Isospectral flows related to Frobenius-Stickelberger-Thiele polynomials

TL;DR

This paper explores isospectral deformations of Frobenius-Stickelberger-Thiele polynomials, revealing integrable lattice structures and connections to peakon equations, thus advancing understanding of their spectral and dynamical properties.

Contribution

It introduces an integrable FST lattice from isospectral flows and links it to the modified Camassa-Holm peakon lattice and interlacing peakon systems.

Findings

Identifies an integrable FST lattice from isospectral deformations.

Maps the mCH peakon lattice to a negative flow of the FST lattice.

Shows degenerate FST lattice cases relate to interlacing peakon ODE systems.

Abstract

The isospectral deformations of the Frobenius-Stickelberger-Thiele (FST) polynomials introduced in [32](Spiridonov et al. Commun. Math. Phys. 272:139--165, 2007 ) are studied. For a specific choice of the deformation of the spectral measure, one is led to an integrable lattice (FST lattice), which is indeed an isospectral flow connected with a generalized eigenvalue problem. In the second part of the paper the spectral problem used previously in the study of the modified Camassa-Holm (mCH) peakon lattice is interpreted in terms of the FST polynomials together with the associated FST polynomials, resulting in a map from the mCH peakon lattice to a negative flow of the finite FST lattice. Furthermore, it is pointed out that the degenerate case of the finite FST lattice unexpectedly maps to the interlacing peakon ODE system associated with the two-component mCH equation studied in [17](Chang et al. Adv. Math. 299:1--35, 2016).