%\documentclass[pdf,contemporain,slideColor,colorBG,accumulate,nototal]{prosper}
\documentclass[pdf,rico,slideColor,colorBG,accumulate,nototal]{prosper}

%\RequirePackage[koi8-r]{inputenc}%unix encoding
%\RequirePackage[T2A]{fontenc}
%\RequirePackage[russian]{babel}
%\RequirePackage{misccorr}
%\usepackage{macros-cp}

%\RequirePackage{color}
\RequirePackage{graphicx}


\renewcommand{\figurename}{\scriptsize Figure}

%\title{Турбулентность и сингулярности\\ в нелинейных волновых системах}
\title{Breaking of Progressive Gravity Surface Wave\\
and Modulation Instability}
\author{Dyachenko A.I.}
\institution{Landau Institute for Theoretical Physics RAS}
%Институт теоретической физики им. Л.Д. Ландау РАН}

\begin{document}
\maketitle

%---------------------------------------------------------------------

%------------------------------------------------------------SLIDE - 2
%\overlays{1}{%
\begin{slide}{Equations}
\small

\vspace*{-10pt}
\mbox{%
\begin{picture}(140,90)
% Edges
  \put(0,0){\line(1,0){140}} \put(0,90){\line(1,0){140}}
  \put(0,0){\line(0,1){90}} \put(140,0){\line(0,1){90}}
% Axises
\put(10,10){\begin{picture}(120,70)%
              \put(0,40){\vector(1,0){120}}
	      \put(60,0){\vector(0,1){70}}
	      \put(110,30){$x$} \put(65,65){$y$}
	      \qbezier[1000](0,0)(20,90)(50,40)
	      \qbezier[1000](50,40)(80,-10)(120,30)
	      \put(7,0){$\eta(x,t)$}
	      \put(117,67){\circle{15}}
	      \put(115,65){$z$}
	    \end{picture}}
\end{picture}}\qquad
\parbox[b]{5cm}{potential irrotational flow
$$\triangle\phi (x,y,t) = 0$$}
\vspace{0.0cm}
\begin{raggedright}
{\tiny Boundary conditions:}
{$\left[\begin{array}{l}
      \frac{\partial\phi}{\partial t} + \frac{1}{2}|\nabla \phi|^2 + 
g\eta = \frac{P}{\rho}, \\
      \frac{\partial\eta}{\partial t} + \eta_x \phi_x = \phi_y
  \end{array}\right\vert$}~at~$y=\eta (x,t)$.

%\fromSlide{2}{%
%  \vspace*{10pt}
  \begin{minipage}{10cm}
\begin{eqnarray}\nonumber%\label{Eq2}
\frac{\partial \phi}{\partial y} &=& 0,  y \rightarrow -\infty, \cr
\frac{\partial \phi}{\partial x} &=& 0,  |x| \rightarrow \infty.
%\phi(x,y) &=& \phi(x+2\pi,y).
\end{eqnarray}
\end{minipage}

\end{raggedright}
\end{slide}
%------------------------------------------------------------------------------

%--------------------------------------------------------------------- SLIDE -3
%\overlays{2}{%
\begin{slide}{Hamiltonian and Lagrangian}
\small

\vspace*{-10pt}
\begin{raggedright}
Hamiltonian $H$ is the total energy of the fluid $H = T + U$
\mbox{%
\begin{minipage}{6cm}
\begin{eqnarray}
T &=& \frac{1}{2} \int_{-\infty}^{\infty} dx \int_{-\eta}^{\eta} 
(\nabla\Phi)^2 dy\,,
\nonumber\\
U &=& \frac{g}{2} \int \eta^{2} dx\,.\nonumber
\end{eqnarray}
\end{minipage}}
\mbox{
\begin{minipage}{4cm}
\begin{eqnarray}
\frac{\partial\eta}{\partial t} &=& \frac{\delta H}{\delta \Psi}\,,\nonumber\\
\frac{\partial\Psi}{\partial t} &=& - \frac{\delta
  H}{\delta\eta}\,,\nonumber\\
\Psi (x,t) &=& \left. \Phi (x,y,t)\right|_{y=\eta}\nonumber
\end{eqnarray}
\end{minipage}}

