Geometric Phases, Reduction and Lie-Poisson Structure for the Resonant Three-wave Interaction

Abstract

Hamiltonian Lie-Poisson structures of the three-wave equations associated with the Lie algebras su(3) and su(2; 1) 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.