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 [3,4] 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 [2],[5],[6] and the fifth order interaction is the first nonvanishing term [1]. 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. [7] 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 [1] 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 [1]. The same formulation of the Hamiltonian formalism and the same symbolic definitions are used.