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\begin{frontmatter}
\title{ Effective Five Wave Hamiltonian for Surface Water Waves.\\ 
{\large Published in Physics Letters A, {\bf 230}, 38 (1997).} }

\author{Y.V.Lvov } \address{ Department of Mathematics, University of
Arizona, Tucson, AZ, 85721,USA\\ Department of Physics, University of
Arizona, Tucson, AZ, 85721,USA} 
\begin{abstract} 
An effective five-wave Hamiltonian for one dimensional water waves on the
surface of an infinitely deep ideal fluid is presented.  The diagrammatic
technique is used. For some particular orientations of wave vectors
the interaction matrix element is exactly zero, while for others it is
given by remarkably simple formulaes. This is another  miracle
of water wave theory. 
\end{abstract} 
\end{frontmatter}

\section{Introduction} 

In this article  five-wave interactions of gravity waves on
the surface of an ideal fluid of infinite depth is studied.  A whole set of
physical processes is described by the five-wave interactions, for
instance, the $2\Leftrightarrow3$ instability, 
the creation of horse-shoe-like
structures \cite{BS1,BS2} and others. An important property of such processes
is that they do not conserve the wave action integral. 

Five wave interactions are of great significance in the one dimensional
case because the amplitude of four wave interactions in the effective
Hamiltonian is exactly equal to zero \cite{DZ},\cite{C1},\cite{C2}
 and the fifth order
interaction is the first nonvanishing term \cite{DLZ}.  In the weakly two
dimensional case then, the narrow spectra is defined by the
combination of one dimensional five wave interactions, and four wave
interactions with small angle.

The mathematical reason for the vanishing of the four wave interaction term 
in the 
one dimensional case is not understood yet. There was even a
suggestion, that such vanishing also occurred in higher orders.
Stiassnie et al. \cite{Sch} showed that a certain fifth order
amplitude was zero. That  observation may have lead some to 
believe that all fifth
order terms were  zero.  This was shown to be wrong by Diachenko, Lvov
and Zakharov \cite{DLZ} for collinear wave vector interaction, thus proving
nonintegrability of the system.  However, the final expression for the collinear
wave vector of the fifth order interaction is simple and compact, which is
yet another unsolved mystery.

In the current work, fifth order matrix elements for all possible
relative orientations of the wave vectors are obtained.  For two
particular orientations the resulting interactions  vanish, and
in all other orientations they are nonzero, but given by remarkably
simple expressions.

This work is a natural continuation of \cite{DLZ}.  The same formulation 
of the Hamiltonian formalism and the same symbolic definitions are used. 
\section{Canonical variables and the Hamiltonian of the problem}
\setcounter{equation}{0}
The basic set of equations describing a two-dimensional
potential flow of an ideal incompressible
fluid with a free surface in a gravity field fluid is
the following one:
\begin{eqnarray} \nonumber
\phi_{xx} + \phi_{zz} &=& 0 \hspace{1cm} (\phi_z\to 0, z\to -\infty), \cr
\eta_t + \eta_x\phi_x &=& \phi_z\bigg|_{z=\eta}\cr
\phi_t + \frac{1}{2}(\phi_x^2 + \phi_z^2) + g\eta &=& {0\bigg|_{z=\eta}};
\end{eqnarray}
here $\eta(x,t)$  is the shape of a surface, $\phi(x,z,t)$  is a potential
function of the flow and $g$ - is a gravitational constant.
	As was shown by Zakharov in\cite{Z68},
the potential on the surface $\psi(x,t) = \phi(x,z,t)\bigg|_{z=\eta}$
and $\eta(x,t)$ are canonically conjugated,
and their Fourier transforms satisfy the equations

