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.
Highlights
In this paper we discuss the ways in which partial differential equation (PDE), difference equations (D∆Es) and partial difference equations (P∆Es) belonging to the same LaxDarboux scheme share the same hierarchy of local conservation laws
Ajacent Lax structures associated with a Darboux transformation which lead to adjacent symmetries of these differential-difference and partial difference equations and are integrable differential-difference equations in their own right
[Sα, Sβ] = 0 ⇒ Sα(Mβ)Mα − Sβ(Mα)Mβ = 0. It leads to the Bianchi lattice which can be regarded as a system of partial difference equations (P∆E) on Z2
Summary
We show that there exists a formal (i.e. in the form of a formal series) gauge transformation which simultaneously diagonalises (or brings to a block-diagonal form) the Lax operators of the Lax structure and Darboux matrices associated with Darboux transformations It provides us with a regular method for recursive derivation of a hierarchy of local conservation laws for the nonlinear differential and difference systems associated with the Lax-Darboux scheme. This paper is based on a lecture courses given by the author in the Bashkir State University (Ufa, 2012) and as a part of MAGIC course on Integrable systems (UK, 2014), a number of conference talks (Ufa, October 2012; Moscow, November 2012 [9]; Cambridge, July 2013 [10]) where the concept of Lax-Darbiux scheme and formal diagonalisation approach were originally presented This method has proven to be useful in a many applications(see for example [5, 11, 12])
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