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 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 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, why they interact in such a way, and why some of them do not interact at all.