Darboux transformations and Recursion operators for differential--difference equations

Abstract

In this paper we review two concepts directly related to the Lax representations: Darboux transformations and Recursion operators for integrable systems. We then present an extensive list of integrable differential-difference equations together with their Hamiltonian structures, recursion operators, nontrivial generalised symmetries and Darboux-Lax representations. The new results include multi-Hamiltonian structures and recursion operators for integrable Volterra type equations, integrable discretization of derivative nonlinear Schrödinger equations such as the Kaup-Newell lattice, the Chen-Lee-Liu lattice and the Ablowitz-Ramani-Segur (Gerdjikov-Ivanov) lattice. We also compute the weakly nonlocal inverse recursion operators.

References in corpus (4)

• Integrability and Symmetries of Difference Equations: the Adler-Bobenko-Suris Case
• Reduction groups and related integrable difference systems of NLS type
• On a discrete analog of the Tzitzeica equation
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Cited by in corpus (21)

• Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere
• Reduction groups and related integrable difference systems of NLS type
• Rational recursion operators for integrable differential-difference equations
• On a method for constructing the Lax pairs for nonlinear integrable equations
• Darboux transformation with Dihedral reduction group
• Recursion and Hamiltonian operators for integrable nonabelian difference equations
• Non-isospectral extension of the Volterra lattice hierarchy, and Hankel determinants
• Perturbative Symmetry Approach for Differential-Difference Equations
• Symmetries, conservation laws and Noether's theorem for differential-difference equations
• On the recursion operators for integrable equations
• Miura-type transformations for lattice equations and Lie group actions associated with Darboux-Lax representations
• Local and non-local multiplicative Poisson vertex algebras and differential-difference equations
• Algebro-geometric solutions to the lattice potential modified Kadomtsev--Petviashvili equation
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Hamiltonian structures for integrable nonabelian difference equations
• Negative flows and non-autonomous reductions of the Volterra lattice
• Bogoyavlensky lattices and generalized Catalan numbers
• Simplifications of Lax pairs for differential-difference equations by gauge transformations and (doubly) modified integrable equations
• On matrix Lax representations and constructions of Miura-type transformations for differential-difference equations
• Matrix Lax pairs under the gauge equivalence relation induced by the gauge group action and Miura-type transformations for lattice equations
• Hamiltonian and recursion operators for a discrete analogue of the Kaup-Kupershmidt equation