Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains

Details Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains

2002-06-28

arxiv.org/abs/nlin/0206049v3

Commun.Math.Phys. 241 (2003) 111-142

Nonlinear Sciences

Exactly Solvable and Integrable Systems

latex2e, 41 pages, no figure; (v2) some minor errors are corrected; (v3) fully revised and shortend, and some results are impr

Scientific paper

10.1007/s00220-003-0929-y

A chain of one-dimensional Schr\"odinger operators connected by successive Darboux transformations is called the ``Darboux chain'' or ``dressing chain''. The periodic dressing chain with period $N$ has a control parameter $\alpha$. If $\alpha \not= 0$, the $N$-periodic dressing chain may be thought of as a generalization of the fourth or fifth (depending on the parity of $N$) Painlev\'e equations . The $N$-periodic dressing chain has two different Lax representations due to Adler and to Noumi and Yamada. Adler's $2 \times 2$ Lax pair can be used to construct a transition matrix around the periodic lattice. One can thereby define an associated ``spectral curve'' and a set of Darboux coordinates called ``spectral Darboux coordinates''. The equations of motion of the dressing chain can be converted to a Hamiltonian system in these Darboux coordinates. The symplectic structure of this Hamiltonian formalism turns out to be consistent with a Poisson structure previously studied by Veselov, Shabat, Noumi and Yamada.

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