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\begin{document}

\begin{frontmatter}

\title{Freak Waves as Nonlinear Stage of Stokes Wave Modulation Instability}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\author[Zakharov]{V.E. Zakharov\thanksref{also1}\thanksref{also2}},~
\author[Tucson,Landau]{V.E. Zakharov\thanksref{also2}}
{
\thanks[also2]{Waves and Solitons LLC, W. Sereno Dr., Gilbert, AZ, 85233, USA} 
\ead{zakharov@math.arizona.edu}
}
\author[Landau]{A.I. Dyachenko\corauthref{cor}}
\corauth[cor]{Corresponding author.}
\ead{alexd@landau.ac.ru}
\author[Landau]{A.O. Prokofiev}
\ead{alexpro@itp.ac.ru}

\address[Tucson]{Department of Mathematics, University of
Arizona, Tucson, AZ, 857201, USA}
\address[Landau]{Landau Institute for Theoretical Physics, 
2 Kosygin str., Moscow, 119334, Russia}

\begin{abstract}
Numerical simulation of evolution of nonlinear gravity waves is presented. Simulation is done 
using two-dimensional code, based on conformal mapping of the fluid to the lower half-plane.  
We have considered two problems: i) modulation instability of wave train and ii) evolution of 
$NLSE$ solitons with different steepness of carrier wave. 
In both cases we have observed formation of freak waves. 
\end{abstract}

\begin{keyword}
free surface, freak wave, modulation instability
\PACS{02.60.Cb, 47.15.Hg, 92.10.Dh, 92.10Hm}
\end{keyword}
\end{frontmatter}

\section{Introduction}

Waves of extremely large size, alternatively called freak, rogue
or giant waves are a well-documented hazards for mariners (see, for
instance \cite{Smith1976}, \cite{Dean1990}, \cite{Chase2003}). These waves are
responsible for loss of many ships and many human lives. Freak
waves could appear in any place of the world ocean (see \cite{Earle1975},
\cite{Mori2002}, \cite{Divinsky2004}); however, in some regions
they are more probable than in the others. One of the regions where
freak waves are especially frequent is the Agulhas current of the
South-East coast of South Africa (see \cite{Gerber1996}, 
\cite{GutshabashLavrenov1986}, \cite{IrvineTilley1988}, \cite{Lavrenov1998}, 
\cite{Mallory1974}). In the paper by \cite{Peregrine1976} it was  suggested that in 
areas of strong current
such as the Agulhas, giant waves could be produced when wave
action is concentrated by reflection into a caustic region.
According to this theory, a variable current acts analogously to
an optic lens to focus wave action. The caustic theory of freak
waves was supported since that time by works of many authors.
Among them \cite{Smith1976}, \cite{GutshabashLavrenov1986}, 
\cite{IrvineTilley1988}, \cite{Sand1990}, \cite{Gerber1987}, \cite{Gerber1993}, 
\cite{KharPel2003}. The
statistics of caustics with application to calculation of the
freak wave formation probability was studied in the paper of 
\cite{WhiteFornberg1998}.

On our opinion, a connection between freak wave generation and
caustics for swell or wind-driven sea is the indisputable fact.
However, this is not the end of the story. Focusing of ocean waves
by an inhomogeneous current is a pure linear effect. Meanwhile, no
doubts that freak waves are essentially nonlinear objects. They
are very steep. In the last stage of their evolution, the
steepness becomes infinite, forming a "wall of water". Before this
moment, the steepness is higher than one for the limiting Stokes
wave. Moreover, a typical freak wave is a single event (see, for
instance \cite{Divinsky2004}. Before breaking it has a crest,
three-four (or even more) times higher than the crests of neighbor
waves. The freak wave is preceded by a deep trough or "hole in the
sea". A characteristic life time of a freak wave is short - ten of
wave periods or so. If the wave period is fifteen seconds, this is
just few minutes. Freak wave appears almost instantly from a
relatively calm sea. Sure, these peculiar features of freak waves
cannot be explained by a linear theory. Focusing of ocean waves
creates only preconditions for formation of freak waves, which is
a strongly nonlinear effect.

It is natural to associate appearance of freak waves with the
modulation instability of Stokes waves. This instability is
usually called after Benjamin and Feir, however, it was first
discovered by \cite{Lighthill1965}. The theory of instability was
developed independently by \cite{BenFeir1967} and by
\cite{Zakh1966}. \cite{Feir1967} was the first who observed the instability
experimentally in 1967.

Slowly modulated weakly nonlinear Stokes wave is described by
nonlinear Shr\"odinger equation ($NLSE$), derived by \cite{Zakh1968}. 
This equation is integrable (see \cite{ZakharovShabat1972}) and
is just the first term in the hierarchy of envelope equations
describing packets of surface gravity waves. The second term in
this hierarchy was calculated by \cite{Dysthe1979}, the next one was
found a few years ago by \cite{TrulsenDysthe1996}. The Dysthe
equation was solved numerically by Ablovitz and his collaborates
(see \cite{Ablovitz2000}).

