Alber, Mark S. and Luther, Gregory G. and Marsden, Jerrold E. and Robbins, Jonathan M. (1999) Geometry and Control of ThreeWave 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. 5580. ISBN 0821809458 http://resolver.caltech.edu/CaltechAUTHORS:20100708091115993

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Abstract
The integrable structure of the threewave equations is discussed in the setting of geometric mechanics. LiePoisson structures with quadratic Hamiltonian are associated with the threewave 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 rigidbody 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 quasiphasematching 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, HewlettPackard 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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Record Number:  CaltechAUTHORS:20100708091115993  
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:20100708091115993  
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ID Code:  18945  
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Deposited By:  Ruth Sustaita  
Deposited On:  09 Jul 2010 16:41  
Last Modified:  26 Dec 2012 12:12 
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