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Geometry and Control of Three-Wave Interactions

Alber, Mark S. and Luther, Gregory G. and Marsden, Jerrold E. and Robbins, Jonathan M. (1999) Geometry and Control of Three-Wave Interactions. In: The Arnoldfest Proceedings of a Conference in Honour of V.I. Arnold for his Sixtieth Birthday. AMS and Fields Institute. American Mathematical Society , Rhode Island, US, pp. 55-80. ISBN 0821809458.

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The integrable structure of the three-wave equations is discussed in the setting of geometric mechanics. Lie-Poisson structures with quadratic Hamiltonian are associated with the three-wave equations through the Lie algebras su(3) and su(2, 1). A second structure having cubic Hamiltonian is shown to be compatible. The analogy between this system and the rigid-body or Euler equations is discussed. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. We show that using piecewise continuous controls, the transfer of energy among three 1 waves can be controlled. The so called quasi-phase-matching control strategy, which is used in a host of nonlinear optical devices to convert laser light from one frequency to another, is described in this context. Finally, we discuss the connection between piecewise constant controls and billiards.

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Additional Information:© 1999, American Mathematical Society. October 13, 1998. MSA was 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. The research of JEM was partially supported by the National Science Foundation and the California Institute of Technology. JMR was 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.
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NSFDMS 9626672
Basic Research Institute in the Mathematical Sciences (BRIMS)UNSPECIFIED
NATOCRG 950897
Department of Mathematics and the Center for Applied Mathematics, University of Notre DameUNSPECIFIED
Hewlett-Packard LabsUNSPECIFIED
Series Name:AMS and Fields Institute
Record Number:CaltechAUTHORS:20100708-091115993
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:18945
Deposited By: Ruth Sustaita
Deposited On:09 Jul 2010 16:41
Last Modified:03 Oct 2019 01:50

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