Integrable chains on algebraic curves

TL;DR

This paper develops a theory of integrable discrete Lax operators on algebraic curves, constructs commuting flows, and explicitly solves them using spectral transforms and theta-functions, revealing new integrable Hamiltonian systems.

Contribution

It introduces a novel class of integrable Hamiltonian systems based on rank 2 Lax operators on variable algebraic curves, expanding the scope of integrable systems theory.

Findings

Flows are linearized by spectral transform

Explicit solutions via theta-functions are obtained

New integrable Hamiltonian systems are identified

Abstract

The discrete Lax operators with the spectral parameter on an algebraic curve are defined. A hierarchy of commuting flows on the space of such operators is constructed. It is shown that these flows are linearized by the spectral transform and can be explicitly solved in terms of the theta-functions of the spectral curves. The Hamiltonian theory of the corresponding systems is analyzed. The new type of completely integrable Hamiltonian systems associated with the space of rank $r=2$ discrete Lax operators on a {\it variable} base curve is found.