Formal Diagonalisation of Lax-Darboux Schemes

Abstract

We discuss the concept of Lax-Darboux scheme and illustrate it on well known examples associated with the Nonlinear Schro¨dinger (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.

Keywords

• formal diagonalisation
• lax-darboux schemes
• nonlinear schr¨odinger equation
• nls