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

Abstract

A chain of one-dimensional Schrödinger 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 $\alpha \ne 0$, the $N$-periodic dressing chain may be thought of as a generalization of the fourth or fifth (depending on the parity of $N$) Painlevé 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.

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

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• Higher order Painlevé equations of type $A_l^{(1)}$
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• Gauge Symmetry Origin of Bäcklund Transformations for Painlevé Equations
• From KP/UC hierarchies to Painleve equations
• Cohomological study on variants of the Mumford system, and integrability of the Noumi-Yamada system
• On new classes of explicit solutions of Dirac, dynamical Dirac and Dirac--Weyl systems with non-vanishing at infinity potentials, their properties and applications
• On Hamiltonian Formalism for Dressing Chain Equations of Even Periodicity
• Lectures on exceptional orthogonal polynomials and rational solutions to Painlevé equations