ i\theta \right] $ (here $\theta =kX_0-\omega
T_0),$ we obtain

\textbf{Order} $\varepsilon ^{1/2},$ $\exp \left[ i\theta \right] $%
\begin{equation}
\left[ C_0\omega ^2+\alpha k^2+\lambda k^2\omega ^2\right] v_{11}=0;
\label{2.7}
\end{equation}

\textbf{Order} $\varepsilon ^{3/2},$ $\exp \left[ i\theta \right] $%
\begin{equation}
-2i\omega \left[ C_0+\lambda k^2\right] \frac{\partial v_{11}}{\partial T_1}%
+2ik\left[ \alpha +\lambda \omega ^2\right] \frac{\partial v_{11}}{\partial
X_1}-2\beta \omega ^2v_{11}^{*}v_{22}=0,  \label{2.8}
\end{equation}

\textbf{Order} $\varepsilon ,\exp \left[ 2i\theta \right] $%
\begin{equation}
-4\left( C_0\omega ^2+\alpha k^2+4\lambda k^2\omega ^2\right) -4\omega
^2\beta v_{11}^2=0;  \label{2.9}
\end{equation}

\textbf{Order} $\varepsilon ^{3/2},$ $\exp \left[ 3i\theta \right] $%
\begin{equation}
-9\left( C_0\omega ^2+\alpha k^2+9\lambda k^2\omega ^2\right) -18\omega
^2\beta v_{11}v_{22}=0;  \label{2.10}
\end{equation}

\textbf{Order} $\varepsilon ^2,$ $\exp \left[ 2i\theta \right] $%
\begin{equation}
\left[ -4\left( C_0\omega ^2+\alpha k^2+4\lambda k^2\omega ^2\right) \right]
v_{42}-8\beta \omega ^2v_{11}^{*}v_{33}=0;  \label{2.11}
\end{equation}

\textbf{Order} $\varepsilon ^2,$ $\exp \left[ 4i\theta \right] $%
\begin{equation}
\left[ -16\left( C_0\omega ^2+\alpha k^2+16\lambda k^2\omega ^2\right)
\right] v_{44}-\beta \omega ^2\left[ 32v_{11}v_{33}+16v_{22}^2\right] =0
\label{2.12}
\end{equation}

\textbf{Order} $\varepsilon ^{5/2},$ $\exp \left[ i\theta \right] $%
\begin{equation}
\left. 
\begin{array}{c}
C_0\left[ \frac{\partial ^2v_{11}}{\partial T_1^2}-2i\omega \frac{\partial
v_{11}}{\partial T_2}\right] +\alpha \frac{\partial ^2v_{11}}{\partial X_1^2}%
-\lambda \left[ -k^2\frac{\partial ^2v_{11}}{\partial T_1^2}+2i\omega k^2%
\frac{\partial v_{11}}{\partial T_2}+4k\omega \frac{\partial ^2v_{11}}{%
\partial T_1\partial X_1}\right. \\ 
\left. -\omega ^2\frac{\partial ^2v_{11}}{\partial X_1^2}\right] +\beta
\left[ -4i\omega \frac{\partial v_{11}^{*}v_{22}}{\partial T_1}-2\omega
^2v_{11}^{*}v_{42}-2\omega ^2v_{22}^{*}v_{33}\right] =0.
\end{array}
\right.  \label{2.13}
\end{equation}

For the nontrivial solution we must have $v_{11}\neq 0.$ Then (2.7) gives 
\begin{equation}
C_0\omega ^2+\alpha k^2+\lambda k^2\omega ^2=0.  \label{2.14}
\end{equation}
(2.14) is the dispersion relation which is illustrated in Figure 2 for the
line parameters 
\begin{equation}
C_s=5C_0=1200pF,\text{ }L=14\mu H,\text{ }C_N=38.4pF,\text{ }0\leq k\leq
1.58,\text{ }\varepsilon =0.1.  \label{2.15}
\end{equation}
\begin{equation*}
\left. 
\begin{array}{c}
\FRAME{itbpxFX}{218.6875pt}{146.375pt}{0pt}{}{}{Figure }{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "T";width 218.6875pt;height 146.375pt;depth 0pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'C:/ARTICLES/H2ZZRTNI.wmf';tempfile-properties "XP";}} \\ 
Fig.2.\text{\emph{Dispersion curve for the linearized version }} \\ 
\text{\emph{of the transmission line shown in Fig. 1.}}
\end{array}
\right.
\end{equation*}

