Geometric phases, reduction and Lie-Poisson structure for the resonant three-wave interaction
Identifieur interne : 000836 ( Main/Curation ); précédent : 000835; suivant : 000837Geometric phases, reduction and Lie-Poisson structure for the resonant three-wave interaction
Auteurs : Mark S. Alber [États-Unis] ; Gregory G. Luther [États-Unis, Royaume-Uni] ; Jerrold E. Marsden [États-Unis] ; Jonathan M. Robbins [Royaume-Uni]Source :
- Physica D: Nonlinear Phenomena [ 0167-2789 ] ; 1998.
Abstract
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.
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DOI: 10.1016/S0167-2789(98)00127-4
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Alber</name><affiliation wicri:level="1"><mods:affiliation>Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA</mods:affiliation><country xml:lang="fr">États-Unis</country><wicri:regionArea>Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556</wicri:regionArea></affiliation><affiliation wicri:level="1"><mods:affiliation>1 Research partially supported by NSF grants DMS 9626672 and 9508711.</mods:affiliation><country xml:lang="fr">États-Unis</country><placeName><region type="state">État du Mississippi</region></placeName></affiliation></author><author><name sortKey="Luther, Gregory G" sort="Luther, Gregory G" uniqKey="Luther G" first="Gregory G." last="Luther">Gregory G. Luther</name><affiliation><mods:affiliation>Corresponding author.</mods:affiliation></affiliation><affiliation wicri:level="1"><mods:affiliation>Control and Dynamical Systems, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA</mods:affiliation><country xml:lang="fr">États-Unis</country><wicri:regionArea>Control and Dynamical Systems, Caltech, 1200 E. California Blvd., Pasadena, CA 91125</wicri:regionArea></affiliation><affiliation wicri:level="1"><mods:affiliation>The Basic Research Institute in the Mathematical Sciences (BRIMS), Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS12 6QZ, UK</mods:affiliation><country xml:lang="fr">Royaume-Uni</country><wicri:regionArea>The Basic Research Institute in the Mathematical Sciences (BRIMS), Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS12 6QZ</wicri:regionArea></affiliation><affiliation><mods:affiliation>∗ Present Address: Engineering Sciences and Applied Mathematics Department, McCormick School of Engineering and Applied Science, Northwestern University, 2145 Sheridan Road, Evanston, II 60208-3125, USA. 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Robbins</name><affiliation wicri:level="1"><mods:affiliation>The Basic Research Institute in the Mathematical Sciences (BRIMS), Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS12 6QZ, UK</mods:affiliation><country xml:lang="fr">Royaume-Uni</country><wicri:regionArea>The Basic Research Institute in the Mathematical Sciences (BRIMS), Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS12 6QZ</wicri:regionArea></affiliation><affiliation wicri:level="1"><mods:affiliation>School of Mathematics, University Walk, Bristol BS8 1TW, UK</mods:affiliation><country xml:lang="fr">Royaume-Uni</country><wicri:regionArea>School of Mathematics, University Walk, Bristol BS8 1TW</wicri:regionArea></affiliation><affiliation><mods:affiliation>3 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.</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Physica D: Nonlinear Phenomena</title><title level="j" type="abbrev">PHYSD</title><idno type="ISSN">0167-2789</idno><imprint><publisher>ELSEVIER</publisher><date type="published" when="1998">1998</date><biblScope unit="volume">123</biblScope><biblScope unit="issue">1–4</biblScope><biblScope unit="page" from="271">271</biblScope><biblScope unit="page" to="290">290</biblScope></imprint><idno type="ISSN">0167-2789</idno></series><idno type="istex">630CDBC6B61D070528D5826101DA645152746C7F</idno><idno type="DOI">10.1016/S0167-2789(98)00127-4</idno><idno type="PII">S0167-2789(98)00127-4</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0167-2789</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">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.</div></front></TEI></record>Pour manipuler ce document sous Unix (Dilib)
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