\[\frac{\partial \psi_k}{\partial t} =
-\frac{\delta \Ha}{\delta \eta_k^*} \hspace{2cm}
  \frac{\partial \eta_k}{\partial t} = \frac{\delta \Ha}{\delta \psi_k^*}.\]
\noindent Here $\Ha=K+U$ is the total energy of the fluid with the following
kinetic and potential energy terms:
\begin{eqnarray} \nonumber
   K = \frac{1}{2}\int\!dx\!\int_{-\infty}^\eta\,v^2\!dz \hspace{1cm}
   U = \frac{g}{2}\int \eta^2\!dx
\end{eqnarray}
\noindent A Hamiltonian can be expanded in an infinite series in powers of
a characteristic wave steepness $k\eta_k <\!< 1$ (\cite{{Z68},{CYS80}})
by using an iterative procedure. All terms up to the fifth order of this series
contribute to the amplitude of the five-wave interaction.
So the Hamiltonian is expressed in terms of complex wave amplitudes $a_k,\  a_k^*$
which satisfies the canonical equation of motion:
\noindent 
\begin{equation} \nonumber
\frac{\partial a_k}{\partial t} + i\frac{\delta \Ha}{\delta a_k^*}=0.
\end{equation}
\noindent here $\omega_k = \sqrt{g|k|}$ -is the dispersion law for
the gravity waves.
\noindent $\Ha$ can be expanded as follow
\begin{eqnarray} \label{HamSer}
\Ha = \Ha_2 + \Ha_3 + \Ha_4 + \Ha_5 + \ldots
\end{eqnarray}
In the normal variable $a_k$ the second order term in the Hamiltonian
acquires the form:
\begin{eqnarray} \nonumber
\Ha_2 = \int \omega_k a_k a_k^* dk
\end{eqnarray}
\noindent The third order term, which describes 
$0\Leftrightarrow 3$ 
(first line) and $1\Leftrightarrow 2$ processes (second line) is:
\begin{eqnarray} \nonumber
\Ha_3 & = &\frac{1}{2}\int\!V^{k_1}_{k_2 k_3}
\{a_{k_1}^* a_{k_2} a_{k_3}+a_{k_1} a_{k_2}^* a_{k_3}^*\}
\delta_{k_1-k_2-k_3}\!dk_1dk_2dk_3+\cr
&+&\frac{1}{6}\int\!U_{k_1 k_2 k_3}
\{a_{k_1} a_{k_2} a_{k_3} + a_{k_1}^* a_{k_2}^* a_{k_3}^*\}
\delta_{k_1+k_2+k_3}\!dk_1dk_2dk_3
\end{eqnarray}
\noindent Fourth order term in the Hamiltonian consists of three terms:
\begin{eqnarray} \nonumber
\Ha_4 = \frac{1}{24}\int &&R_{k_1 k_2 k_3 k_4}
(a_{k_1}a_{k_2}a_{k_3}a_{k_4}+a_{k_1}^*a_{k_2}^*a_{k_3}^*a_{k_4}^*)\times
\delta_{k_1+k_2+k_3+k_4}\!dk_1dk_2dk_3dk_4\cr
+ \frac{1}{6}\int &&G^{k_1}_{k_2 k_3 k_4}
(a_{k_1}^*a_{k_2}a_{k_3}a_{k_4}+a_{k_1}a_{k_2}^*a_{k_3}^*a_{k_4}^*)\times
\delta_{k_1-k_2-k_3-k_4}\!dk_1dk_2dk_3dk_4\cr
+  \frac{1}{4}\int&& W^{k_1 k_2}_{k_3 k_4}
a_{k_1}^*a_{k_2}^*a_{k_3}a_{k_4}
\delta_{k_1+k_2-k_3-k_4}dk_1dk_2dk_3dk_4
\end{eqnarray}
\noindent describing different types of  wave interactions 
(first line is 
${4\Leftrightarrow 0} $, second line is ${3\Leftrightarrow 1} $, last line is 
${2\Leftrightarrow 2} $
interactions).

Among the different terms of the fifth order,  the only  term
corresponding to the process $2\Leftrightarrow3$ is considered:
\begin{eqnarray} \nonumber
\Ha_5 = \!\frac{1}{12}\int Q^{k_1 k_2 k_3}_{k_4 k_5}
\{a_{k_1}^* a_{k_2}^* a_{k_3}^* a_{k_4} a_{k_5}+\!c.c.\}
\delta_{k_1+k_2+k_3-k_4-k_5}\!dk_1dk_2dk_3dk_4dk_5
\end{eqnarray}
Here $V^{k_1}_{k_2,k_3}, U_{k_1,k_2,k_3},R_{k_1 k_2 k_3 k_4}, 
G^{k_1}_{k_2 k_3 k_4}, W^{k_1 k_2}_{k_3 k_4}, Q^{k_1 k_2 k_3}_{k_4 k_5}$
are interaction matrix elements of third, fourth and fifth order.
  