Using the dispersion relation (2.14), equations (2.9)-(2.12) give 
\begin{equation}
v_{22}=-\frac \beta {3\lambda k^2}v_{11}^2;  \label{2.16}
\end{equation}
\begin{equation}
v_{33}=\frac{\beta ^2}{12\lambda ^2k^4}v_{11}^3;  \label{2.17}
\end{equation}
\begin{equation}
v_{42}=-\frac{\beta ^3}{108\lambda ^2k^8\omega ^2}\left| v_{11}\right|
^2v_{11}^2;  \label{2.18}
\end{equation}
\begin{equation}
v_{44}=-\frac{\beta ^3}{54\lambda ^3k^6}v_{11}^4  \label{2.19}
\end{equation}
respectively.

Solving for $\partial v_{11}/\partial T_1$ from (2.9) and using (2.16), we
obtain 
\begin{equation*}
\frac{\partial v_{11}}{\partial T_1}=\frac{C_0}\alpha \frac{\omega ^3}{k^3}%
\frac{\partial v_{11}}{\partial X_1}-\frac{i\beta \omega ^3}{\alpha k^2}%
v_{11}^{*}v_{22}
\end{equation*}

where $V_g=-\frac{C_0}\alpha \left( \frac \omega k\right) ^3=\frac{C_0\sqrt{%
-\alpha }}{(C_0+\lambda k^2)^{3/2}}$ is the group velocity. Hence 
\begin{equation}
\left. 
\begin{array}{l}
\frac{\partial ^2v_{11}}{\partial T_1^2}=\frac{C_0^2}{\alpha ^2}\frac{\omega
^6}{k^6}\frac{\partial ^2v_{11}}{\partial X_1^2}-C_0\frac{i\beta \omega ^6}{%
\alpha ^2k^5}\frac \partial {\partial X_1}\left( v_{11}^{*}v_{22}\right) -%
\frac{i\beta \omega ^3}{\alpha k^2}\frac \partial {\partial T_1}\left(
v_{11}^{*}v_{22}\right) \\ 
\frac{\partial ^2v_{11}}{\partial T_1\partial X_1}=\frac{C_0}\alpha \frac{%
\omega ^3}{k^3}\frac{\partial ^2v_{11}}{\partial X_1^2}-\frac{i\beta \omega
^3}{\alpha k^2}\frac \partial {\partial X_1}\left( v_{11}^{*}v_{22}\right) .
\end{array}
\right.  \label{2.20}
\end{equation}
Combining (2.20) and (2.13), and using (2.16)-(2.18), we obtain, in terms of
the original variables $t$ and $x$, 
\begin{equation}
i\frac{\partial v_{11}}{\partial t}+P\frac{\partial ^2v_{11}}{\partial x^2}%
+iQ_1\frac \partial {\partial t}\left( \left| v_{11}\right| ^2v_{11}\right)
+iQ_2\frac \partial {\partial x}\left( \left| v_{11}\right| ^2v_{11}\right)
+Q_3\left| v_{11}\right| ^4v_{11}=0,  \label{2.21}
\end{equation}
where 
\begin{equation}
P=P(k)=-\frac 12\omega ^{\prime \prime }=-\frac{3C_0\lambda \sqrt{-\alpha }k%
}{2\left( C_0+\lambda k^2\right) ^{5/2}};  \label{2.22}
\end{equation}
\begin{equation}
Q_1=Q_1(k)=-\frac{\beta ^3\varepsilon }{2\lambda k^2\left( C_0+\lambda
k^2\right) };  \label{2.23}
\end{equation}
\begin{equation}
Q_2=Q_2(k)=\frac{\beta ^2\varepsilon \left( C_0+4\lambda k^2\right) }{%
3\lambda k^3\left( C_0+\lambda k^2\right) ^2};  \label{2.24}
\end{equation}
\begin{equation}
Q_3=Q_3(k)=-\frac{\beta ^4\varepsilon ^2\sqrt{-\alpha }}{12\lambda
^3k^5\left( C_0+\lambda k^2\right) ^{3/2}}.  \label{2.25}
\end{equation}
Using the transformation 
\begin{equation}
\xi =x-Q_2Q_1^{-1}t,\qquad \tau =t,\qquad v_{11}(\xi ,\tau )=u(\xi ,\tau
)\exp \left[ iQ_2P^{-1}Q_1^{-1}\xi /2\right] ,  \label{2.26}
\end{equation}
we write (2.21) in the final form 
\begin{equation}
i\frac{\partial u}{\partial \tau }+P\frac{\partial ^2u}{\partial \xi ^2}%
+iQ_1\frac \partial {\partial \tau }\left( \left| u\right| ^2u\right)
-\gamma u+Q_3\left| v_{11}\right| ^4v_{11}=0,  \label{2.27}
\end{equation}
where $\gamma =-Q_2^2P^{-1}Q_1^{-2}/4.$

