Integrable boundary conditions and modified Lax equations

TL;DR

This paper systematically studies integrable boundary conditions in classical models, deriving modified Lax pairs that depend on the classical r-matrix and boundary matrix, for both discrete and continuum cases.

Contribution

It provides a systematic method to identify modified Lax pairs for boundary integrable models based on the classical r-matrix and boundary matrix.

Findings

Derived modified Lax pairs for discrete models

Derived modified Lax pairs for continuum models

Connected boundary conditions with classical r-matrix structure

Abstract

We consider integrable boundary conditions for both discrete and continuum classical integrable models. Local integrals of motion generated by the corresponding transfer matrices give rise to time evolution equations for the initial Lax operator. We systematically identify the modified Lax pairs for both discrete and continuum boundary integrable models, depending on the classical r-matrix and the boundary matrix.