The Atiyah-Hitchin Bracket for the Cubic Nonlinear Schrodinger Equation. I. General Potentials

Details The Atiyah-Hitchin Bracket for the Cubic Nonlinear Schrodinger Equation. I. General Potentials The Atiyah-Hitchin Bracket for the Cubic Nonlinear Schrodinger Equation. I. General Potentials

2001-11-08

arxiv.org/abs/math-ph/0111017v6

IMRP Int. Math. Res. Pap. 2006, 17683, 1-60

Physics

Mathematical Physics

23 pages, 2 figures

Scientific paper

This is the first in a series of papers on Poisson formalism for the cubic nonlinear Schr\"{o}dinger equation with repulsive nonlinearity and its relation to complex geometry. In this paper we study general continuous potentials. We demonstrate that the Weyl functions of the corresponding auxiliary Dirac spectral problem carry a natural Poisson structure. We call it the Atiyah--Hitchin Poisson bracket. We show that the Poisson bracket on the phase space is the image of the Atiyah--Hitchin bracket on Weyl functions under the inverse spectral transform.

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