Darboux transformations and Recursion operators for differential--difference equations

TL;DR

This paper reviews Darboux transformations and recursion operators for integrable differential-difference equations, presenting new multi-Hamiltonian structures, discretizations, and inverse recursion operators for various lattice equations.

Contribution

It introduces new multi-Hamiltonian structures, recursion operators, and discretizations for integrable differential-difference equations, expanding the understanding of their algebraic structures.

Findings

New multi-Hamiltonian structures for Volterra type equations

Discretizations of derivative nonlinear Schrödinger equations

Computed weakly nonlocal inverse recursion operators

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 Schr\"odinger 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.