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Theoretical and Mathematical Physics

, Volume 177, Issue 3, pp 1606–1654 | Cite as

Darboux transformations and recursion operators for differential-difference equations

  • F. Khanizadeh
  • A. V. MikhailovEmail author
  • Jing Ping Wang
Article

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 

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Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • F. Khanizadeh
    • 1
  • A. V. Mikhailov
    • 2
    Email author
  • Jing Ping Wang
    • 1
  1. 1.School of Mathematics, Statistics, and Actuarial ScienceUniversity of KentKentUK
  2. 2.Applied Mathematics DepartmentUniversity of LeedsLeedsUK

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