Since the first work of \cite{Smith1976}, many authors tried  to
explain the freak wave formation in terms of NLSE and its
generalizations, like Dysthe equation. A vast scientific
literature is devoted to this subject. The list presented below is
long but incomplete: \cite{Peregrine1983}, \cite{Peregrine1988}, \cite{Tanaka1990},
\cite{TrulsenDysthe1996},
\cite{TrulsenDysthe1997}, \cite{Trulsen2000}, \cite{TKDV2000},
\cite{Ablovitz2000}, \cite{Onorato2000a}, 
\cite{Onorato2000b}, \cite{Onorato2001}, \cite{Onorato2002}.

One cannot deny some advantages achieved by the use of the
envelope equations. Results of many authors agree in one important
point: nonlinear development of modulation instability leads to
concentration of wave energy in a small spatial region. This is a
"hint" regarding possible formation of freak wave. On the other hand,
it is clear that the freak wave phenomenon cannot be explained in
terms of envelope equations. Indeed, $NLSE$ and its
generalizations are derived by expansion in series on powers of
parameter $\lambda\simeq{1\over Lk}$, where $k$ is a wave number,
$L$ is a length of modulation. For real freak wave $\lambda\sim 1$
and any "slow modulation expansion" fails. However, the analysis
in the framework of the NLS-type equations gives some valuable
information about formation of freak waves.

Modulation instability leads to decomposition of initially
homogeneous Stokes wave into a system of envelope solitons (more
accurately speaking - quasi-solitons (\cite{ZakharovKuznetsov1998},
\cite{ZDP2004}). This state can be called "solitonic
turbulence", or, more exactly "quasisolitonic turbulence".  In the
framework of pure NLSE, solitonic turbulence is "integrable".
Solitons are stable , they scatter on each other elastically. However, even
in this simplest scenario, spatial distribution of wave energy
displays essential intermittency. More exact Dysthe equation is
not integrable. In this model solitons can merge, this effect
increases spatial intermittency and leads to establishing of
chaotic intense modulations of energy density. So far this model
cannot explain formation of freak waves with $\lambda\sim 1$.

This effect can be explained if the envelope solutions of a
certain critical amplitude are unstable, and can collapse. In the
framework of 1-D focusing NLSE solitons are stable, thus solitons
instability and the collapse must have a certain threshold in
amplitude. Instability of a soliton of large amplitude and further
collapse could be a proper theoretical explanation of the freak
wave origin.

This scenario was observed in numerical experiment on the
heuristic one-dimensional Maida-McLaughlin Tabak (MMT) model 
(see \cite{MMT1997}) of
one-dimensional wave turbulence \cite{ZDP2004}.  At a
proper choice of parameters this model mimics gravity waves on the
surface of deep water. In the experiments described in the cited
paper instability of a moderate amplitude monochromatic wave leads
first to creation of a chain of solitons, which merge due to
inelastic interaction into one soliton of a large amplitude. This
soliton sucks energy from neighbor waves, becomes unstable and
collapse up to $\lambda\sim 1$, producing the freak wave. We
believe that this mechanism of freak wave formation is universal.

The most direct way to prove validity of the outlined above
scenario for freak wave formation is a direct numerical solution
of Euler equation, describing potential oscillations of ideal
fluid with a free surface in a gravitational field. This solution
can be made by the methods published in several well-known articles,
\cite{Dommer87, West1987, CG2001}. 
Here we use another method, based on conformal
mapping. It shoul be mention that idea to exploit comformal mapping
for unsteady flows was presented in \cite{Ovs1973}, and later in \cite{MOI1981, CS1998}. 
Method used in this article has origin in (\cite{DKSZ1996}, has been using in
\cite{ZDV2002}, and was finally formulated in \cite{D2001}. This method is
applicable in $1+1$ geometry, it includes conformal mapping of
fluid bounded by the surface to the lower half-plane together with
"optimal" choice of variables, which guarantees well-posedness of
the equations (\cite{D2005}) and existence of smooth, unique solution of the equations for
a finite time (\cite{Shamin2006}). 
Here we would like to stress that one of the main goal 
of this paper is to demonstrate effectiveness of the conformal variables to simulate 
exact 2D potential flow with a free boundary. Earlier, fully nonlinear numerical
experiments regarding wave breaking, freak wave formation, comparison with
weakly nonlinear model (such as Nonlinear Shredinger equation) were done in the papers 
\cite{DoldPer, Tanaka1990, BT1998, HPD1999, CG2002}. From the other hand, using conformal approach 
we have studied in the paper \cite{ZDV2002}, the nonlinear stage of
modulation instability for Stokes waves of steepness $\mu = ka
\simeq 0.3$ and $\mu=0.1$.


In the present article we perform similar experiment for waves of
 steepness $\mu \simeq 0.15$. This experiment could be
considered as a simulation of a realistic situation. If a typical
steepness of the swell is $0.06\div 0.07$, in caustic area it
could easily be two-three times more. In the new experiment, we
start with the Stokes waver train, perturbed by a long wave with
twenty time less amplitude. We observe development of modulation
instability and finally, the  explosive formation of the freak
wave that is pretty similar to waves observed in natural
experiments.


