Geodesic
Darboux transformations and recursion operators for differential--difference equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 177 (2013) no. 3, pp. 387-440

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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.
Keywords: symmetry, recursion operator, bi-Hamiltonian structure, Lax representation, integrable equation.
Mots-clés : Darboux transformation 
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     author = {F. Khanizadeh and A. V. Mikhailov and Jing Ping Wang},
     title = {Darboux transformations and recursion operators for differential--difference equations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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F. Khanizadeh; A. V. Mikhailov; Jing Ping Wang. Darboux transformations and recursion operators for differential--difference equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 177 (2013) no. 3, pp. 387-440. http://geodesic.mathdoc.fr/item/TMF_2013_177_3_a1/