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International Mathematics Research Papers (2006) Vol. 2006 : article ID 17683, 60 pages, doi:10.1155/IMRP/2006/17683
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Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

The Atiyah-Hitchin bracket and the cubic nonlinear Schrödinger equation

K. L. Vaninsky

Department of Mathematics, College of Natural Science, Michigan State University East Lansing, MI 48824, USA E-mail address: vaninsky{at}math.msu.edu

We study Poisson formalism for the cubic nonlinear Schrödinger equation and its relation to complex geometry. In the first part, for general continuous potentials, we demonstrate that the Weyl function of the auxiliary Dirac spectral problem carries natural Poisson structure. This is the Atiyah-Hitchin Poisson bracket. We show that the Poisson bracket on the phase space is an image of the Atiyah-Hitchin bracket on Weyl function under the inverse spectral transform. In the second part, we consider periodic potentials. The spectral problem for the periodic auxiliary Dirac operator leads to a hyperelliptic Riemann surface. We introduce on this Riemann surface a meromorphic function. We call it the Weyl function, since it is closely related to the classical Weyl function. We show that the pair "Riemann surface + Weyl function" carries a natural Poisson structure. We call it the deformed Atiyah-Hitchin bracket. This deformed bracket can be taken as a starting point for construction of the Poisson formalism.



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