The Hamiltonian $\Ha$ in the normal variables $a_k$ is too complicated to
work with. Our purpose is to simplify the Hamiltonian to the form:
\begin{eqnarray} \label{Ham4_5}
\!&\Ha& = \int \omega_k b_k b_k^* dk + \frac{1}{4}\int T^{k_1 k_2}_{k_3 k_4}
b_{k_1}^*b_{k_2}^*b_{k_3}b_{k_4}
\delta_{k_1+k_2-k_3-k_4}dk_1dk_2dk_3dk_4+ \cr
%
\!&+&\!\!\frac{1}{12}\!\int\! T^{k_1 k_2 k_3}_{k_4 k_5}\!
\{b_{k_1}^* b_{k_2}^* b_{k_3}^* b_{k_4} b_{k_5}+\!c.c.\}
\delta_{k_1+k_2+k_3-k_4-k_5}\!dk_1dk_2dk_3dk_4dk_5
\end{eqnarray}
\noindent 
One of the ways to do that is to perform a canonical transformation
\cite{Z74}, \cite{KRS90}
\begin{eqnarray} \label{transformation}
a_k &=& b_k + \int \Gamma^{k}_{k_1 k_2}b_{k_1}b_{k_2}\delta_{k-k_1-k_2} -
	   2\int \Gamma^{k_2}_{k k_1}b^*_{k_1}b_{k_2}\delta_{k+k_1-k_2} +\cr
&+&\int \Gamma_{k k_1 k_2}b^*_{k_1}b^*_{k_2}\delta_{k+k_1+k_2} +
\int B^{k k_1}_{k_2 k_3}b^*_{k_1}b_{k_2}b_{k_3}\delta_{k+k_1-k_2-k_3}+\ldots
\end{eqnarray}
\noindent where the $\Gamma$'s and $B$'s are determined 
in such a way that the transformation is canonical, and that 
the transformed Hamiltonian has the form (\ref{Ham4_5}).
The transformation (\ref{transformation}) is canonical up to
the terms of order of $|b_k|^3$. 

On the resonant manifold 
$\omega_{k} + \omega_{k_1} = \omega_{k_2} + \omega_{k_3} \ \ \ 
\vec k + \vec k_1 = \vec k_2 + \vec k_3$ there are two types of resonances - 
trivial and nontrivial. Trivial resonances are
$k_2 = k_1,\hspace{.5cm} k_3 = k,\ \ or\hspace{.5cm}k_3 =
k_1,\hspace{.5cm} k_2=k$. Nontrivial resonances may be parameterized as
\begin{eqnarray} \label{nontriv4}
k   &=& a(1+\zeta)^2, \hspace{0.5cm}k_1 = a(1+\zeta)^2 \zeta^2, \cr
k_2 &=& -a\zeta^2, \hspace{1.0cm}k_3 = a(1+\zeta+\zeta^2)^2
\end{eqnarray}

It was shown in \cite{DZ},\cite{C1,C2} that on the nontrivial manifold (\ref{nontriv4}) 
$T^{k,k_1}_{k_2,k_3}\equiv 0$, i.e. four-wave processes do not produce
``new wave vectors'', and that system is integrable to this degree of accuracy.
This was the main motivation for investigating fifth order  interactions. 

To find $T^{k k_1 k_2}_{k_3 k_4}$ one can calculate the terms of the
order of $b^3$ and $b^4$ in the canonical transformation
(\ref{transformation}). This very cumbersome procedure was fulfilled
by V.Krasitskii\cite{KRS94}, but the resulting expressions are so
complicated that they can hardly  be used for any practical purpose. Here 
the method of Feinman diagrams presented in \cite{ZSH}, \cite{DLZ}
 is used.

