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Darboux transformations and recursion operators for differential–difference equations

F. Khanizadeh^a, A. V. Mikhailov^b, Jing Ping Wang<sup>a

a</sup> School of Mathematics, Statistics and Actuarial Science, University of Kent, UK
^b Applied Mathematics Department, University of Leeds, UK

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

Keywords: symmetry, recursion operator, bi-Hamiltonian structure, Darboux transformation, Lax representation, integrable equation.

Received: 15.05.2013

DOI: 10.4213/tmf8550

 English version: Theoretical and Mathematical Physics, 2013, 177:3, 1606–1654

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