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Geometric Phases, Reduction and LiePoisson Structure for the Resonant ThreeWave Interaction
Alber, Mark S.; Luther, Greg G.; Marsden, Jerrold E.
HPLBRIMS9715
Keyword(s): geometric phase; nonlinear waves; integrable systems; three wave; Lax equation; LiePoisson
Abstract: Hamiltonian LiePoisson structures of the threewave 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 nonlinearwave 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 threewave 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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