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Spectral Curve, Darboux Coordinates and Hamiltonian Structure of Periodic Dressing Chains

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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 $\alpha \not= 0$, the $N$-periodic dressing chain may be thought of as a generalization of the fourth or fifth (depending on the parity of $N$) 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.

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  1. Department of Fundamental Sciences, Faculty of Integrated Human Studies, Kyoto University, Yoshida, Sakyo-ku, Kyoto, 606-8501, Japan

    Kanehisa Takasaki

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  1. Kanehisa Takasaki
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Communicated by L. Takhtajan

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Takasaki, K. Spectral Curve, Darboux Coordinates and Hamiltonian Structure of Periodic Dressing Chains. Commun. Math. Phys. 241, 111–142 (2003). https://doi.org/10.1007/s00220-003-0929-y

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