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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

DOI: https://doi.org/10.4213/tmf8550

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

Bibliographic databases:

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

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    This publication is cited in the following articles:
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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
    3. H.-Q. Zhao, “Darboux transformations and new explicit solutions for the relativistic Volterra lattice”, Appl. Math. Lett., 38 (2014), 79–83  crossref  mathscinet  zmath  isi  scopus
    4. A. V. Mikhailov, G. Papamikos, J. P. Wang, “Darboux transformation with dihedral reduction group”, J. Math. Phys., 55:11 (2014), 113507  crossref  mathscinet  zmath  adsnasa  isi  scopus
    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
    6. R.-G. Zhou, J. Chen, “Two hierarchies of new differential-difference equations related to the Darboux transformations of the Kaup–Newell hierarchy”, Commun. Theor. Phys., 63:1 (2015), 1–6  crossref  mathscinet  zmath  adsnasa  isi  scopus
    7. A. V. Mikhailov, “Formal diagonalisation of Lax–Darboux schemes”, Model. i analiz inform. sistem, 22:6 (2015), 795–817  mathnet  crossref  mathscinet  elib
    8. S. Konstantinou-Rizos, A. V. Mikhailov, P. Xenitidis, “Reduction groups and related integrable difference systems of nonlinear Schrodinger type”, J. Math. Phys., 56:8 (2015), 082701  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. I. T. Habibullin, A. R. Khakimova, M. N. Poptsova, “On a method for constructing the Lax pairs for nonlinear integrable equations”, J. Phys. A-Math. Theor., 49:3 (2016), 035202  crossref  mathscinet  zmath  adsnasa  isi  scopus
    10. G. Berkeley, S. Igonin, “_orig miura-type transformations for lattice equations and lie group actions associated with darboux-lax representations”, J. Phys. A-Math. Theor., 49:27 (2016), 275201  crossref  mathscinet  zmath  isi  elib  scopus
    11. A. V. Mikhailov, G. Papamikos, J. P. Wang, “Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere”, Lett. Math. Phys., 106:7 (2016), 973–996  crossref  mathscinet  zmath  isi  elib  scopus
    12. I. T. Habibullin, A. R. Khakimova, “Invariant manifolds and Lax pairs for integrable nonlinear chains”, Theoret. and Math. Phys., 191:3 (2017), 793–810  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    13. L. Peng, “Symmetries, conservation laws, and Noether's theorem for differential-difference equations”, Stud. Appl. Math., 139:3 (2017), 457–502  crossref  mathscinet  zmath  isi  scopus
    14. N. Liu, X.-Y. Wen, “Dynamics and elastic interactions of the discrete multi-dark soliton solutions for the Kaup-Newell lattice equation”, Mod. Phys. Lett. B, 32:7 (2018), 1850085  crossref  mathscinet  isi  scopus
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    16. X.-X. Xu, M. Xu, “A family of integrable different-difference equations, its Hamiltonian structure, and Darboux–Bäcklund transformation”, Discrete Dyn. Nat. Soc., 2018, 4152917  crossref  mathscinet  isi  scopus
    17. X.-M. Chen, X.-B. Hu, F. Mueller-Hoissen, “Non-isospectral extension of the Volterra lattice hierarchy, and Hankel determinants”, Nonlinearity, 31:9 (2018), 4393–4422  crossref  mathscinet  zmath  isi  scopus
    18. N. Liu, X.-Y. Wen, Ya. Liu, “Fission and fusion interaction phenomena of the discrete kink multi-soliton solutions for the Chen-Lee-Liu lattice equation”, Mod. Phys. Lett. B, 32:19 (2018), 1850211  crossref  mathscinet  isi  scopus
    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
    21. H.-T. Wang, X.-Y. Wen, “Dynamics of multi-soliton and breather solutions for a new semi-discrete coupled system related to coupled NLS and coupled complex mKdV equations”, Mod. Phys. Lett. B, 32:28 (2018), 1850340  crossref  mathscinet  isi  scopus
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  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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