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

Alber, Mark S.; Luther, Greg G.; Marsden, Jerrold E.


Keyword(s): geometric phase; nonlinear waves; integrable systems; three wave; Lax equation; Lie-Poisson

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 have important implications for nonlinear-wave applications 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.

24 Pages

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