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

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\renewcommand{\figurename}{\scriptsize Figure}

%\title{Турбулентность и сингулярности\\ в нелинейных волновых системах}
\title{On the Free Surface Hydrodynmics \\(with surface tension)}
\author{\ }
\institution{\ }
%Институт теоретической физики им. Л.Д. Ландау РАН}

\begin{document}
\maketitle

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

%------------------------------------------------------------SLIDE - 2
%\overlays{1}{%
\begin{slide}{Basic 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}
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%--------------------------------------------------------------------- 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}
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%------------------------------------------------------------SLIDE - 4
%\overlays{1}{%
\begin{slide}{Explicit Equations}
\small

\vspace*{-10pt}
\begin{raggedright}
$Z$ - {\it the conformal mapping}, \hspace{0.5cm} 
$\Phi$ - {\it complex velocity potential}.
$$z = x+i y,\hspace{1em}  \hspace{1cm} \Phi = \Psi + \hat H\Psi.$$
%\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 $\hat P(f) = \frac{1}{2}( 1 + i\hat H )(f)$.

\hspace{6cm} {\small Ovsyannikov,1973}
\end{raggedright}
\end{slide}
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%------------------------------------------------------------SLIDE -5
%\overlays{1}{%
\begin{slide}{Cubic Equations}
\small

\vspace*{-10pt}
\begin{raggedright}
$$
R = \frac{1}{z_w}, \hspace{5em}\Phi_w = -i V z_w.
$$
\vspace{1cm}
%\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) -\cr
 & & \hspace{3cm} -2\sigma R \hat P'( \sqrt{R}' \sqrt{\bar R} - \sqrt{\bar R}' \sqrt{R}).
\end{eqnarray}
\vspace{0.5cm}
{\scriptsize Complex transport velocity} $U$ {\scriptsize is defined via} $\hat P$ 
$$ U = \hat P(V\bar R + \bar V R).$$

\end{raggedright}
\end{slide}
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%-----------------------------------------------------------SLIDE - 6
\begin{slide}{Freak Wave}
\small
\vspace*{-20pt}
\begin{raggedright}
\begin{figure}[!h]
\includegraphics[width=9cm,height=7cm]{Fig2/Zoom.eps}
\vspace{-10pt}
\caption{{\tiny Close view} }
\end{figure}
%
\end{raggedright}
\end{slide}
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\end{document}
%---------------------------------------------------------------------- SLIDE -
\begin{slide}{EIA4}
\small

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