\section{Basic equations}
\setcounter{equation}{0}

Suppose that incompressible fluid covers the domain
\begin{equation}\label{Z_domain}
- \infty <y < \eta(x,t).
\end{equation}
The flow is potential, hence
\begin{equation}\label{pot_flow}
V= \nabla \phi, \hspace{1cm} \nabla V =0, \hspace{1cm} \nabla^2
\phi =0.
\end{equation}
Let $\psi = \left. \phi \right|_{y=\eta}$ be the potential at the
surface and $H = T+U$ be the total energy. The terms
\begin{equation}\label{Ham}
T = -\frac{1}{2}\int_{-\infty}^{\infty} \psi \phi_n {\rm d} x,
\end{equation}
\begin{equation}
U = \frac{g}{2}\int_{-\infty}^{\infty} \eta^2(x,t){\rm d}x,
\end{equation}
are correspondingly kinetic and potential parts of the energy, $g$
is a gravity acceleration and $\phi_n$ is a normal velocity at the
surface. The variables $\psi$ and $\eta$ are canonically
conjugated; in these variables Euler equation of hydrodynamics
reads
\begin{equation}\label{Ham_Eq}
\frac{\partial \eta}{\partial t} = \frac{\delta H}{\delta
\psi},\hspace{1cm} \frac{\partial \psi}{\partial t} =
-\frac{\delta H}{\delta \eta}.
\end{equation}

One can perform  the conformal transformation to map the domain
that is filled with fluid,
\[-\infty < x < \infty,\hspace{1cm}
-\infty < y < \eta(x,t),\hspace{1cm} z = x +iy\] in $z$-plane to
the lower half-plane
\[-\infty <u <-\infty,\hspace{1cm} -\infty <
v <0,\hspace{1cm} w = u +iv \]
 in $w$-plane. Now, the shape of
surface $\eta(x,t)$ is presented by parametric equations
\[y=y(u,t),\; x=x(u,t),\] where $x(u,t)$ and $y(u,t)$ are related
through Hilbert transformation
\begin{equation}\label{unitar}
y = \hat H \left ( x(u,t)-u \right ), \hspace{2cm} x(u,t)  = u -\hat H y(u,t).
\end{equation}
Here
\[\hat H(f(u)) = P.V.\frac{1}{\pi}\int_{-\infty}^{\infty}
\frac{f(u'){\rm d }u'}{u'-u}.\]

Equations (\ref{Ham_Eq}) minimize the action,
\begin{equation}
S = \int L dt,
\end{equation}
\begin{equation}\label{Lagrangian}
L = \int{ \psi \frac{\partial\eta}{\partial t} dx} - H.
\end{equation}
Lagrangian $L$ can be expressed as follows,
%
\begin{eqnarray}\label{Lagrange}
L = \int_{-\infty}^{\infty} \psi (y_t x_u - x_t y_u)du &+&
\frac{1}{2}\int_{-\infty}^{\infty} \psi \hat{H} \psi_{u} du -
\frac{g}{2}\int_{-\infty}^{\infty}y^2 x_u du+\cr &+&
\int_{-\infty}^{\infty}f\left(y - \hat{H}(x-u)\right) du\,.
\end{eqnarray}
%
Here $f$ is the Lagrange multiplier which imposes the relation
(\ref{unitar}). Minimization of action in conformal variables
leads to implicit equations (see \cite{DKSZ1996})
\begin{eqnarray}\label{implicit}
y_t x_u - x_t y_u &=& - \hat{H}\psi_u\cr \psi_t y_u - \psi_u y_t +
g y y_u &+& \hat H(\psi_t x_u - \psi_u x_t + g y x_u) = 0.
\end{eqnarray}
System (\ref{implicit}) can be resolved with respect to the time
derivatives. Omitting the details, we present only the final
result
\begin{eqnarray}\label{EqPHI-Z}
Z_t &=& i U Z_u,\cr \Phi_t &=& i U \Phi_u - B +i g(Z-u).
\end{eqnarray}
Here
\begin{equation}\label{defF}
\Phi = 2\hat P\psi
\end{equation}
is a complex velocity potential,
$U$ is a complex transport velocity:
\begin{equation}\label{defu}
U = 2 \hat P \left(\frac{-\hat H \psi_u}{|z_u|^2} \right)
\end{equation}
and
\begin{equation}\label{defb}
B = \hat P \left(\frac{|\Phi_u|^{2}}{|z_u|^2} \right) = \hat P
\left(|\Phi_z|^{2}\right ).
\end{equation}
In~(\ref{defF}), (\ref{defu})~and~(\ref{defb}) $\hat P$ is the projector
operator generating a function that is analytical in a lower
half-plane 
$$
\hat P(f)=\frac{1}{2} \left(1+ i\hat H \right)f.
$$
In the equations (\ref{EqPHI-Z}) 
$$z(w)\rightarrow w,\hspace{1cm}
\Phi(w)\rightarrow 0,\hspace{1cm} \mbox{at } v\rightarrow
-\infty.
$$ 
All functions $z$, $\Phi$, $U$ and $B$ are analytic ones in
the lower half-plane $v < 0$.


