Hamiltonian Lie-Poisson structures of the three-wave equations
associated with the Lie algebras
and
are derived and shown to be compatible. Poisson
reduction is performed using the method of invariants and geometric
phases associated with the reconstruction are calculated. These
results can be applied to applications of nonlinear-waves
in, for instance, nonlinear optics. Some of the general structures
presented in the latter part of this paper are implicit in the
literature; our purpose is to put the three-wave interaction in the
modern setting of geometric mechanics and to explore some new things,
such as explicit geometric phase formulas, as well as some old things,
such as integrability, in this context.
