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Teoreticheskaya i Matematicheskaya Fizika, 2013, Volume 177, Number 3, Pages 387–440
DOI: https://doi.org/10.4213/tmf8550 (Mi tmf8550) 		 
This article is cited in 57 scientific papers (total in 57 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
Full-text PDF (746 kB) Citations (57)
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DOI: https://doi.org/10.4213/tmf8550
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.
Received: 15.05.2013
English version:
Theoretical and Mathematical Physics, 2013, Volume 177, Issue 3, Pages 1606–1654
DOI: https://doi.org/10.1007/s11232-013-0124-z
Bibliographic databases:
Document Type: Article
Language: Russian
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
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    • This publication is cited in the following 57 articles:
    1. Alexander V. Mikhailov, Jing P. Wang, Encyclopedia of Mathematical Physics, 2025, 162  crossref
    2. Wenjia Wu, Bao Wang, “Recursion operator of integrable lattices associated with Cauchy bi-orthogonal polynomials”, Phys. Scr., 100:4 (2025), 045217  crossref
    3. V.E. Adler, “Bogoyavlensky Lattices and Generalized Catalan Numbers”, Russ. J. Math. Phys., 31:1 (2024), 1  crossref
    4. Jin Liu, Da‐jun Zhang, Xuehui Zhao, “Symmetries of the DΔmKP hierarchy and their continuum limits”, Stud Appl Math, 152:1 (2024), 404  crossref
    5. V. E. Adler, “Negative flows and non-autonomous reductions of the Volterra lattice”, Open Communications in Nonlinear Mathematical Physics, Special Issue in Memory of... (2024)  crossref
    6. Sergei Igonin, “Simplifications of Lax pairs for differential–difference equations by gauge transformations and (doubly) modified integrable equations”, Partial Differential Equations in Applied Mathematics, 11 (2024), 100821  crossref
    7. Edoardo Peroni, Jing Ping Wang, “Hamiltonian and recursion operators for a discrete analogue of the Kaup-Kupershmidt equation”, Open Communications in Nonlinear Mathematical Physics, Special Issue in Memory of... (2024)  crossref
    8. Evgeny Chistov, Sergei Igonin, “On matrix Lax representations and constructions of Miura-type transformations for differential-difference equations”, Partial Differential Equations in Applied Mathematics, 2024, 101014  crossref
    9. Fangcheng Fan, Weikang Xie, “A generalized integrable lattice hierarchy related to the Ablowitz–Ladik lattice: Conservation law, Darboux transformation and exact solution”, Rev. Math. Phys., 35:10 (2023)  crossref
    10. Wei-Kang Xie, Fang-Cheng Fan, “Soliton, breather, rogue wave and continuum limit in the discrete complex modified Korteweg-de Vries equation by Darboux-Bäcklund transformation”, Journal of Mathematical Analysis and Applications, 525:2 (2023), 127251  crossref
    11. Fang-Cheng Fan, Zhi-Guo Xu, “Breather and rogue wave solutions for the generalized discrete Hirota equation via Darboux–Bäcklund transformation”, Wave Motion, 119 (2023), 103139  crossref
    12. Fan F., “Discrete N-Fold Darboux Transformation and Infinite Number of Conservation Laws of a Four-Component Toda Lattice”, Int. J. Geom. Methods Mod. Phys., 19:03 (2022), 2250040  crossref  mathscinet  isi
    13. Alexander V. Mikhailov, Vladimir S. Novikov, Jing Ping Wang, “Perturbative Symmetry Approach for Differential–Difference Equations”, Commun. Math. Phys., 393:2 (2022), 1063  crossref
    14. Xiaoxue Xu, Cewen Cao, Da-jun Zhang, “Algebro-geometric solutions to the lattice potential modified Kadomtsev–Petviashvili equation”, J. Phys. A: Math. Theor., 55:37 (2022), 375201  crossref
    15. Matteo Casati, Jing Ping Wang, “Hamiltonian Structures for Integrable Nonabelian Difference Equations”, Commun. Math. Phys., 392:1 (2022), 219  crossref
    16. Qiulan Zhao, Muhammad Arham Amin, “Explicit solutions of rational integrable differential-difference equations”, Partial Differential Equations in Applied Mathematics, 5 (2022), 100338  crossref
    17. O. Dafounansou, D.C. Mbah, F.L. Taussé Kamdoum, M.G. Kwato Njock, “Darboux transformations for the multicomponent vector solitons and rogue waves of the multiple coupled Kundu–Eckhaus equations”, Wave Motion, 114 (2022), 103041  crossref
    18. Casati M., Wang J.P., “Recursion and Hamiltonian Operators For Integrable Nonabelian Difference Equations”, Nonlinearity, 34:1 (2021), 205–236  crossref  mathscinet  isi
    19. Igonin S. Kolesov V. Konstantinou-Rizos S. Preobrazhenskaia M.M., “Tetrahedron Maps, Yang-Baxter Maps, and Partial Linearisations”, J. Phys. A-Math. Theor., 54:50 (2021), 505203  crossref  mathscinet  isi
    20. Fan F.-Ch., Wen X.-Y., “A Generalized Integrable Lattice Hierarchy Associated With the Toda and Modified Toda Lattice Equations: Hamiltonian Representation, Soliton Solutions”, Wave Motion, 103 (2021), 102727  crossref  mathscinet  isi
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