Formal diagonalisation of Lax-Darboux schemes

TL;DR

This paper introduces a formal diagonalisation method for Lax-Darboux schemes, linking various integrable systems and generating their conservation laws through a unified transformation.

Contribution

It presents a novel transformation that diagonalises all elements of the Lax-Darboux scheme simultaneously, enabling the derivation of conservation laws for multiple integrable systems.

Findings

Existence of a transformation that diagonalises the entire Lax-Darboux scheme

Generation of conservation laws for associated integrable systems

Connections established between conservation laws of different systems

Abstract

We discuss the concept of Lax-Darboux scheme and illustrate it on well known examples associated with the Nonlinear Schrodinger (NLS) equation. We explore the Darboux links of the NLS hierarchy with the hierarchy of Heisenberg model, principal chiral field model as well as with differential-difference integrable systems (including the Toda lattice and differential-difference Heisenberg chain) and integrable partial difference systems. We show that there exists a transformation which formally diagonalises all elements of the Lax-Darboux scheme simultaneously. It provides us with generating functions of local conservation laws for all integrable systems obtained. We discuss the relations between conservation laws for systems belonging to the Lax-Darboux scheme.