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

Alber, Mark S.; Luther, Gregory G.; Marsden, Jerrold E. ; Robbins, Jonathan M.

HPL-BRIMS-98-08

Keyword(s):three-wave interaction; geometric phases; reduction; Lie-Poisson structure

Abstract: Hamiltonian Lie-Poisson structures of the three-wave equations associated with the Lie algebras su(3) and su(2,1) are delivered 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 integrability, in this context. Notes:Mark S. Alber, Dept. of Mathematics,University of Notre Dame, Notre Dame, IN 46556. Gregory G. Luther, Engineering Sciences and Applied Mathematics Department, McCormick School of Engineering and Applied Science, Northwestern University, 2145 Sheridan Road, Evanston, Il 60208-3125. Jerrold E. Marsden, Control and Dynamical Systems 107-81, Caltech, Pasadena, CA 91125.

26 Pages

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