Geometric phases, reduction and Lie-Poisson structure for the resonant three-wave interaction

Hamiltonian Lie-Poisson structures of the three-wave equations associated w ith the Lie algebras su(3) and su(2,1) are derived and shown to be compatib le. Poisson reduction is performed using the method of invariants, and geom etric phases associated with the reconstruction are calculated. These resul ts can be applied to applications of nonlinear-waves in, for instance, nonl inear optics. Some of the general structures presented in the latter part o f this paper are implicit in the literature; our purpose is to put the thre e-wave interaction in the modern setting of geometric mechanics and to expl ore some new things, such as explicit geometric phase formulas, as well as some old things, such as integrability, in this context. Copyright (C) 1998 Elsevier Science B.V.