Corpus ID: 237593037

A discrete Darboux-Lax scheme for integrable difference equations

  title={A discrete Darboux-Lax scheme for integrable difference equations},
  author={Xenia Fisenko and S. Konstantinou-Rizos and Pavlos Xenitidis},
We propose a discrete Darboux–Lax scheme for deriving auto-Bäcklund transformations and constructing solutions to quad-graph equations that do not necessarily possess the 3D consistency property. As an illustrative example we use the Adler–Yamilov type system which is related to the nonlinear Schrödinger (NLS) equation [19]. In particular, we construct an auto-Bäcklund transformation for this discrete system, its superposition principle, and we employ them in the construction of the oneand two… Expand

Figures from this paper


Integrable discretisations for a class of nonlinear Schrodinger equations on Grassmann algebras
Integrable discretisations for a class of coupled (super) nonlinear Schrodinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra.Expand
Direct linearization approach to discrete integrable systems associated with ℤ𝒩 graded Lax pairs
  • Wei-jie Fu
  • Medicine, Mathematics
  • Proceedings of the Royal Society A
  • 2020
The link between the Fordy–Xenitidis (FX) discrete systems in coprime case and linear integral equations in certain form is established, which reveals solution structure of these equations. Expand
A Constructive Approach to the Soliton Solutions of Integrable Quadrilateral Lattice Equations
Scalar multidimensionally consistent quadrilateral lattice equations are studied. We explore a confluence between the superposition principle for solutions related by the Bäcklund transformation, andExpand
On the Lagrangian formulation of multidimensionally consistent systems
Multidimensional consistency has emerged as a key integrability property for partial difference equations (PΔEs) defined on the ‘space–time’ lattice. It has led, among other major insights, to aExpand
Darboux transformations, finite reduction groups and related Yang–Baxter maps
In this paper, we construct Yang–Baxter (YB) maps using Darboux matrices which are invariant under the action of finite reduction groups. We present six-dimensional YB maps corresponding to DarbouxExpand
Supersymmetric KdV equation: Darboux transformation and discrete systems
For the supersymmetric KdV equation, a proper Darboux transformation is presented. This Darboux transformation leads to the B\"{a}cklund transformation found early by Liu and Xie \cite{liu2}. TheExpand
Entwining Yang–Baxter maps related to NLS type equations
We construct birational maps that satisfy the parametric set-theoretical Yang–Baxter equation and its entwining generalisation. For this purpose, we employ Darboux transformations related toExpand
Second Order Integrability Conditions for Difference Equations: An Integrable Equation
Integrability conditions for difference equations admitting a second order formal recursion operator are presented and the derivation of symmetries and canonical conservation laws are discussed. In aExpand
Symbolic Computation of Lax Pairs of Partial Difference Equations using Consistency Around the Cube
A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (PΔEs) is reviewed and previously unknown Lax pairs are presented for P� ΔEs recently derived by Hietarinta. Expand
Reduction groups and related integrable difference systems of nonlinear Schrödinger type
We extend the reduction group method to the Lax-Darboux schemes associated with nonlinear Schrodinger type equations. We consider all possible finite reduction groups and construct corresponding LaxExpand