Recently we found that equations (\ref{EqPHI-Z}) were derived in 
\cite{Ovs1973}, and we call them here Ovsyannikov's equations, $OE$.
Implicit equations (\ref{implicit}) were not known until 1994, 
so we call them $DKSZ$-equations.

Note, that equation (\ref{implicit}) can be used to obtain the Lagrangian
description of surface dynamics. Indeed, from (\ref{implicit}) one can get
\begin{equation}\label{Funk}
\Psi=\partial^{-1} \hat H (y_t x_u -x_t y_u)
\end{equation}
Plugging (\ref{Funk}) into (\ref{Lagrangian}) one can express Lagrangian $L$
only in terms of surface elevation. This result was independently
obtained by \cite{Balk96}. In \cite{D2001} equations (\ref{EqPHI-Z}) were transformed to a 
simple form, which is convenient both for numerical simulation and
analytical study. By introducing of new variables
\begin{equation}\label{CHANGE}
R = \frac {1}{Z_w},\hspace{1cm} \mbox{and } V = i\Phi_z =
i{\Phi_w\over Z_w}
\end{equation}
one can transform system (\ref{EqPHI-Z}) into the following one
\begin{eqnarray}\label{R-V}
R_t &=& i(U R_w - R U_w), \cr V_t &=& i(U V_w - R B_w) + g(R-1).
\end{eqnarray}
Now complex transport velocity $U$ and $B$
\begin{eqnarray}\label{UandB}
U &=& \hat P(V\bar R + \bar V R)\cr B &=& \hat P(V\bar V).
\end{eqnarray}
Thereafter, we will call equations (\ref{R-V}), (\ref{UandB})
Dyachenko equations, $DE$.

Both $DKSZ$-equations (\ref{implicit}) and $OE$ (\ref{EqPHI-Z}) have the same
constants of motion
\begin{equation}\label{MotionE}
H=-\int_{-\infty}^{\infty} \Psi \hat H \Psi_u du + \frac{g}{2}
\int_{-\infty}^{\infty} y^2 x_u dy ,
\end{equation}
the same total mass of fluid
\begin{equation}\label{Mass}
M=\int_{-\infty}^{\infty} y x_u du ,
\end{equation}
and the same horizontal momentum
\begin{equation}\label{Momentum}
P_x=\int_{-\infty}^{\infty}\Psi y_u du.
\end{equation}
The Dyachenko equations (\ref{R-V}), (\ref{UandB}) have the same
integrals. To express them in terms of $R$ and $V$, one has to
perform the integration
\begin{equation}\label{energyRV}
Z = u + \int_{-\infty}^u \frac{du}{R}, \hspace{1cm} \Phi = -i
\int_{-\infty}^u \frac{V}{R}du.
\end{equation}



\section{Freak waves as a result of modulation instability}
\setcounter{equation}{0}

{
The Stokes wave is unstable with respect to 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{Zakh1968}) and in \cite{BenFeir1967}.

The presented technique based on the conformal mapping makes
possible to study modulation instability in a very compact way.
It is convenient to use the Dyachenko equation (\ref{R-V}),
(\ref{UandB}). 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}
u\rightarrow 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 progressive wave (with subscript $0$) the following relation is valid:
\begin{equation}\nonumber
R_0=1-\frac{iV_0}{c_0}.
\end{equation}
For the perturbation $\delta V$ and $\delta r$ one can derive 
linear system against the stationary solution
\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}).
 
After simple calculations one can obtain dispersion relation for $\Omega$:
\begin{equation}
\Omega^2=\frac{1}{8}\left(-A^2 \kappa^2 +\frac{1}{8}
\kappa^4\right).
\end{equation}
Here $A$ is the amplitude of the first Fourier harmonic of the Stokes wave train:
$$
r = A e^{-iu}+ \dots
$$
The result that was obtained by \cite{Zakh1966, Zakh1968}, 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
modulation and other instabilities with any arbitrary accuracy.
}

Here we study modulation instability of uniform wave train of Stokes wave.
Question of great interest is the nonlinear stage of the instability.
Here and everywhere below we do simulation in periodic domain $L=2\pi$ and
$$
g = 1.
$$
%
Wavetrain of the amplitude $a$ with wavenumber $k_0$ is unstable with respect to large 
scale modulation $\delta k$. Growth rate of the instability $\gamma$ is 
\begin{equation}\label{MODGROW}
\gamma = \frac{\omega_0}{2}\left (\left (\frac{\delta k}{k_0}\right )^2
(a k_0)^2 - \frac{1}{4} \left (\frac{\delta k}{k_0}\right )^4 \right )^{\frac{1}{2}}.
\end{equation}
Here $\omega_0$ is the linear dispersion relation for gravity wave
$$
\omega_0 = \sqrt{g k_0}.
$$

