Microscopic conservation laws for integrable lattice models

TL;DR

This paper develops microscopic conservation laws for integrable lattice models like Toda and Ablowitz-Ladik, providing discrete analogues of conservation objects crucial in continuous integrable systems.

Contribution

It introduces microscopic conservation laws for discrete integrable systems, extending concepts previously used in continuous models to lattice models.

Findings

Conservation of the perturbation determinant under lattice dynamics

Discrete analogues of Green's function trace conservation laws

Revisiting classical microscopic conservation laws in a discrete setting

Abstract

We consider two discrete completely integrable evolutions: the Toda Lattice and the Ablowitz-Ladik system. The principal thrust of the paper is the development of microscopic conservation laws that witness the conservation of the perturbation determinant under these dynamics. In this way, we obtain discrete analogues of objects that we found essential in our recent analyses of KdV, NLS, and mKdV. In concert with this, we revisit the classical topic of microscopic conservation laws attendant to the (renormalized) trace of the Green's function.