First one introduce the so called formal classical scattering matrix which
relates the asymptotic states of the system ``before'' and ``after''
interactions: \begin{eqnarray} \nonumber c^{+}_k = \hat
S[c^{-}_k] \end{eqnarray} \noindent 
By ``for
$\hat
S[c^{-}_k]$ is a nonlinear operator which can be presented
as a series in powers of $c^{-}, {c^{-}}^{*}$.  
It has the following form \begin{eqnarray}
\label{ScMat} \hat S[c^{-}_k] = c^{-}_k &-& \sum_{n+m\ge3}^{}
\frac{2\pi i}{(n-1)!m!}\int S_{n
m}(k,k_1,\ldots,k_{n-1};k_n,\ldots,k_{n+m-1})\times\cr
&\times&\delta_{k+k_1+\ldots +k_{n-1}-k_n-\ldots -k_{n+m-1}}
\delta_{\omega_k+\omega_{k_1}+\ldots +\omega_{k_{n-1}}
-\omega_{k_n}-\ldots -\omega_{k_{n+m-1}}}\times\cr
&\times&{c^-}^*_{k_1}\ldots {c^-}^*_{k_{n-1}}c^-_{k_n}\ldots
c^-_{k_{n+m-1}} dk_1\ldots dk_{n+m-1} \end{eqnarray} 
We will treat this series as formal one and will not care about their
convergence \cite{DLZ,ZM}.
The functions
$S_{n m}$ are the elements of the scattering matrix. They are defined
on the resonant manifolds \begin{eqnarray} \label{resN} \vec k+\vec k_1+\ldots
+\vec k_{n-1} &=&\vec k_n+\ldots +\vec k_{n+m-1}
\cr \omega_k+\omega_{k_1}+\ldots
+\omega_{k_{n-1}} &=& \omega_{k_n}+\ldots +\omega_{k_{n+m-1}}
\end{eqnarray} Note, that the value of the matrix element $S_{n m}$
on the resonant manifold (\ref{resN}) is invariant with respect to
the  canonical transformation (\ref{transformation}) and that there is a
simple algorithm for calculation of the matrix elements.  The element
$S_{n m}$ is a finite sum of the terms which can be expressed through the
coefficients of the Hamiltonians $H_i, i\le {n+m}$. Each term can be
marked by a certain Feinman diagram taken in a "tree" approximation,
i.e. having no internal loops. To 
calculate 
$T^{k k_1 k_2}_{k_3 k_4}$ one calculates the
first nonzero elements of the scattering matrix
for the Hamiltonian (\ref{HamSer}) and for the Hamiltonian
(\ref{Ham4_5}).  Because these two Hamiltonians are connected by
the canonical transformation (\ref{transformation}), the results must
coincide.  
The first nontrivial element of the scattering matrix in the
one-dimensional case is
\begin{eqnarray} \nonumber
S_{3 2}(k,k_1,k_2,k_3,k_4) = T^{k k_1 k_2}_{k_3 k_4}
\end{eqnarray}
\noindent 
If $S_{3 2}(k,k_1,k_2,k_3,k_4)$ is calculated 
in terms of the initial Hamiltonian (\ref{HamSer})
it consists of 81 terms, 
with  60 
diagrams combining three third order interactions
 (one of such diagrams with 
the corresponding expressions is shown below):
\\
{
%picture1
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\begin{picture}(48,24)(0,10)
%\input{pic1}
%\input{pic1-48}
\multiput(13,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[b]}
\multiput(11,12)(-2,2){6}{\oval(2,2)[t,r]}
\multiput(11,14)(-2,2){6}{\oval(2,2)[b,l]}
\put(0,0){\line(1,1){12}}
\multiput(25,12)(4,0){2}{\oval(2,1.6)[t]}
\multiput(27,12)(4,0){1}{\oval(2,1.6)[b]}
\put(30,12){\line(1,0){6}}
\multiput(24,13)(0,4){3}{\oval(1.6,2)[l]}
\multiput(24,15)(0,4){2}{\oval(1.6,2)[r]}
\put(18,12){\line(1,0){6}}
\multiput(37,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(37,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(37,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(37,10)(2,-2){6}{\oval(2,2)[t,r]}
%\put(12,12){\circle*{1.5}}
%\put(24,12){\circle*{1.5}}
%\put(36,12){\circle*{1.5}}
\put(3,0){$q$}
\put(10.85,10.85){\vector(1,1){1}}
\put(10.85,13.15){\vector(1,-1){1}}
\put(24,22){\vector(0,1){2}}
\put(47.1,23){\vector(1,1){1}}
\put(47.1,1){\vector(1,-1){1}}
\put(3,24){$p$}
\put(25,24){$k_1$}
\put(42,24){$k_2$}
\put(42,0){$k_3$}
\put(22.5,12){\vector(1,0){1}}
\put(34.5,12){\vector(1,0){1}}
\put(17,16){\makebox(0,0){$p+q$}}
\put(31,16){\makebox(0,0){$k_2+k_3$}}
\end{picture}

\eb
\[ {{V^{{\it k_2} + {\it k_3}}_{{\it k_2},{\it k_3}}\,
      V^{p + q}_{{\it k_1},{\it k_2} + {\it k_3}}\,
	V^{p+q}_{p,q}}\over
    {\left( \omega_{{\it k_2}} + \omega_{{\it k_3}} -
	\omega_{{\it k_2} + {\it k_3}} \right) \,
      \left( \omega_{p} + \omega_{q} - \omega_{p + q} \right) }}\]
\ep