%\fromSlide{2}{
\vspace{1cm}
Action $S =  \int L dt$
$$
   \delta S = 0,
$$
Lagrangian
$$
L = \int{ \Psi \frac{\partial\eta}{\partial t}dx}-H\,.
$$

\end{raggedright}
\end{slide}
%------------------------------------------------------------------------------


%--------------------------------------------------------------------- SLIDE -4
\begin{slide}{Conformal Mapping}

\vspace*{-8pt}
Let us apply the conformal mapping of the domain on the plane $z=x + iy$,
$$
-\infty < x < \infty,\quad -\infty < y \leq \eta (x,t)\,,
$$
to the lower half-plane,
$$
-\infty < u < \infty,\quad  -\infty < v\leq 0\,,
$$
on the plane  $\omega = u + iv$.

\makebox[5cm]{%
\begin{picture}(140,90)
% Edges
  \put(0,0){\line(1,0){140}} \put(0,90){\line(1,0){140}}
  \put(0,0){\line(0,1){90}} \put(140,0){\line(0,1){90}}
% Axises
\put(10,10){\begin{picture}(120,70)%
              \put(0,40){\vector(1,0){120}}
	      \put(60,0){\vector(0,1){70}}
	      \put(110,30){$x$} \put(65,65){$y$}
	      \qbezier[1000](0,0)(20,90)(50,40)
	      \qbezier[1000](50,40)(80,-10)(120,30)
	      \put(7,0){$\eta(x,t)$}
	      \put(117,67){\circle{15}}
	      \put(115,65){$z$}
	    \end{picture}}
\end{picture}}
\makebox[5cm]{%
\begin{picture}(140,90)
% Edges
  \put(0,0){\line(1,0){140}} \put(0,90){\line(1,0){140}}
  \put(0,0){\line(0,1){90}} \put(140,0){\line(0,1){90}}
% Axises
\put(10,10){\begin{picture}(120,70)%
              \put(0,40){\vector(1,0){120}}
	      \put(60,0){\vector(0,1){70}}
	      \put(110,30){$u$} \put(65,65){$v$}
	      \qbezier[200](0,39.0)(60,39.0)(115,39.0)
	      \qbezier[200](0,40.5)(60,40.5)(115,40.5)
	      \put(117,67){\circle{15}}
	      \put(115,65){$w$}
	    \end{picture}}
\end{picture}}
\end{slide}
%------------------------------------------------------------------------------


%--------------------------------------------------------------------- SLIDE -5
\begin{slide}{Conformal Mapping}

\vspace{-8pt}
%
After this transformation, the shape of the surface is given
parametrically by
%
$$
y=y(u,t), \quad x=u+\tilde x (u,t)\,.
$$
%
Functions  $y$ and $\tilde x$ are coupled by the relations:
$$
y=\hat{H}\tilde{x}\,\hspace{2.5cm}\tilde{x} = -\hat{H}y\, .
$$
Here $\hat{H}$ is the Hilbert transformation,
%
$$
\hat{H} f(u) = \frac{1}{\pi} P.V.\int_{-\infty}^{\infty}\frac{f(u')}
{(u'-u)} du'\,.
$$
For Fourier harmonics $y_k = i\mbox{sign}(k) x_k$.
\end{slide}
%---------------------------------------------------------------------

%------------------------------------------------------------SLIDE - 6
%\overlays{1}{%
\begin{slide}{Implicit Equations}
\small

\vspace*{-10pt}
\begin{raggedright}
Lagrangian can be expressed as follows,
{\scriptsize 
$$
L = \int_{-\infty}^{\infty} \Psi (y_t x_u - x_t y_u)du 
+ \frac{1}{2}\int_{-\infty}^{\infty}
\left (\Psi \hat{H} \Psi_{u} du - g y^2 x_u\right )du 
+ \int_{-\infty}^{\infty}f(y - \hat{H}\tilde{x}) du\,.
$$}
%
Here $f$ is the Lagrange multiplier. Hamilton's principle,
$$
\frac{\delta S}{\delta \Psi} = 0\,,\quad\frac{\delta S}{\delta y} = 0, \quad\hbox{and}\quad
\frac{\delta S}{\delta x} = 0\,.
$$

