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

Alber, Mark S. and Luther, Gregory G. and Marsden, Jerrold E. and Robbins, Jonathan M. (1998) Geometric Phases, Reduction and Lie-Poisson Structure for the Resonant Three-wave Interaction. Technical Reports, HPL-BR. Hewlett-Packard Company .

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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.

Item Type:Report or Paper (Report)
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Additional Information:© 1994-2000 Hewlett-Packard Company. Spring, 1995; this version, January 30, 1998. Research partially supported by NSF grants DMS 9626672 and 9508711. GGL gratefully acknowledges support from BRIMS, Hewlett-Packard Labs and from NSF DMS under grants 9626672 and 9508711. Research partially supported by the Department of Energy under Contract DE-FG0395- ER25251. Research partially supported by NSF grant DMS 9508711, NATO grant CRG 950897 and by the Department of Mathematics and the Center for Applied Mathematics, University of Notre Dame
Funding AgencyGrant Number
NSFDMS 9626672
NSFDMS 9508711
Basic Research Institute in the Mathematical Sciences (BRIMS) Hewlett-Packard LabsUNSPECIFIED
Department of Energy (DOE)DE-FG0395-ER25251
NATOCRG 950897
University of Notre Dome, Department of Mathematics and the Center for Applied MathematicsUNSPECIFIED
Subject Keywords:Geometric phase, integrable systems, Lax equation, Lie-Poisson, nonlinear waves, reduction, three wave interaction
Series Name:Technical Reports
Issue or Number:HPL-BR
Classification Code:PACS: 4.30.Nk, 3.40.Kf, 42.65.-k, 3.40.Gc, 52.35.Mw
Record Number:CaltechAUTHORS:20100707-143741392
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:18932
Deposited By: Ruth Sustaita
Deposited On:09 Jul 2010 17:27
Last Modified:03 Oct 2019 01:49

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