\documentstyle{elsart} \newcommand{\Ha}{\mbox{$\cal H$}} \newcommand{\Pa}{\mbox{$\cal P$}} \newcommand{\Ps}{\mbox{$\cal \Psi$}} \newcounter{newcnt} \renewcommand{\theequation}{\thesection.\arabic{equation}} \begin{document} \def \eb {\begin{flushright} \vskip -1.2cm \begin{minipage}{9cm}} \def \ep {\end{minipage} \end{flushright}} \def \o {\omega} \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 \setlength{\unitlength}{.1cm} \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. 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V.E.Zakharov, (Springer-Verlag, 1991), 190. \bibitem{ZM} Zakharov,V.E. and Manakov S.V. {\em JETP\/}, {\bf 42}, (1975), 842. \end{thebibliography} \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 & & \\ & $a>0;0\o_4,\o_5;$& \\ $k_4$ 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}