%\fromSlide{1}{%
%\vspace*{10pt}
gives the following equations,
\begin{eqnarray}\nonumber
y_t x_u - x_t y_u &=& - \hat{H}\Psi_u\,,\cr
\Psi_t x_u - \Psi_u x_t + g y x_u &=& \hat H(\Psi_t y_u - \Psi_u y_t + g y y_u)\,.
\end{eqnarray}
\hspace{5cm} Ovsyannikov, 1973

\end{raggedright}
\end{slide}
%---------------------------------------------------------------------

%------------------------------------------------------------SLIDE - 7
%\overlays{1}{%
\begin{slide}{Explicit Equations}
\small

\vspace*{-10pt}
\begin{raggedright}
If {\it conformal mapping} has been applied then it is naturally
introduce complex analytic functions
$$z = x+i y,\hspace{1em} \mbox{\tiny and complex velocity potential}
\hspace{1em} \Phi = \Psi + \hat H\Psi.$$
%\fromSlide{1}{%
\vspace*{-30pt}
\begin{eqnarray}\nonumber
z_t &=& i U z_u, \cr
\Phi_t &=& i U \Phi_u - \hat P(\frac{|\Phi_u|^2}{|z_u|^2}) +i g(z-u).
\end{eqnarray}
$U$ {\scriptsize is a complex transport velocity:}
\vspace{-10pt}
$$U = \hat P(\frac{-\hat H \Psi_u}{|z_u|^2}).\hspace{6em} u \rightarrow w$$

Projector operator $\hat P(f) = \frac{1}{2}( 1 + i\hat H )(f)$.

\end{raggedright}
\end{slide}
%---------------------------------------------------------------------

%-----------------------------------------------------------SLIDE - 8 
\begin{slide}{Classical variables $\Psi$, $\eta$}
\small

\vspace*{-10pt}
\begin{raggedright}
$$
H = \frac{1}{2} \int_{-\infty}^{\infty} \Psi\hat G(\eta)\Psi dx
\hspace{1em} +\hspace{1em} \frac{g}{2} \int_{-\infty}^{\infty} \eta^2 dx  
$$
\vspace{10pt}
Normal complex variable $a_k$:
$$
\eta_k =  \sqrt{\frac{\omega_k}{2g}}(a_k+a^*_{-k}) \hspace{.7cm}
\psi_k =  -i\sqrt{\frac{2g}{\omega_k}}(a_k-a^*_{-k}) \hspace{3em}
\omega_k = \sqrt{gk}
$$
$a_k$ satisfies the equation 
$$
\frac{\partial a_k}{\partial t} + i\frac{\delta H}{\delta a_k^*}=0,
$$
$$
\hspace{2em} a_k \rightarrow b_k, \hspace{3em}
\mbox{\scriptsize excluding nonresonant terms.}
$$
\end{raggedright}
\end{slide}
%---------------------------------------------------------------------

%-----------------------------------------------------------SLIDE - 9 
%\overlays{2}{%
\begin{slide}{Classical variables $b_k$}
\small

\vspace*{-10pt}
\begin{raggedright}
Hamiltonian without cubic terms:
$$
H=\int\!\omega_k b_k b_k^*\!dk + 
\frac{1}{2}\int\! T_{k k_1}^{k_2 k_3}b_k^* b_{k_1}^* b_{k_2}b_{k_3}
\delta_{k+k_1-k_2-k_3}\!dk\!dk_1\!dk_2\!dk_3 + \ldots;
$$
\vspace{10pt}
$$
k + k_1  =  k_2 + k_3,\hspace{5em}
\omega_k + \omega_{k_1} =  \omega_{k_2} + \omega_{k_3},
$$
\centerline{$T_{k k_1}^{k_2 k_3} = 0$!}
%\fromSlide{2}{%

