\section{Modulation instability of Stokes waves}
\label{ModulInstStkesWaves}
\setcounter{equation}{0}

{The Stokes wave is unstable with respect to growing of
long-scale
modulation. This remarkable fact was first established 
in  \cite{Lighthill1965}, 
who calculated a growth-rate of}
instability in the limit of long-wave perturbation. As far as
Lighthill's growth-rate coefficient was proportional to the wave
number of perturbation length, the result was in principle
incomplete: somewhere at short scales the instability must be
arrested. The complete form of the growth-rate coefficient was
found independently in \cite{Zakh1966}, \cite{Z68}) and in \cite{BenFeir1967}. The
original Zakharov's work of 1966 was published in Russian journal
 and it's English translation appeared next year, when the article of Benjamin
 and Feir was already published; so the instability was called by
Benjamin and Feir. Actually, it is not unfair because it was Fair
who first observed this instability in experiment.

The presented technique based on the conformal mapping makes
possible to study modulation instability in a very compact way.
It is convenient to use cubic equations (\ref{EqRV}),
with
\begin{equation}\label{UandB}
U = \hat P(V\bar R + \bar V R).
\end{equation}
Let $g=1,\,\,k=1$. To study instability of the
Stokes wave, propagating with the velocity $c>1$, one has to go to
the moving reference frame by the following change of variables:
\begin{equation}
y=u-ct,\quad \tau=t,\quad R=1-\frac{iV}{c}+ r .
\end{equation}

Then the Dyachenko equations take the form:
\begin{eqnarray}\label{D_small}
&&\frac{\partial}{\partial t}\left(r-\frac{iV}{c}\right)+c
r'=i(\tilde u r' -r\tilde u')\nonumber\\
&&\frac{\partial V}{\partial t}=i(VV' -B')-\frac{V}{c} B'
-\frac{ig}{c} V + gr + i\tilde u V'\\
&&\tilde u =\bar p (\bar r V + r\bar V)\nonumber
\end{eqnarray}
For the stationary wave we have
\[r=0,\quad \tilde u=0, \quad
\frac{\partial r}{\partial t}=0,\quad \frac{\partial V}{\partial
t}=0,
\]
so, for its description we obtain the following equation
\begin{equation}\label{STAT_D}
c V_u =i(VV'-B')-\frac{V}{c}\, B' -\frac{ig}{c}\,V .
\end{equation}
The solution of this equation can be found in the form of Fourier
series. Again, we put for simplicity $g=1, u=1$. Then
\begin{equation}
V=\sum_{n=1}^{\infty} V_n e^{-inu}
\end{equation}
Coefficients $V_n$ can be found either directly for equation (\ref{STAT_D})
or expressed in terms of Fourier coefficients $a_n$ of complex
elevation $z$. Indeed,
\[
V=-ic \frac{\sum_{n=1}^{\infty} n a_n
e^{-inu}}{1+\sum_{n=1}^{\infty} n a_n e^{-inu}}
\]
Thus
\begin{eqnarray}
&&V_1=-ica_1\nonumber\\
&&V_2=-ic(2a_2 -a_1^2).
\end{eqnarray}

Linearization of system (\ref{D_small}) against the stationary solution
leads to equations
\begin{eqnarray}\label{LinearD}
&&\frac{\partial}{\partial t}\left(r-i\frac{\delta V}{c}\right) +c
r_u=0,\\
&&\left(\frac{\partial}{\partial t}+c\frac{\partial}{\partial
u}\right) \delta V=i\frac{\partial}{\partial u}(V_0 \delta
V-\delta B)-\frac{V_0}{c} \delta B_u - \frac{ig}{c}\delta V + gr
.\nonumber
\end{eqnarray}
System (\ref{LinearD}) contains all information about stability of the
Stokes wave.