\begin{itemize}
\item
The shape of Stokes progressive wave is given by:
$$
y = \frac{c^2}{2g}(1-\frac{1}{|Z_u|^2}),
$$
while $\Phi$ is related to the surface as
$$\Phi = -c(Z-u), \hspace{5em} V = ic(R-1).$$
The amplitude of the wave $\frac{h}{L}$ is the parameter for initial condition.
(For the sharp peaked limiting wave $\frac{h}{L} \simeq 0.141$)
\item
Put 100 such waves with small perturbation in the periodic domain of $2\pi$.
\end{itemize}
In such a way we prepared initial wave train with the steepness $\mu \simeq 0.095$
Main Fourier harmonic of this wave train is $k=100$. 
Similar problem was studied in \cite{SongBanner2002}. But instead of long wavetrain
they studied evolution of small group of waves.

For perturbation small value
for Fourier harmonic with $k_p=1$ was set. So, that
$$
R_k = R_k^{\mbox{unperturbed}} + 0.05 R_{100} \exp^{-ik_p u}.
$$
Surface profile of this initial condition is shown in Figure~\ref{TRAIN_I}

\begin{figure}[!h]
\includegraphics[angle=0,width=0\PW]{./FIG1/m0.07-initial-Z.eps}
\caption{\label{TRAIN_I} Initial profile of the wave train}
\end{figure}
Fourier spectrum of the initial condition is shown in Figure~\ref{TRAIN_I_R} and 
Figure~\ref{TRAIN_I_V}.

\begin{figure}[!h]
\includegraphics[angle=0,width=\PW]{./FIG1/m0.07-initial-R.eps}
\caption{\label{TRAIN_I_R} Fourier coefficients $|R_k|$ for initial condition 
($\mu \simeq 0.095$).}
\end{figure}

\begin{figure}[!h]
\includegraphics[angle=0,width=\PW]{./FIG1/m0.07-initial-V.eps}
\caption{\label{TRAIN_I_V} Fourier coefficients $|V_k|$ for initial condition 
($\mu \simeq 0.095$).}
\end{figure}

After sufficient large time, which is more than 1300 wave periods one can observe
freak wave formation, as it is shown in Figure~\ref{FREAK}. Freak wave grows from
mean level of waves to its maximal value for several wave periods, than vanishes  or 
breaks.

\begin{figure}[!h]
\includegraphics[angle=0,width=\PW]{./FIG1/m0.07-end-Z.eps}
\caption{\label{FREAK} Freak wave on the surface profile. $T=802.07$}
\end{figure}

Detailed view at the freak wave at the moment of maximal amplitude is shown in 
Figure~\ref{FREAK_ZOOM}. This set of experiments is similar to that of 
\cite{DoldPer, Tanaka1990}.
The difference is that we were able to increase the accuracy of the simulation, and consider
much longer wavetrains. Also (due to using conformal mapping) we can simulate breaking 
with multivalued surface profile.
Acccuracy in the simulation is very important because of the freak 
wave appears in a very subtle manner on the phase relations between Fourier harmonics of the
surface. Moreover, for shorter wavetrains threshold of modulation 
instability increases, and breaking doesn't happen even for large steepness. In our 
experiments we have observed threshold of steepness for wave breaking a little less than in 
\cite{Tanaka1990}, but above $\mu =0.1$. Still, surface profile from \cite{Tanaka1990} (Fig.5)
is very similar to the picture in Figure~\ref{FREAK_ZOOM} with $\mu=0.095$.


\begin{figure}[!h]
\includegraphics[angle=0,width=\PW]{./FIG1/m0.07-end-Z11.eps}
\caption{\label{FREAK_ZOOM} Zoom in surface profile at $T=802.07$}
\end{figure}
During numerical simulation of the final stage of freak wave formation, resolution must be
increased to resolve high curvature of the surface profile. To do this we have ben increasing
number of Fourier harmonics, which reached $2^{20}$ at the end ($T=802.07$). Fourier 
coefficients
of $R_k$ are shown in Figure~\ref{FREAK_FOURIER}.


\begin{figure}[!h]
\includegraphics[angle=0,width=\PW]{./FIG1/m0.07-end-R.eps}
\caption{\label{FREAK_FOURIER} Fourier coefficients $|R_k|$ at $T=802.07$.}
\end{figure}

If amplitude of the wave train is large, than freak wave may eventually break.
Such a picture is presented in the Figure~\ref{FREAK_BREAK}, which corresponds to the other
numerical simulation with the initial steepness $\mu \simeq 0.14$.

\begin{figure}[!h]
\includegraphics[angle=-90,width=\PW]{./FIG1/Break458_842.eps}
\caption{\label{FREAK_BREAK} Profile of breaking wave.}
\end{figure}



\section{Exact equations and nonlinear Shr\"odinger approximation}

Evolution of  weakly nonlinear Stokes wavetrain can be described by
nonlinear Shr\"odinger equation ($NLSE$), derived by \cite{Zakh1968}.
This equation is integrable (see \cite{ZakharovShabat1972}) and
is just the first term in the hierarchy of envelope equations
describing packets of surface gravity waves. The second term in
this hierarchy was calculated by \cite{Dysthe1979}, the next one was
found a few years ago by \cite{TrulsenDysthe1996}. The Dysthe
equation was solved numerically by Ablovitz and his collaborates
(see \cite{Ablovitz2000}).

