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Darboux transformations and recursion operators for differential-difference equations

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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.

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Authors and Affiliations

  1. School of Mathematics, Statistics, and Actuarial Science, University of Kent, Kent, UK

    F. Khanizadeh & Jing Ping Wang

  2. Applied Mathematics Department, University of Leeds, Leeds, UK

    A. V. Mikhailov

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  1. F. Khanizadeh
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  2. A. V. Mikhailov
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  3. Jing Ping Wang
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Correspondence to A. V. Mikhailov.

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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 177, No. 3, pp. 387–440, December, 2013.

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Khanizadeh, F., Mikhailov, A.V. & Wang, J.P. Darboux transformations and recursion operators for differential-difference equations. Theor Math Phys 177, 1606–1654 (2013). https://doi.org/10.1007/s11232-013-0124-z

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