\


\


and also   20 diagrams combining third- 
and fourth order interactions
(one of such diagrams with 
the corresponding expressions is shown below):\\

%picture61
%\input{pic61-72}
\begin{picture}(36,24)(0,10)

\multiput(2.5,12)(4,0){3}{\oval(2,1.6)[t]}
\multiput(4.5,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(12,13)(0,4){3}{\oval(1.6,2)[r]}
\multiput(12,15)(0,4){2}{\oval(1.6,2)[l]}

\multiput(13,12)(4,0){2}{\oval(2,1.6)[b]}
\multiput(15,12)(4,0){1}{\oval(2,1.6)[t]}
\put(18,12){\line(1,0){6}}

\multiput(25,12)(2,2){6}{\oval(2,2)[t,l]}
\multiput(25,14)(2,2){6}{\oval(2,2)[b,r]}
\multiput(25,12)(2,-2){6}{\oval(2,2)[b,l]}
\multiput(25,10)(2,-2){6}{\oval(2,2)[t,r]}
\put(12,0){\vector(0,1){12}}
\put(12,22){\vector(-1,1){1}}

\put(13,0){$q$}


\put(11,12.8){\vector(1,-1){1}}
\put(35.1,23){\vector(1,1){1}}
\put(35.1,1){\vector(1,-1){1}}
\put(23,12){\vector(1,0){1}}
\put(1,8){$p$}
\put(13,24){$k_1$}
\put(30,24){$k_2$}
\put(30,0){$k_3$}
\put(19,16){\makebox(0,0){$k_2+k_3$}}
\end{picture}

\eb
\[ {{V^{{\it k_2} + {\it k_3}}_{{\it k_2},{\it k_3}}\,
      W^{p,q}_{{\it k_1},{\it k_2} + {\it k_3}}}\over
    {\omega_{\it k_2}+ \omega_{\it k_3} - \omega_{{\it k_2} + {\it k_3}}}}
   \]
\ep
}

\

and also the fifth order vertex itself.

We use "Mathematica 2.2" for performing the analytical and numerical
calculations of this paper.  Initially, the expression for $T^{1,2}_{3,4,5}$ occupies 1
Megabyte of computer memory, but we were able to 
simplify it to the form presented below. For some of the
orientations, $T^{1,2}_{345}$ is equal to zero. We verify 
this fact by 
 computing
$T^{12}_{345}$ numerically on 100 random points of the resonant manifold 
(\ref{res}) and get zero with accuracy of $10^{-90}$.
\section{Results} We present here the results of calculation of the 
matrix
element on the resonant manifold \begin{eqnarray} k_1 + k_2 + k_3 = p +
q \ \ \ \ \ \omega_{k_1} + \omega_{k_2} + \omega_{k_3} = \omega_{p}
+ \omega_{q} \ \ \ \ \ \ \ \o_k=\sqrt{g |k|} \label{res}\end{eqnarray} 
There are five topologically different configurations for the 
${k_1,k_2,k_3}\to{p,q}$ interaction on (\ref{res}) 
for arbitrary signs of wave vectors in 1D:
\begin{enumerate} \item{ {\it All wave vectors positive.} }
\item {\it Positive $p$ and $q$,
and one of the ${k_1,k_2,k_3}$  negative}.  
\item {\it Positive $p$
and $q$, and two of the ${k_1,k_2,k_3}$ negative}.  \item {\it $p$
$q$ with  different signs, ${k_1,k_2,k_3}$ positive}.  \item {\it
$p$ $q$ with different signs, and one of ${k_1,k_2,k_3}$  negative}
\end{enumerate} 
The results of this calculations is presented below, and summarized
in the table at the end of the article.
Some of the final expressions  are naturally less symmetric than
others, because the symmetry of expression reflects the 
symmetry of the wave-vector setup.
{ \begin{enumerate} 
\item {The answer is
given in \cite{DLZ}
\begin{eqnarray}  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)}\\&=&
\frac{2}{g^{1/2}\pi^{3/2}}\frac{\o_1^{5/2}\o_2^{5/2}\o_3^{5/2}
\o_4^{3/2}\o_5^{3/2}}
{{max(\o_1^2,\o_2^2,\o_3^2)g^4} } \end{eqnarray} } \item 
Choose $k_3$ to be negative, and $k_1>k_2$.  One can parameterize the 
resonant manifold by \begin{eqnarray} \o_1=c(a+a b + b),\ \ \ \ \
k_1=\o_1^2/g;\cr \o_2=c(a b -1), \ \ \ \ \ k_2=\o_2^2/g; \cr \o_3=c,\ \ \
\ \ k_3=-\o_3^2/g ;\cr \o_4=c(a+1)b, \ \ \ \ \ p=\o_4^2/g;\cr
\o_5=c(b+1)a,\ \ \ \ \ q=\o_5^2/g;\cr\end{eqnarray} where $c>0$, $\ \ \
a,b>0$ with $a b >1$