Since the first work of \cite{Smith1976}, many authors tried  to
explain the freak wave formation in terms of NLSE and its
generalizations, like Dysthe equation. A vast scientific
literature is devoted to this subject. The list presented below is
long but incomplete: \cite{Ablovitz2000}, \cite{Onorato2000a}, 
\cite{Onorato2000b}, \cite{Onorato2001}, \cite{Onorato2002},
\cite{Peregrine1983}, \cite{Peregrine1988}, \cite{TrulsenDysthe1996},
\cite{TrulsenDysthe1997}, \cite{Trulsen2000}, \cite{TKDV2000}, \cite{CG2002}.

One cannot deny some advantages achieved by the use of the
envelope equations. Results of many authors agree in one important
point: nonlinear development of modulation instability leads to
concentration of wave energy in a small spatial region. This is a
"hint" regarding possible formation of freak wave. On the other hand,
it is clear that the freak wave phenomenon cannot be explained in
terms of envelope equations. Indeed, $NLSE$ and its
generalizations are derived by expansion in series on powers of
parameter $\lambda\simeq{1\over Lk}$, where $k$ is a wave number,
$L$ is a length of modulation. For real freak wave $\lambda\sim 1$
and any "slow modulation expansion" fails. At this point interesting question rises:
what happens to $NLSE$ approximation when increasing the steepness of the carrier wave?
In particular, we study "exact" soliton solutions for $NLSE$ placed in the 
exact equations (\ref{R-V}). 

Such type of problem was considered in the \cite{HPD1999}, but with low resolution,
and small length of periodic carrier. Also in \cite{CG2002}
%, CFGK2006} 
numerical solutions for
envelope equation was compared with "almost" exact equations.

For the equations (\ref{R-V}) $NLSE$ model can be derived for the envelope of $R$.
$$
R = 1 + R_1 e^{-ik_0 u -\omega_0 t} + \dots
$$
$$
i{R_1}_t + \frac{1}{8}\frac{\omega_0}{k_0^2}{R_1}_{uu} +\frac{1}{2}\omega_0 k_0^2|R_1|^2 R_1 =0.
$$
Initial conditions consist of "linear wave carrier" $\e^{-i k_0 u}$, modulated in accordance
with soliton solution for $NLSE$:
\begin{eqnarray}\label{NLSE_I}
R(u) &=& 1 + s_0\frac{\e^{-i k_0 u}}{\cosh{(\lambda k_0 u)}},\cr
V(u) &=& -i c_0 s_0 \frac{\e^{-i k_0 u}}{\cosh{(\lambda k_0 u)}}.
\end{eqnarray}
Here $s_0$ is the steepness of the carrier wavetrain, $c_0$ - phase velocity of the carrier.

First comparison of fully nonlinear model for water wave with $NLSE$ was
done in \cite{CG2002} for the wave carrier with the steepness $\mu\simeq 0.091$.
For such steepness there was a good agreement between two models, but only for the short time.
After finite time weakly nonlinear model ($NLSE$) ceases to be valid.
  
In our work  we want to study the situation with larger and smaller stepness, to find out how
$NLSE$ approximation breaks.

\subsection{Small steepness}


First experiment was intended to observe how $NLSE$ works. In the initial conditions
(\ref{NLSE_I}) we used
$$
s_0 \simeq 0.07, \hspace{0.5cm}  \lambda = 0.1, \hspace{0.5cm} k_0 = 100.
$$
Initial surface of fluid is shown in Figure~\ref{SOLITON_007_I}.

\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/cosh0.1-initial-Z.eps}
\caption{\label{SOLITON_007_I} {Initial surface profile like for $NLSE$ soliton with $\mu\simeq 0.07$.}}
\end{figure}

After couple of thousands wave periods soliton changes a little, as it is seen in
Figure~\ref{SOLITON_007_A}:

\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/cosh0.1-end-Z.eps}
\caption{\label{SOLITON_007_A} Surface profile like for $NLSE$ 
soliton with $\mu\simeq 0.07$ at T=1500.}
\end{figure}

Also in the Figure~\ref{SOLITON_007_R_I} and Figure~\ref{SOLITON_007_R_A} Fourier spectra
of the soliton at both moments of time are presented.

\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/cosh0.1-initial-R.eps}
\caption{\label{SOLITON_007_R_I} Fourier harmonics of the initial soliton with $\mu\simeq 0.07$.}
\end{figure}

\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/cosh0.1-end-R.eps}
\caption{\label{SOLITON_007_R_A} Fourier harmonics of the soliton with $\mu\simeq 0.07$ at T=1500.}
\end{figure}
So, one can can see that for the steepness $\mu\leq 0.07$ $NLSE$ model is quite reasonable.