Thus the resulting equation (2.27) that describes the evolution of a
wavepacket is a complex envelope equation that involves higher order
nonlinearities. We call this equation the Modified Ginzburg-Landau Equation.

For the line parameters (2.15) we plot the following coefficient of the
spatial dispersion's curve. 
\begin{equation*}
\ \ \left. 
\begin{array}{c}
\FRAME{itbpxFX}{218.6875pt}{146.375pt}{0pt}{}{}{Figure }{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "T";width 218.6875pt;height 146.375pt;depth 0pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'C:/ARTICLES/H2ZZRTIL.wmf';tempfile-properties "XP";}} \\ 
Fig.\text{ 3. \emph{Coefficient of the spatial dispersion's curve.}}
\end{array}
\right.
\end{equation*}
It is seen from Figure 3 that the coefficient of the spatial dispersion is
always negative when $0\leq k\leq 1.58.$

In the next section we study the Benjamin-Feir instability of the
monochromatic wave solutions.

\section{The Benjamin-Feir instability}

To study the Benjamin-Feir instability of the monochromatic wave solutions,
we first express $u$ in the polar form 
\begin{equation}
u(\xi ,\tau )=a(\xi ,\tau )\exp \left[ ib(\xi ,\tau )\right] .  \label{3.1}
\end{equation}
Substituting (3.1) into (2.27) and separating imaginary and real parts we
obtain 
\begin{equation}
\left( 1+3Q_1a^2\right) \frac{\partial a}{\partial \tau }+P\left( 2\frac{%
\partial a}{\partial \xi }\frac{\partial b}{\partial \xi }+a\frac{\partial
^2b}{\partial \xi ^2}\right) =0,  \label{3.2}
\end{equation}
\begin{equation}
\left( a+Q_1a^3\right) \frac{\partial b}{\partial \tau }+P\left( a\left( 
\frac{\partial b}{\partial \xi }\right) ^2-\frac{\partial ^2a}{\partial \xi
^2}\right) +\gamma a-Q_3a^5=0.  \label{3.3}
\end{equation}

If the wave has a fixed, single wavenumber, then $P=-\frac 12\frac{d^2\omega 
}{dk^2}=0$ and system (3.2),(3.3) reduces to 
\begin{equation*}
\left( 1+3Q_1a^2\right) \frac{\partial a}{\partial \tau }=0,\text{ \qquad }%
\left( a+Q_1a^3\right) \frac{\partial b}{\partial \tau }+\gamma a-Q_3a^5=0
\end{equation*}
whose solutions are 
\begin{equation}
a=a_0,\qquad b=\frac{Q_3a_0^4-\gamma }{1+Q_1a_0^2}\tau +\text{constant,}
\label{3.4}
\end{equation}
where $a_0$ is constant.