The result depends upon the sign of $\o_2-\o_3$.  \begin{itemize}
\item case when $\o_2<\o_3$. In this case $\o_{k_1}>\o_{k_4},
\o_{k_5} > \o_{k_3}> \o_{k_2}$ \begin{eqnarray} T^{k_1
k_2 k_3}_{k_4 k_5} = \frac{1}{g^{9/2}\pi^{3/2}}
\o_{k_1}^{3/2}\o_{k_2}^{11/2} \o_{k_3}^{1/2}\o_{k_4}^{1/2}
\o_{k_5}^{1/2}
\end{eqnarray} \item case of $\o_2>\o_3$. Here
$\o_{k_1}> \o_{k_4}, \o_{k_5}> \o_{k_2}> \o_{k_3}$ \begin{eqnarray}
 T^{k_1 k_2 k_3}_{k_4 k_5} = \frac{1}{g^{9/2}\pi^{3/2}}
{\o_{k_1}^{3/2}\o_{k_2}^{3/2} \o_{k_3}^{5/2}\o_{k_4}^{1/2}\o_{k_5}^{1/2}}
 (2\o_{k_2}^2-\o_{k_3}^2) \end{eqnarray} \end{itemize} 
\item{\large \bf zero} \item{\bf \large zero } \item chose $q<0$ and
$k_3<0$ and positive $p$, $k_1,\ \ k_2$.  Then for any positive
$\o_1,\ \ \o_2, \ \ \o_3$ one can find $\o_4,\ \ \o_5$ satisfying
resonant condition (\ref{res}) to be
$$\o_4=\frac{\o_1^2+\o_1\o_2+\o_2^2+\o_1\o_3+\o_2\o_3}{\o_1+\o_2+\o_3}$$
$$\o_5=\frac{(\o_1+\o_3)(\o_2+\o_3)}{\o_1+\o_2+\o_3}$$ 
One can see from this parameterization, that $\o_4>\o_1,\ \o_5>\o_3,
\ \o_4>/o_2$. Choose $\o_{k_1}>\o_{k_2}$.
Then there are three 
variants of relations between
$\o_{k_1},\o_{k_2},\o_{k_3}$.
One of them $\o_{k_3}>\o_{k_1}>\o_{k_2}$
does not fix the relation between $\o_{k_4}$ and $\o_{k_5}$, so there are
four different cases of relations between $\o$'s for which the  
fifth order 
matrix element can be calculated. 
\begin{enumerate} \item If $\o_1>\o_2>\o_3$ then
$\o_4>\o_5$ and  
$${T^{k_1,k_2,k_3}_{k_4,k_5} =
-\frac{\o_{k_1}^{1/2}\o_{k_2}^{1/2}\o_{k_3}^{11/2}
\o_{k_4}^{3/2}\o_{k_5}^{1/2}} {\pi^{3/2}g^{9/2}
}}$$ 
\item If  $\o_1>\o_3>\o_2$ then $\o_4>\o_5$ and 
$$T^{k_1,k_2,k_3}_{k_4,k_5}
=\frac{\o_{k_1}^{1/2}\o_{k_2}^{5/2}\o_{k_3}^{3/2}\o_{k_4}^{3/2}
\o_{k_5}^{1/2}(\o_{k_2}^2-2\o_{k_3}^2)}
{{\pi^{3/2}g^{9/2} }}$$ 
\item If   $\o_{k_3}>\o_{k_1}>\o_{k_2}$ and  $\o_4>\o_5>$ then 
$$
T^{k_1,k_2,k_3}_{k_4,k_5}
=\frac{
\o_{k_1}^{1/2}\o_{k_2}^{1/2}\o_{k_3}^{3/2}\o_{k_4}^{3/2}\o_{k_5}^{1/2}
(\o_1^4+\o_2^4-2\o_1^2\o_3^2-2\o_2^2\o_3^2+\o_3^4)}
{{\pi^{3/2}g^{9/2}}}$$ 
\item  If  $\o_{k_3}>\o_{k_1}>\o_{k_2}$ and $\o_5>\o_4$, then
$$
T^{k_1,k_2,k_3}_{k_4,k_5} =-\frac{2}
{\pi^{3/2}g^{9/2}}
\o_1^{5/2}\o_2^{5/2}\o_3^{3/2}\o_4^{3/2}\o_5^{1/2} $$ \end{enumerate}  \end{enumerate}
\section{Conclusion} 
All the matrix elements for the fifth order interaction in one dimension
are calculated in this work for surface waves on top of an ideal fluid
of infinite depth.   The expressions obtained are
astonishingly simple.   For some particular orientation of the wave vectors the 
the interaction matrix element is equal to {\it zero}. 
We think that  this fact has  a deep physical meaning and that
there should be simpler ways of getting these results.