\vspace{20pt}
\centerline{$T^{k_1 k_2 k_3}_{k_4 k_5} = \frac{2}{g^{1/2}\pi^{3/2}}
\sqrt{\frac{\omega_{k_1} \omega_{k_2} \omega_{k_3}}{\omega_{k_4} \omega_{k_5}}}
\frac{k_1 k_2 k_3 k_4 k_5}{max(k_1,k_2,k_3)}
$}

\end{raggedright}
\end{slide}
%---------------------------------------------------------------------

%------------------------------------------------------------SLIDE -10
%\overlays{1}{%
\begin{slide}{Cubic Equations}
\small

\vspace*{-10pt}
\begin{raggedright}
It turned out, that the equations can be simplified just by 
changing variables. Introduce instead of $z(w,t)$ 
and $\Phi(w,t)$ another analytic functions $R(w,t)$ and $V(w,t)$ 
$$
R = \frac{1}{z_w}, \hspace{5em}\Phi_w = -i V z_w.
$$
%\fromSlide{1}{%
\vspace*{-20pt}
\begin{eqnarray}\nonumber
R_t &=& i\left [ UR' - U'R\right ], \cr
V_t &=& i\left [ UV' - R \hat P(V\bar V)'\right ] + g(R-1).
\end{eqnarray}
{\scriptsize Complex transport velocity} $U$ {\scriptsize is defined via} $\hat P$ 
$$ U = \hat P(V\bar R + \bar V R).$$

\end{raggedright}
\end{slide}
%---------------------------------------------------------------------

%------------------------------------------------------------SLIDE -11
%\overlays{1}{%
\begin{slide}{Setup the problem}
\small
\vspace*{-10pt}
\begin{raggedright}
\begin{itemize}
\item
The shape of stationary progressive wave is given by:
$$
y = \frac{c^2}{2g}(1-\frac{1}{|z_w|^2}),
$$
while $\Phi$ is related to the surface as
$$\Phi = -c(z-w), \hspace{5em} V = ic(R-1).$$
The amplitude of the wave $\frac{h}{L} \simeq 0.088$, \\
\vspace{10pt}
{\tiny For the sharp peaked limiting wave} $\frac{h}{L} \simeq 0.141$.
\end{itemize}
%\fromSlide{1}{%
\vspace*{-20pt}

\end{raggedright}
\end{slide}
%---------------------------------------------------------------------

%-----------------------------------------------------------SLIDE - 12
%\overlays{2}{%
\begin{slide}{Setup the problem}
\small
\vspace*{-10pt}
\begin{raggedright}
\begin{itemize}
\item
Put 10 such waves in the periodic domain of $2\pi$.
\item
Add long scale perturbation to this wave train
$$
V_k = ic(R_k-1) + \epsilon \delta_{1,k}.
$$
\end{itemize}
%\fromSlide{2}{%
\vspace*{-10pt}
\begin{figure}[!h]
\includegraphics[angle=-00,width=10.5cm]{Fig/XY00000.eps}
%\vspace{4mm}
\caption{{\tiny Initial profile of the wave train}}
\end{figure}

\end{raggedright}
\end{slide}
%---------------------------------------------------------------------

%-----------------------------------------------------------SLIDE - 13
\begin{slide}{Simulation}
\small
\vspace*{-10pt}
\begin{raggedright}
\begin{figure}[!h]
\includegraphics[angle=-00,width=10.5cm]{Fig/XY00900.eps}
\vspace{4mm}
\caption{\tiny Typical profile of the wave train}
\end{figure}
\end{raggedright}
\end{slide}
%---------------------------------------------------------------------

%-----------------------------------------------------------SLIDE - 14
\begin{slide}{Simulation}
\small
\vspace*{-20pt}
\begin{raggedright}
%
\begin{figure}[!h]
\includegraphics[angle=-00,width=10.0cm]{Fig/RV00900.eps}
\vspace{4mm}
\caption{\tiny Typical spectra of $|R_k|^2$ and $|V_k|^2$.}
\end{figure}
\end{raggedright}
\end{slide}
%---------------------------------------------------------------------

%-----------------------------------------------------------SLIDE - 15
\begin{slide}{Wave breaks}
\small
\vspace*{-20pt}
\begin{raggedright}
%
 \begin{figure}[!h]
\includegraphics[angle=00,width=10.5cm]{Fig/XY01214.eps}
\vspace{4mm}
\caption{\tiny Wave breaks.}
\end{figure}
\end{raggedright}
\end{slide}
%---------------------------------------------------------------------