Another numerical experiment showing effective simulation with equations (\ref{R-V})
along with applicability $NLSE$ model for moderate steepness, $\mu\simeq 0.085$, is the 
collision of two solitons.

In the Figure~\ref{SOLITON_COL_I} initial condition is shown:
\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/coshv0.12-initial-Z.eps}
\caption{\label{SOLITON_COL_I} {Initial surface profile of two $NLSE$ solitons with $\mu\simeq 0.085$.}}
\end{figure}

Moment of collision is shown in the 
Figure~\ref{SOLITON_COL_C}:~\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/coshv0.12-end-Z.eps}
\caption{\label{SOLITON_COL_C} {Two $NLSE$ solitons with $\mu\simeq 0.085$. collide at T=30.8}}
\end{figure}

and detailed view showing carrier wavetrain under the envelope is in the 
Figure~\ref{SOLITON_COL_C1}.
\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/coshv0.12-end-Z1.eps}
\caption{\label{SOLITON_COL_C1} {Detailed view of two colliding $NLSE$ solitons with $\mu\simeq 
0.085$ at T=30.8.}}
\end{figure}

 
After second collision (recall that boundary conditions are periodic) solitons
are plotted in the Figure~\ref{SOLITON_COL_C2}:
\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/coshv0.12-after-Z.eps}
\caption{\label{SOLITON_COL_C2} {Two $NLSE$ solitons with $\mu\simeq 0.085$. after two 
collisions at T=250.0.}}
\end{figure}

Fourier spectra of these two solitons at the moments of time $T=0.05, 30.8, 250.0$
are shown in Figure~\ref{SOLITON_COL_S_I}, ~\ref{SOLITON_COL_S_C}, ~\ref{SOLITON_COL_S_C2}.
\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/coshv0.12-initial-R.eps}
\caption{\label{SOLITON_COL_S_I} {Fourier spectrum of the initial surface profile of two $NLSE$ 
solitons with $\mu\simeq 0.085$.}}
\end{figure}
\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/coshv0.12-end-R.eps}
\caption{\label{SOLITON_COL_S_C} {Fourier spectrum of two colliding $NLSE$ 
solitons with $\mu\simeq 0.085$. at $T=30.8$}}
\end{figure}
\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/coshv0.12-after-R.eps}
\caption{\label{SOLITON_COL_S_C2} {Fourier spectrum of two $NLSE$ 
solitons with $\mu\simeq 0.085$. at $T=250.0$}}
\end{figure}


\subsection{Large steepness}

Now let's turn to the higher steepness of the carrier,
$$
\mu = 0.1.
$$
In the Figure~\ref{SOLITON_010_I} there is initial condition:~\begin{figure*}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/cosh0.14-initial-Z.eps}
\caption{\label{SOLITON_010_I} {Initial surface profile like for $NLSE$ soliton with $\mu\simeq 
0.10$.}}
\end{figure*}

Again, after couple of thousands wave periods soliton changes a little, as it is seen in
Figure~\ref{SOLITON_010_A}:~\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/cosh0.14-end-Z.eps}
\caption{\label{SOLITON_010_A} Surface profile like for $NLSE$ 
soliton with $\mu\simeq 0.10$ at T=2345.}
\end{figure}

In the Figure~\ref{SOLITON_010_R_I} and Figure~\ref{SOLITON_010_R_A} Fourier spectra
of the soliton at both moments of time are presented.

\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/cosh0.14-initial-R.eps}
\caption{\label{SOLITON_010_R_I} Fourier harmonics of the initial soliton with $\mu\simeq 0.10$.}
\end{figure}

\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/cosh0.14-end-R.eps}
\caption{\label{SOLITON_010_R_A} Fourier harmonics of the soliton with $\mu\simeq 0.10$ at T=2345.}
\end{figure}
From this pictures one can see that for steepness $\mu\simeq 0.10$ some corrections
to the $NLSE$ model are desirable. Dysthe equations are exactly intended for that situation. 

But what happens when further increasing the steepness?
Below we consider the case of the steepness of the carrier
$$
\mu = 0.14.
$$
In the Figure~\ref{SOLITON_014_I} there is initial condition:
\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/cosh0.2-initial-Z.eps}
\caption{\label{SOLITON_014_I} {Initial surface profile like for $NLSE$ soliton with $\mu\simeq 0.14$.}}
\end{figure}

Very fast, after couple of dozen wave periods soliton drastically changes, as it is seen in
Figure~\ref{SOLITON_014_A}:

\begin{figure}[!h]
\includegraphics[angle=-0,width=\PW]{./FIG1/cosh0.2-end-Z.eps}
\caption{\label{SOLITON_014_A} Surface profile like for $NLSE$ 
soliton with $\mu\simeq 0.14$ at T=38.4.}
\end{figure}
One can see freak wave at the surface (in Figure~\ref{SOLITON_014_A1}):
\begin{figure}[!t]
\includegraphics[angle=-0,width=\PW]{./FIG1/cosh0.2-end-Z1.eps}
\caption{\label{SOLITON_014_A1} Zoomed surface profile near freak wave $\mu\simeq 0.14$ at T=38.4.}
\end{figure}