It natural to define the local wavenumber $k$ as the $\xi $ derivative of
the total phase and the local frequency as the negative of the $\tau $
derivative of the total phase $\theta =k_0\xi -\omega _0\tau +b(\xi ,\tau )$%
\begin{equation*}
k=k_0+b_\xi ,\qquad \omega =\omega _0-b_\tau .
\end{equation*}
Note that 
\begin{equation}
k_\tau +\omega _\xi =b_{\xi \tau }-b_{\tau \xi }=0,  \label{3.5}
\end{equation}
which expresses the conservation of the number of waves. We will write the
change in wavenumber $b_\xi $ as $K.$ Now equation (3.2) gives 
\begin{equation}
\frac \partial {\partial \tau }\left( 2a^2+3Q_1a^4\right) +4P\frac \partial
\xi \left( a^2K\right) =0,  \label{3.6}
\end{equation}
which is the equation of conservation of wave action. On the other hand
equation (3.3) gives 
\begin{equation*}
a\left( 1+Q_1a^2\right) b_\tau +P\left( aK^2-a_{\xi \xi }\right) +\gamma
a-Q_3a^5=0,
\end{equation*}
which when differentiating with respect to $\xi $ gives 
\begin{equation}
\left. 
\begin{array}{c}
a^2\left( 1+Q_1a^2\right) ^2K_\tau +a_\xi \left( 1+3Q_1a^2\right) \left[
Q_3a^5-\gamma a+P\left( a_{\xi \xi }-aK^2\right) \right] \\ 
+Pa\left( 1+Q_1a^2\right) \left( a_\xi K^2+2aKK_\xi -a_{\xi \xi \xi }\right)
+a\left( 1+Q_1a^2\right) \left( \gamma -5Q_3a^4\right) a_\xi =0
\end{array}
\right.  \label{3.7}
\end{equation}
Equation (3.7) is the relation for conservation of waves, since 
\begin{equation*}
\omega =\omega _0+\frac{\gamma +P\left( K^2-\frac{a_{\xi \xi }}a\right)
-Q_3a^4}{1+Q_3a^2}.
\end{equation*}

Next the monochromatic wave solution (3.4) means that 
\begin{equation*}
k=k_0,\qquad \omega =\omega _0-\frac{Q_3a_0^4-\gamma }{1+Q_1a_0^2}.
\end{equation*}
This is the Stokes wave. Test it linear stability by setting 
\begin{equation}
a=a_0+\tilde a,\qquad K=\tilde K,  \label{3.8}
\end{equation}
where $\tilde a$ is assumed to be infinitesimal. Substituting (3.8) into
(3.6) and (3.7) and keeping only linear terms in perturbation quantities, we
obtain 
\begin{equation*}
\tilde a_\tau =-\frac{Pa_0}{1+3Q_1a_0^2}\tilde K_\xi ,
\end{equation*}
\begin{equation*}
K_\tau =\frac P{a_0\left( 1+Q_1a_0^2\right) }\tilde a_{\xi \xi \xi }+\frac{%
2\left( Q_1Q_3a_0^4+2Q_3a_0^2+Q_1\gamma \right) a_0^2}{a_0\left(
1+Q_1a_0^2\right) ^2}\tilde a_\xi ,
\end{equation*}
or 
\begin{equation}
\tilde a_{\tau \tau }=-\frac{P\left( 1+Q_1a_0^2\right) ^{-2}}{\left(
1+3Q_1a_0^2\right) }\left[ P\left( 1+Q_1a_0^2\right) \tilde a_{\xi \xi \xi
\xi }+2a_0^2\left( Q_1Q_3a_0^4+2Q_3a_0^2+Q_1\gamma \right) \tilde a_{\xi \xi
}\right] .  \label{3.9}
\end{equation}

Therefor if $\tilde a\propto \exp \left[ il\xi +\Omega \tau \right] ,$%
\begin{equation}
\Omega ^2=-\frac{\left( 1+Q_1a_0^2\right) ^{-2}k^2}{\left(
1+3Q_1a_0^2\right) }\left[ \left( 1+Q_1a_0^2\right) P^2l^2-2a_0^2P\left(
Q_1Q_3a_0^4+2Q_3a_0^2+Q_1\gamma \right) \right] .  \label{3.10}
\end{equation}

Because $\beta =-C_N<0,$ it follows from (2.22)-(2.25) that $P(k)<0,$ $%
Q_1(k)>0,$ $Q_3(k)<0,$ and $\gamma =-Q_2^2P^{-1}Q_1^{-2}/4>0.$ Therefore we
have the following results

\begin{theorem}
If 
\begin{equation}
Q_1Q_3a_0^4+2Q_3a_0^2+Q_1\gamma <0,  \label{3.11}
\end{equation}
the monochromatic wave solution (3.4) will be unstable to long waves in the
range 
\begin{equation}
0<l^2<\frac{2a_0^2\left( Q_1Q_3a_0^4+2Q_3a_0^2+Q_1\gamma \right) }{P\left(
1+Q_1a_0^2\right) }.  \label{3.12}
\end{equation}
\end{theorem}

(3.11) is the Benjamin-Feir instability criterion to the MGLE in the
electrical mono-inductance transmission line. This new result is different
from the Lange and Newell criterion for the Stokes wave$^{27,28)}$ by the
presence of the amplitude $a_0$ of the monochromatic wave.

