O(2)‐lnvariant Hamiltonians on ℂ⁴

Abstract

A normal-form theory and a group-theoretic classification for periodic solutions of 0(2)-invariant Hamiltonians on C 4 is developed. The theory applies to Hamiltonian systems with an 0(2) spatial symmetry that also have a linear-mode interaction. Our motivation is the classic (m, n) mode-interaction problem for capillary-gravity waves. It is well known that the addition of surface-tension effects to irrotational water waves results in a countable infinity of values of the surface-tension coefficient at which two traveling waves of differing wavelength travel at the same speed. However, recognizing the reflectional symmetry in space, the linearized problem is actually spanned by four traveling waves. In other words there is an 0(2) symmetry in space. A classification theorem for group-invariant Hamiltonian systems, based on a listing of the isotropy subgroups and their fixed-point spaces, is used to show that there are between seven and fourteen classes of periodic solutions in 0(2)-invariant Hamiltonian systems with a mode interaction. The results are used to interpret, from a group-theoretic viewpoint, the classic Wilton ripple.