In the Figure~\ref{SOLITON_014_R_I} and Figure~\ref{SOLITON_014_R_A} Fourier spectra
of the soliton at both moments of time are presented. They demonstrate the quality of
the numerical simulation.

\begin{figure}[!t]
\includegraphics[angle=-0,width=\PW]{./FIG1/cosh0.2-initial-R.eps}
\caption{\label{SOLITON_014_R_I} Fourier harmonics of the initial soliton with $\mu\simeq 0.14$.}
\end{figure}

\begin{figure}[!t]
\includegraphics[angle=-0,width=\PW]{./FIG1/cosh0.2-end-R.eps}
\caption{\label{SOLITON_014_R_A} Fourier harmonics of the soliton with $\mu\simeq 0.14$ at T=38.4.}
\end{figure}
From the last case, with the steepness $\mu=0.14$, one can see that envelope approximation
completely fails. Such event as one single crest (freak wave) can not be described in terms
of wave envelope.





\section{Do freak waves appear from quasisolitonic turbulence?}
\setcounter{equation}{0}

Let us summarize the results of our numerical experiments.
Certainly, they reproduce the most apparent features of freak
waves: single wave crests of very high amplitude, exceeding the
significant wave height more than three times, appear from
"nowhere" and reach full height in a very short time, less than
ten periods of surrounding waves. The singular freak wave is
proceeded by the area of diminished wave amplitudes. Nevertheless,
the central question about the physical mechanism of freak waves
origin is still open.

In our experiments, the freak wave appears as a result of
development of modulation instability, and it takes a long time
for the onset of instability  to create a freak wave. Indeed, the
level of perturbation in our last experiment is relatively high.
The two-three inverse growth-rate is enough to reach the state of
full-developed instability, when the initial Stokes wave is
completely decomposed. Meanwhile, the freak wave appears only
after fifteenth inverse growth-rates of instability. What happens
 after developing of instability but before formation of freak
wave?

During this relatively long period of time, the state of fluid
surface can be characterized as quasisolitonic turbulence, that
consists of randomly located quasi-solitons of different amplitudes
moving with different group velocities. Numerical study of interaction
of envelope soliton was done in \cite{CG2002}. Such interaction leads to 
formation of wave with large amplitude. Here we can think in term of 
quasisolitonic turbulence. Such turbulence was studied in the recent work of Zakharov, Dias and
Pushkarev (\cite{ZDP2004}) in a framework of so-called
defocusing $MMT$ model.
\begin{equation}\label{MMT_defocus}
i\frac{\partial\Psi}{\partial t}=\left|\frac{\partial}{\partial
x}\right|^{1/2} \,\Psi +\left|\frac{\partial}{\partial
x}\right|^{3/4}\left(\left|\left|\frac{\partial}{\partial
x}\right|^{3/4}\Psi\right|^2 \left|\frac{\partial}{\partial
x}\right|^{3/4}\,\Psi\right)
\end{equation}

This is a heuristic model description of gravity surface waves in
deep water. In this model, quasi-solitons of small amplitude are
stable, interact inelastically and can merge. Above some critical
level quasi-solitons of large amplitude are unstable. They collapse
in finite time forming very short wave pulses, which can be
considered as models of freak waves. Equation \ref{MMT_defocus} has the exact
solution:
\begin{eqnarray}
\Psi&=&A\,e^{ikx -i\omega t}\nonumber\\
\omega&=&k^{1/2}\left(1+k^{5/2}\,A^2\right).
\end{eqnarray}
This solution can be constructed as a model of the Stokes wave and
is unstable with respect to modulation instability. Development
of this instability was studied numerically. On the first stage,
the unstable monochromatic wave decomposes to a system of almost
equal quasi-solitons. Then, the quasisolitonic turbulence is
formed: quasi-solitons move chaotically, interact with each other,
and merge. Finally they create one large quasi-soliton, which
exceeds threshold of instability and collapses, creating a freak
wave.

One can think that a similar scenario of freak wave formation is
realized in a real sea. We like to stress that the key point in
this scenario is the quasisolitonic turbulence and not the Stokes
wave. The Stokes wave is just a "generator" of this turbulence.
The quasisolitonic turbulence can appear as a result of
instability of narrow spectral distributions of gravity waves.

The formulated above concept is so far a hypothesis, which has to
be confirmed by future numerical experiments.

\section*{Acknowledgments}

This work was supported by the US Army Corps of Engineers Grant W912HZ-05-P-0351, by ONR Grant 
N00014-03-1-0648, NSF Grant DMS 0404577, RFBR Grant
06-01-00665, the Program "Fundamental Problems in Nonlinear
Dynamics"  from the RAS Presidium, and Grant "Leading Scientific
Schools of Russia".




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