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TMF, 2001, Volume 129, Number 2, Pages 163–183 (Mi tmf528)  

This article is cited in 20 scientific papers (total in 20 papers)

Lagrangian Chains and Canonical Bäcklund Transformations

V. E. Adlera, V. G. Marikhinb, A. B. Shabatb

a Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences
b L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: We consider Darboux transformations for operators of arbitrary order and construct the general theory of Bäcklund transformations based on the Lagrangian formalism. The dressing chain for the Boussinesq equation and its reduction are demonstrative examples for the suggested general theory. We also discuss the well-known Bäcklund transformations for classical soliton equations.

DOI: https://doi.org/10.4213/tmf528

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English version:
Theoretical and Mathematical Physics, 2001, 129:2, 1448–1465

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Received: 06.06.2001

Citation: V. E. Adler, V. G. Marikhin, A. B. Shabat, “Lagrangian Chains and Canonical Bäcklund Transformations”, TMF, 129:2 (2001), 163–183; Theoret. and Math. Phys., 129:2 (2001), 1448–1465

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Shabat, AB, “Discretization of the Schrodinger spectral problem”, Inverse Problems, 18:4 (2002), 1003  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    2. Alonso, LM, “Towards a theory of differential constraints of a hydrodynamic hierarchy”, Journal of Nonlinear Mathematical Physics, 10:2 (2003), 229  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    3. Cieslinski, JL, “Darboux covariant equations of von Neumann type and their generalizations”, Journal of Mathematical Physics, 44:4 (2003), 1763  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    4. A. K. Svinin, “Invariant Submanifolds of the Darboux–Kadomtsev–Petviashvili Chain and an Extension of the Discrete Kadomtsev–Petviashvili Hierarchy”, Theoret. and Math. Phys., 141:2 (2004), 1542–1561  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    5. S. B. Leble, “Necessary Covariance Conditions for a One-Field Lax Pair”, Theoret. and Math. Phys., 144:1 (2005), 985–994  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. F. Musso, A. B. Shabat, “Elementary Darboux Transformations and Factorization”, Theoret. and Math. Phys., 144:1 (2005), 1004–1013  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. V. E. Adler, A. B. Shabat, “Dressing chain for the acoustic spectral problem”, Theoret. and Math. Phys., 149:1 (2006), 1324–1337  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. Vsevolod E. Adler, Alexey B. Shabat, “On the One Class of Hyperbolic Systems”, SIGMA, 2 (2006), 093, 17 pp.  mathnet  crossref  mathscinet  zmath
    9. Filipuk, GV, “The symmetric fourth Painlevé hierarchy and associated special polynomials”, Studies in Applied Mathematics, 121:2 (2008), 157  crossref  mathscinet  zmath  isi  scopus  scopus
    10. R. N. Garifullin, A. B. Shabat, “The structure of polynomial conservation laws”, Theoret. and Math. Phys., 161:3 (2009), 1590–1598  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    11. A. B. Shabat, Z. S. El'kanova, “Commuting differential operators”, Theoret. and Math. Phys., 162:3 (2010), 276–285  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    12. A. B. Shabat, “Simmetricheskie mnogochleny i zakony sokhraneniya”, Vladikavk. matem. zhurn., 14:4 (2012), 83–94  mathnet
    13. Balakhnev M.Yu., Demskoi D.K., “Auto-Backlund Transformations and Superposition Formulas for Solutions of Drinfeld-Sokolov Systems”, Appl. Math. Comput., 219:8 (2012), 3625–3637  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    14. V. G. Marikhin, “Action as an invariant of Bäcklund transformations for Lagrangian systems”, Theoret. and Math. Phys., 184:1 (2015), 953–960  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    15. Hietarinta J., Joshi N., Nijhoff F., “Discrete Systems and Integrability”, Discrete Systems and Integrability, Cambridge Texts in Applied Mathematics, Cambridge Univ Press, 2016, 1–445  mathscinet  zmath  isi
    16. David Hobby, Ekaterina Shemyakova, “Classification of Multidimensional Darboux Transformations: First Order and Continued Type”, SIGMA, 13 (2017), 010, 20 pp.  mathnet  crossref
    17. Li S., Shemyakova E., Voronov T., “Differential Operators on the Superline, Berezinians, and Darboux Transformations”, Lett. Math. Phys., 107:9 (2017), 1689–1714  crossref  mathscinet  zmath  isi  scopus  scopus
    18. Caparros Quintero A., Hernandez Heredero R., “Formal Recursion Operators of Integrable Nonevolutionary Equations and Lagrangian Systems”, J. Phys. A-Math. Theor., 51:38 (2018), 385201  crossref  isi  scopus
    19. G. S. Mauleshova, “The dressing chain and one-point commuting difference operators of rank 1”, Siberian Math. J., 59:5 (2018), 901–908  mathnet  crossref  crossref  isi
    20. S. V. Smirnov, “Factorization of Darboux–Laplace transformations for discrete hyperbolic operators”, Theoret. and Math. Phys., 199:2 (2019), 621–636  mathnet  crossref
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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