The modulation instability is described by a perturbation
presented as a sum of following harmonics:
\[
\delta V, r \equiv e^{-i\kappa u} , e^{(1\pm\kappa)u
-in\kappa},\quad  n=1,\ldots ,\quad \kappa<1 .
\]
In the leading order of nonlinearity one can put
\begin{eqnarray}\label{LeadingO}
&&r=p_1 e^{-i(1+\kappa)u} + p_2 e^{-i(1-\kappa)u} ,\nonumber\\
&&V=q_1 e^{-i(1+\kappa)u} + q_2 e^{-i(1-\kappa)u} .
\end{eqnarray}
Plugging (\ref{LeadingO}) to (\ref{LinearD}) one obtains closed equations to $p_1, p_2,
q_1, q_2$:
\begin{eqnarray}
&&\dot{p}_1 -\frac{i}{c}\,\, \dot{q}_1=ic(1+\kappa) p_1 ,\nonumber\\
&&\dot{\bar p}_2 +\frac{i}{c}\,\, \dot{\bar q}_2 =-ic (1-\kappa)
p_2 ,\\
&&\dot q_1 -i\left(\frac{1}{c}-c(1+\kappa)\right)\,q_1 -p_1 =V_2
(1+\kappa)\bar q_2 ,\nonumber\\
&&\dot{\bar q}_2 -i\left(\frac{1}{c}-c(1-\kappa)\right)\,\bar q_2
-\bar p_2 = V_2(1-\kappa)\,q_1 .\nonumber
\end{eqnarray}
Here $V_2$ is the amplitude of second harmonics. Assuming
$p_1,q_1,\bar p_2, \bar q_2 \simeq e^{i(\Omega +\kappa c)t}$, one
gets the following equation for $\Omega$:
\begin{equation}\label{EqOmega}
\left[(\Omega -c)^2 -1-\kappa\right]\left[(\Omega
+c)^2-1+\kappa\right]=(c^2 -\Omega^2)(\frac{1}{c}-c)^2
(1-\kappa^2)
\end{equation}
To obtain this equation we put
\begin{equation}\label{Note1}
|V_2|^2=\left(\frac{1}{c}-c\right)^2.
\end{equation}
This condition appears from the natural physical requirement: if
$\kappa=0$, then $\Omega=0$ is a solution of (\ref{EqOmega}). In our
approximation $V_2=-ic a_1^4,\,\,c^2=1+a_1^2$, and condition
(\ref{Note1}) is satisfied. Assuming that $\Omega\ll 1$, one can simplify
(\ref{EqOmega}) to quadratic equation, that is equivalent to the equation
derived in \cite{Z68}:
\begin{equation}\label{EqOmega2}
\left(\Omega -c+\sqrt{1+\kappa}\right)\left(\Omega
-+c-\sqrt{1-\kappa}\right)=(c-1)^2\,(1-\kappa^2).
\end{equation}
 Assuming $a_1=A$, we put
\[
c=1+\frac{1}{2}\,A^2,\,\,\,\sqrt{1+\kappa}\simeq
1+\frac{1}[{2}\kappa
-\frac{1}{8}\kappa^2,\,\,\,\sqrt{1-\kappa}\simeq
1-\frac{1}{2}\kappa -\frac{1}{8}\kappa^2 ,
\]
and get from (\ref{EqOmega2}) the standard growth-rate of modulation
instability
\begin{equation}
\Omega^2=\frac{1}{8}\left(-A^2 \kappa^2 +\frac{1}{8}
\kappa^4\right),
\end{equation}
the result that was obtained by 
\cite{Zakh1966, Z68}, and by 
\cite{BenFeir1967}. Lighthill in 1965 found long-wave asymptotic of
the instability growth-rate,
\begin{equation}
\Omega^2=-\frac{1}{8}\,A^2\,\kappa^2 ,
\end{equation}
with the maximum value of the growth-rate,
\begin{equation}
\Omega^2=-\frac{1}{4}\,A^2\,\kappa^2 ,
\end{equation}
achieved at
\begin{equation}
\kappa^2=4\,A^2 .
\end{equation}

The technique developed above makes possible to study the
mod and other instabilities with any arbitrary accuracy.