%-----------------------------------------------------------SLIDE - 16
\begin{slide}{Slope of the surface}
\small
\vspace*{-40pt}
\begin{raggedright}
%
\begin{figure}[!h]
\includegraphics[angle=00,width=9.0cm]{Fig/DER01214.eps}
\vspace{4mm}
\caption{\tiny Sharp wedge with the angle {\Large $\frac{\pi}{3}$}}
\end{figure}
\end{raggedright}
\end{slide}
%---------------------------------------------------------------------

%-----------------------------------------------------------SLIDE - 17
\begin{slide}{Power spectrum}
\small
\vspace*{-20pt}
\begin{raggedright}
\begin{figure}[!h]
\includegraphics[width=9cm]{Fig/RV01214.eps}
\vspace{-10pt}
\caption{{\tiny Spectrum of} $|R_k|^2$ {\tiny and} $k^{-{8\over 3}}$
  and $k^{-{10\over 3}}$.}
\end{figure}
\end{raggedright}
\end{slide}
%---------------------------------------------------------------------

%-----------------------------------------------------------SLIDE - 18
\begin{slide}{Weaker Waving}
\small
\vspace*{-20pt}
\begin{raggedright}
\begin{figure}[!h]
\includegraphics[width=9cm]{Fig2/XY00001.eps}
\vspace{-10pt}
\caption{{\tiny Initial slightly perturbed wave train} }
\end{figure}
%
\begin{figure}[!h]
\includegraphics[width=9cm]{Fig2/XY00202.eps}
\vspace{-10pt}
\caption{{\tiny Freak wave appears } }
\end{figure}
\end{raggedright}
\end{slide}
%---------------------------------------------------------------------

%-----------------------------------------------------------SLIDE - 19
%\overlays{2}{%
\begin{slide}{? ? ?}
\small
\vspace*{-10pt}
\begin{raggedright}
Sharp peaked stationary gravity wave at the crest 
$$
z \sim i\frac{c^2}{2g} + \left (
\frac{9}{4}\frac{c^2}{g}\right )^{\frac{1}{3}}[-iw^2]^\frac{1}{3} + \epsilon
w^\alpha + \dots\hspace{5em}(\sqrt[3]{-i} = e^{-i\frac{\pi}{6}})
$$
This is well known result of Stokes with the angle - $\frac{2\pi}{3}$.\\
For breaking wave numerics shows - $\frac{\pi}{3}$ wedge.
%\fromSlide{2}{%
\vspace*{-10pt}
$$
z \sim i A(t) + B(t)[-i(w - a(t))(w - b(t))]^\frac{1}{3} + \dots
$$

Root branch that corresponds to the numerics is
$$
z = \cases{-i|u|^{1/3},\hspace{5em} \mbox{ \tiny\emph{missing real part}} &$u \leq 0$;\cr
           |u|^{1/3} ({\sqrt{3}\over 2} - {i\over 2} ), &$u > 0$,}
$$

\end{raggedright}
\end{slide}
%---------------------------------------------------------------------

%-----------------------------------------------------------SLIDE - 20
%\overlays{1}{%
\begin{slide}{? ? ?}
\small
\vspace*{-10pt}
\begin{raggedright}
Wave profile in $(x,y)$ coordinates:
$$
\cases{y^3 = x, &$x \leq 0$;\cr
        y =  \frac{1}{\sqrt{3}}x, &$x > 0$.}
$$
``Overturning'' takes place only near
the wave top, main shape of the wave looks like Stokes wave.
Distance between branch points $a$  and $b$ is small
with
respect to wavelength.
%\fromSlide{1}{%
\vspace*{-10pt}

\end{raggedright}
\end{slide}
%------------------------------------------------------------------------------


\end{document}
%---------------------------------------------------------------------- SLIDE -
\begin{slide}{EIA4}
\small

\vspace*{-12pt}
\end{slide}
%------------------------------------------------------------------------------