This article answers the question, of {\it how} waves interact in the case
of one dimensional waves on the surface of the infinitely deep ideal
fluid.  It is still to be explained, {\it why} they interact in such a
way, and {\it why} some of them do not interact at all.

\section{Acknowledgment} The 
author would like to express gratitude to Dr. Vladimir Zakharov for
the formulation of the problem and fruitful discussions. The author
would also like to thank the research group of the Center for
Computing and Information Technology at the University of Arizona for
providing access to a Scalable Power computer.

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\def \modina {{$  T^{k_1 k_2k_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)}$}} 
\def \modinb {{$  
= \frac{2}{g^{1/2}\pi^{3/2}}\frac{\o_1^{5/2}\o_2^{5/2}\o_3^{5/2}
\o_4^{3/2}\o_5^{3/2}}
{{max(\o_1^2,\o_2^2,\o_3^2)g^4}} $}}
\def \aodin {$ \o_1=c(a+a b + b),$}
\def \aodind {$ \ k_1\propto\o_1^2$}
\def \adva {$\o_2=c(a b -1), k_2\propto\o_2^2$}
\def \atri {$\o_3=c, k_3\propto-\o_3^2$}
\def \achet {$\o_4=c(a+1)b, p\propto\o_4^2$}
\def \apat {$\o_5=c(b+1)a, q\propto\o_5^2$}
\def \confdvaodin{$T^{k_1
k_2 k_3}_{p,q} = \frac{1}{g^{9/2}\pi^{3/2}}
{\o_{k_1}^{3/2}\o_{k_2}^{11/2} \o_{k_3}^{1/2}\o_{k_4}^{1/2}\o_{k_5}^{1/2}}
    $}
\def \confdvadva{$ 
 T^{k_1 k_2 k_3}_{k_4 k_5} = $
}
\def \confdvadvad{$ 
 = \frac{1}{g^{9/2}\pi^{3/2}}
{\o_{k_1}^{3/2}\o_{k_2}^{3/2} \o_{k_3}^{5/2}\o_{k_4}^{1/2}\o_{k_5}^{1/2}
(2\o_{k_2}^2-\o_{k_3}^2) }
$ 
}
\def \omegatri 
{
$\o_4=\o_1+\frac{\o_2^2+\o_2\o_3}{\o_1+\o_2+\o_3}$
}
\def \omegachetire{
 $\o_5=\frac{(\o_1+\o_3)(\o_2+\o_3)}{\o_1+\o_2+\o_3}$ }
\def \confpatodin{$T^{k_1,k_2,k_3}_{k_4,k_5} =-\frac{1}{{\pi^{3/2}g^{9/2}}}
{\o_{k_1}^{1/2}\o_{k_2}^{1/2}\o_{k_3}^{11/2}\o_{k_4}^{3/2}\o_{k_5}^{1/2}} 
$} 
\def \vecoo {$k_1=\o_1^2/g,k_2=\o_2^2/g,$}
\def \veco {$k_3=-\o_3^2/g;$}
\def \vecd {$k_4=-\o_4^2/g,k_5=\o_5^2/g$}
\def \confpatdva{$T^{k_1,k_2,k_3}_{k_4,k_5}
=\frac{1}{{{\pi^{3/2}g^{9/2}}}}\times$}
\def \confpatdvad{$
\o_{k_1}^{1/2}\o_{k_2}^{5/2}\o_{k_3}^{3/2}\o_{k_4}^{3/2}\o_{k_5}^{1/2}(\o_{k_2}^2-2\o_{k_3}^2)$}
\def \confpattri {
$
T^{k_1,k_2,k_3}_{k_4,k_5}=\frac{1}{\pi^{3/2}g^{9/2}}
\o_{k_1}^{1/2}\o_{k_2}^{1/2}\o_{k_3}^{3/2}\o_{k_4}^{3/2}\o_{k_5}^{1/2}
$}
\def \confpattrid {
$
\times
{
(\o_1^4+\o_2^4-2\o_1^2\o_3^2-2\o_2^2\o_3^2+\o_3^4)}$}
\def \confpatchetire {$
T^{k_1,k_2,k_3}_{k_4,k_5} =-\frac{1}{\pi^{3/2}g^{9/2}}
{2\o_1^{5/2}\o_2^{5/2}\o_3^{3/2}\o_4^{3/2}\o_5^{1/2}   }