For the line parameters (2.15) we plot $Q_1Q_3a_0^4+2Q_3a_0^2+Q_1\gamma $ as
the function of the amplitude $a_0$ or/and of function of the wavenumber $k.$%
\begin{equation*}
\left. 
\begin{array}{c}
\FRAME{itbpxFX}{218.6875pt}{146.375pt}{0pt}{}{}{Figure }{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "T";width 218.6875pt;height 146.375pt;depth 0pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'C:/ARTICLES/H2ZZRTHW.wmf';tempfile-properties "XP";}} \\ 
Fig.\text{ 4}\emph{.}\text{ Curve of }Q_1Q_3a_0^4+2Q_3a_0^2+Q_1\gamma \text{
as function of }a_0\text{ with }k=0.1.
\end{array}
\right.
\end{equation*}
\begin{equation*}
\left. 
\begin{array}{c}
\FRAME{itbpxFX}{218.6875pt}{146.375pt}{0pt}{}{}{Figure }{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "T";width 218.6875pt;height 146.375pt;depth 0pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'C:/ARTICLES/H2ZZRTS5.wmf';tempfile-properties "XP";}} \\ 
Fig.\text{ }5.\text{ Curve of }Q_1Q_3a_0^4+2Q_3a_0^2+Q_1\gamma \text{ } \\ 
\text{as function of }k\text{ }\left( 0.09\leq k\leq 1.58\right) \text{with }%
a_0=1000\text{.}
\end{array}
\right.
\end{equation*}
\begin{equation*}
\left. 
\begin{array}{c}
\FRAME{itbpxFX}{218.6875pt}{146.375pt}{0pt}{}{}{Figure }{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "T";width 218.6875pt;height 146.375pt;depth 0pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'C:/ARTICLES/H2ZZRT4M.wmf';tempfile-properties "XP";}} \\ 
\begin{array}{c}
Fig.\text{ }6\emph{.}\text{ Curve of }Q_1Q_3a_0^4+2Q_3a_0^2+Q_1\gamma \text{ 
} \\ 
\text{as function of }k\text{ }\left( 0<k\leq 0.05\right) \text{with }%
a_0=1000\text{.}
\end{array}
\end{array}
\right.
\end{equation*}

Figure 4 shows that for the wavenumber $k=0.1,$ condition $(3.11)$ occurs
for all $a_0>a_{0c}\simeq 240.$ For these values of $a_0,$ the monochromatic
wave solutions corresponding to the fixed wavenumber $k=0.1$ are
modulational unstable. All the monochromatic wave solutions associated to $%
k=0.1$ with any amplitude $a_0<a_{0c}$ are stable.

It is seen from Figures 5 and 6 that for any fixed wavenumber $0<k<0.05,$
the corresponding monochromatic wave solution with amplitude $a_0=1000$ is
modulational unstable, while any monochromatic wave solution corresponding
to the wavenumber $0.09\leq k\leq 1.58$ with amplitude $a_0=1000$ is
modulational stable.

Figure 4-6 show that $Q_1Q_3a_0^4+2Q_3a_0^2+Q_1\gamma $ as function of $k$
or/and $a_0$ changes its sign for particular value of $k$ or/and $a_0.$

\section{Conclusion}

In this paper mono-inductance LC circuit is considered and envelope
modulation is reduced to the modified Ginzburg-Landau equation.
Benjamin-Feir instability for the MGL equation is analyzed. As far as we
know there have been no such Stoves wave analysis relating to LC circuit. As
in most cases, the linear part of the modulation equation (coefficient of
spatial dispersion) is fixed, that is,$P=\frac{-1}2\frac{d^2\omega }{dk^2}.$ 
$Q_1Q_3a_0^4+2Q_3a_0^2+Q_1\gamma <0$ is necessary condition and not
sufficient condition for the instability. It should be noted that $%
Q_1Q_3a_0^4+2Q_3a_0^2+Q_1\gamma >0$ is sufficient condition for the
stability. In fact, if this last condition is satisfied then for every real $%
l$, $\Omega $ will be always pure imaginary and $\tilde a\propto \exp \left[
ik\xi +\Omega \tau \right] $ will be bounded. In most cases the criterion of
the instability does not depend on the amplitude of the monochromatic wave.
But for our MGL equation, the said criterion depends on the amplitude $a_0.$
This fact allows us to construct an unstable monochromatic wave for a given
wavenumber.

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\end{document}