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Energy dependent Schrödinger operators and complex Hamiltonian systems on Riemann surfaces

Alber, Mark S. and Luther, Gregory G. and Marsden, Jerrold E. (1997) Energy dependent Schrödinger operators and complex Hamiltonian systems on Riemann surfaces. Nonlinearity, 10 (1). pp. 223-241. ISSN 0951-7715. doi:10.1088/0951-7715/10/1/015.

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We use so-called energy-dependent Schrödinger operators to establish a link between special classes of solutions on N-component systems of evolution equations and finite dimensional Hamiltonian systems on the moduli spaces of Riemann surfaces. We also investigate the phase-space geometry of these Hamiltonian systems and introduce deformations of the level sets associated to conserved quantities, which results in a new class of solutions with monodromy for N-component systems of PDEs. After constructing a variety of mechanical systems related to the spatial flows of nonlinear evolution equations, we investigate their semiclassical limits. In particular, we obtain semicalssical asymptotics for the Bloch eigenfunctions of the energy dependent Schrödinger operators, which is of importance in investigating zero-dispersion limits of N-component systems of PDEs.

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Additional Information:© 1997 IOP Publishing Ltd. Received 4 April 1995, in final form 16 August 1996. Print publication: Issue 1 (January 1997). Recommended by M. Jimbo. The research carried out by MSA was partially supported by NSF grants DMS 9403861 and 9508711. GGL gratefully acknowledges support from BRIMS, Hewlett-Packard Labs and from NSF DMS under grant 9508711. The research carried out by JEM was partially supported by NSF grant DMS 9633161.
Subject Keywords:Korteweg-Devries equation, small dispersion limit, integrable systems, geometric phases, formalism, monodromy, PDES
Issue or Number:1
Record Number:CaltechAUTHORS:ALBnonlin97
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:551
Deposited By: Tony Diaz
Deposited On:19 Aug 2005
Last Modified:08 Nov 2021 19:03

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