$}
{
\def \parodin{$\o_1    =  a (p^2 - q^2 +1 - 2p)$}
\def \pardva{$\o_2  =  a (p^2 - q^2 +1 + 2p)$}
\def \partri{$\o_3  =  4 a$}
\def \parchet{$\o^4  =  a (p^2 - q^2 +3 - 2q)$}
\def \parpat{$\o_5  =  a (p^2 - q^2 +3 + 2q)$}
\def \srodin{ $\o_1>\o_2>\o_3;$ }
\def \srodind{ $\o_4>\o_5;$ }
\def \srdva { $\o_1>\o_3>\o_2;$}
\def \srdvad { $\o_4>\o_5;$}
\def \srtri {  $\o_3>\o_1>\o_2;$}
\def \srtrid {  $\o_{4}>\o_{5};$}
\def \srchet{   $\o_{3}>\o_{1}>\o_{k_2};$}
\def \srchetd{   $\o_{k_5}>\o_{k_4};$}
\oddsidemargin 0cm
\evensidemargin 0cm
\textwidth 17cm
%\begin{document}
%\special{landscape}
\scriptsize
\newpage
\pagestyle{empty}
\begin{tabular}{||l|l|l|l||}
\hline
\hline
Configu & Parametrization & Relation & Matrix element value \\
ration& & & \\
\hline
\hline
all     &\parodin  &  &  \modina \\
		     &\pardva  &  &          \\
vectors		     &\partri  &  &          \\
		     &\parchet  &  &  \modinb\\
positive	     	     & \parpat &  &          \\	
      	     & $a>0;0<p,q<1; $ &  &          \\	
    	     & $|p\pm q|<1; $ &  &          \\	
\hline
 Positive                & \aodin  &$ \o_1>\o_4,\o_5;$& \\
   $k_4$ and $k_5$,	 & \aodind  & $\o_4,\o_5>\o_3>$ & \\
          and one of     & \adva  & $>\o_2;$   &  \confdvaodin       \\
   $k_1,k_2,k_3$  	 & \atri  & & \\ 
	     \cline{3-4}	
     negativee	 & \achet         & $\o_1>\o_4,\o_5;$& \confdvadva \\
	 Choose 	 &  \apat         & $\o_4,\o_5>\o_2>$       &  \confdvadvad         \\ 
   $k_1>k_2$             &  $a,b>0,a b>1$ &  $>\o_3;$ & \\ 
  \hline 
Positive   &   & & \\
  $k_4$ and $k_5$  &   & & \\
and two of  &   & & \\
$k_1,k_2,k_3$  &  & & {\bf  ZERO} \\
    negative   &   & & \\
\hline 
$k_4$ and $k_5$   &   & & \\
 have  &                 &          &                  \\
different    &   & & \\
signs       &   & & \\
$k_1,k_2,k_3$  &  & & {\bf  ZERO} \\
    positive   &   & & \\
\hline 
  $k_4$ and $k_5$&		   & \srodin       &       \\          
   have different&  \omegatri&  \srodind&  \confpatodin  \\
     	\cline{3-4}
	signs               & & \srdva       &       \\          
one of  & \omegachetire &\srdvad &\confpatdva  \\
  &  & &\confpatdvad  \\
\cline{3-4}	
	$k_1,k_2,k_3$  &\vecoo         & \srtri&    \confpattri         \\
	  negative  & \veco& \srtrid& \confpattrid\\
 \cline{3-4}	 
 & & \srchet  & \\
   choose  &\vecd &   \srchetd  &  \confpatchetire \\
 $k_1>k_2$  &                 &          &                  \\	
\hline
\hline
\end{tabular}
\end{document} 











