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 α. If α = 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 × 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.

Painlevé equations from Darboux chains: I.PIII–PV

“…This Lax pair already appears in [16], in connection with reductions of the Drinfeld-Sokolov hierarchy. Also, the transformations (61) and (63) combined, amount to the Mellin transformation that is used (to the same effect) in [21] in order to connect Lax pairs arising in the context of dressing chains to those that appear in the work by Noumi et al…”

Universal characters, integrable chains and the Painlevé equations

“…In general, for an integer N 3, the N-periodic closing of the chain coincides with the (higher order) Painlevé equation of type A (1) N−1 , proposed by Noumi-Yamada; see [1,15,16,19,26,27]. The aim of the present article is to show certain relationships among the universal characters, the Darboux chains, and the Painlevé equations.…”

“…without loss of generality. As shown in [19] (and see also [1,26]), through the change of variables:…”

“…This is closely interconnected with the spectral theory of Schrödinger operators; in fact, governs a sequence of Schrödinger operators connected with the neighbours by the Darboux transformations; see [17][18][19]26,27].…”

Hamiltonian Structure of PI Hierarchy

“…Their idea was generalized by Sklyanin [27] to a wide range of integrable systems including quantum integrable systems. On the other hand, spectral Darboux coordinates were also applied to isomonodromic systems [28,29,30,31].…”