Spectral curves and discrete Painlev\'e equations

TL;DR

This paper explores how spectral curves derived from Lax matrices characterize discrete Painlevé equations through their isomonodromic deformations, providing a Hamiltonian framework for understanding these nonlinear difference equations.

Contribution

It demonstrates that discrete Painlevé equations can be succinctly described via the characteristic equations of their Lax matrices, linking spectral curves to their integrable structure.

Findings

Spectral curves are biquadratic in Painlevé variables.

Discrete isomonodromic deformations are characterized by Lax matrix spectral equations.

Hamiltonian structure underpins the spectral description of discrete Painlevé equations.

Abstract

It is well known that isomonodromic deformations admit a Hamiltonian description. These Hamiltonians appear as coefficients of the characteristic equations of their Lax matrices, which define spectral curves for linear systems of differential and difference systems. The characteristic equations in the case of the associated linear problems for various discrete Painlev\'e equations is biquadratic in the Painlev\'e variables. We show that the discrete isomonodromic deformations that define the discrete Painlev\'e equations may be succinctly described in terms of the characteristic equation of their Lax matrices.