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TMF, 2013, Volume 177, Number 3, Pages 387–440 (Mi tmf8550)  

This article is cited in 25 scientific papers (total in 25 papers)

Darboux transformations and recursion operators for differential–difference equations

F. Khanizadeha, A. V. Mikhailovb, Jing Ping Wanga

a 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


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

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Received: 15.05.2013

Citation: F. Khanizadeh, A. V. Mikhailov, Jing Ping Wang, “Darboux transformations and recursion operators for differential–difference equations”, TMF, 177:3 (2013), 387–440; Theoret. and Math. Phys., 177:3 (2013), 1606–1654

Citation in format AMSBIB
\by F.~Khanizadeh, A.~V.~Mikhailov, Jing~Ping~Wang
\paper Darboux transformations and recursion operators for differential--difference equations
\jour TMF
\yr 2013
\vol 177
\issue 3
\pages 387--440
\jour Theoret. and Math. Phys.
\yr 2013
\vol 177
\issue 3
\pages 1606--1654

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    2. G. M. Beffa, “Hamiltonian evolutions of twisted polygons in parabolic manifolds: the Lagrangian Grassmannian”, Pac. J. Math., 270:2 (2014), 287–317  crossref  mathscinet  zmath  isi  scopus
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    5. W. Fu, D.-J. Zhang, R.-G. Zhou, “A class of two-component Adler–Bobenko–Suris lattice equations”, Chin. Phys. Lett., 31:9 (2014), 090202  crossref  isi  elib  scopus
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    19. I. T. Habibullin, A. R. Khakimova, “A direct algorithm for constructing recursion operators and Lax pairs for integrable models”, Theoret. and Math. Phys., 196:2 (2018), 1200–1216  mathnet  crossref  crossref  adsnasa  isi  elib
    20. I. T. Habibullin, A. R. Khakimova, “On the recursion operators for integrable equations”, J. Phys. A-Math. Theor., 51:42 (2018), 425202  crossref  isi  scopus
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  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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