The AtiyahHitchin bracket for the cubic nonlinear Schrodinger equation. ii. Periodic potentials
Abstract
This is the second in a series of papers on Poisson formalism for the cubic nonlinear Schrödinger equation with repulsive nonlinearity. In this paper we consider periodic potentials. The inverse spectral problem for the periodic auxiliary Dirac operator leads to a hyperelliptic Riemann surface $\G$. Using the spectral problem 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 discussed in the first paper. We show that the pair $(\G,¶)$ carries a natural Poisson structure. We call it the deformed AtiyahHitchin bracket. The Poisson bracket on the phase space is the image of the deformed AtiyahHitchin bracket under the inverse spectral transform.
 Publication:

arXiv eprints
 Pub Date:
 March 2004
 arXiv:
 arXiv:mathph/0403031
 Bibcode:
 2004math.ph...3031V
 Keywords:

 Mathematical Physics;
 Mathematics  Symplectic Geometry;
 35Q53
 EPrint:
 38 pages, 6 figures