Darboux transformations and recursion operators for differential-difference equations |
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Authors: | F Khanizadeh A V Mikhailov Jing Ping Wang |
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Institution: | 1. School of Mathematics, Statistics, and Actuarial Science, University of Kent, Kent, UK 2. Applied Mathematics Department, University of Leeds, Leeds, UK
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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. |
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