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On geometric phases for soliton equations

Alber, M. S. and Marsden, J. E. (1992) On geometric phases for soliton equations. Communications in Mathematical Physics, 149 (2). pp. 217-240. ISSN 0010-3616. doi:10.1007/BF02097623.

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This paper develops a new complex Hamiltonian structure forn-soliton solutions for a class of integrable equations such as the nonlinear Schrödinger, sine-Gordon and Korteweg-de Vries hierarchies of equations that yields, amongst other things, geometric phases in the sense of Hannay and Berry. For example, one of the possible soliton geometric phases is manifested by the well known phase shift that occurs for interacting solitons. The main new tools are complex angle representations that linearize the corresponding Hamiltonian flows on associated noncompact Jacobi varieties. This new structure is obtained by taking appropriate limits of the differential equations describing the class of quasi-periodic solutions. A method of asymptotic reduction of the angle representations is introduced for investigating soliton geometric phases that are related to the presence of monodromy at singularities in the space of parameters. In particular, the phase shift of interacting solitons can be expressed as an integral over a cycle on an associated Riemann surface. In this setting, soliton geometric asymptotics are constructed for studying geometric phases in the quantum case. The general approach is worked out in detail for the three specific hierarchies of equations mentioned. Some links with tau-functions, the braid group and geometric quantization are pointed out as well.

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Additional Information:© Springer-Verlag 1992. Received: 1 April 1991. Revised: 27 March 1992. We would like to thank the referee for excellent suggestions. We want to express our deep gratitude to D. McLaughlin for very helpful discussions. Communicated by A. Jaffe
Issue or Number:2
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:18953
Deposited By: Ruth Sustaita
Deposited On:09 Jul 2010 15:39
Last Modified:08 Nov 2021 23:48

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