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SIGMA, 2008, Volume 4, 077, 14 pages (Mi sigma330)  

This article is cited in 23 scientific papers (total in 23 papers)

On Miura Transformations and Volterra-Type Equations Associated with the Adler–Bobenko–Suris Equations

Decio Leviab, Matteo Petreraab, Christian Scimiternaab, Ravil Yamilovc

a INFN — National Institute of Nuclear Physics
b Università degli Studi Roma Tre
c Ufa Institute of Mathematics, 112 Chernyshevsky Str., Ufa 450077, Russia

Abstract: We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler–Bobenko–Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Bäcklund transformations for some particular cases of the discrete Krichever–Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate Bäcklund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.

Keywords: Miura transformations; generalized symmetries; ABS lattice equations


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ArXiv: 0802.1850
MSC: 37K10; 37L20; 39A05
Received: August 29, 2008; in final form October 30, 2008; Published online November 8, 2008

Citation: Decio Levi, Matteo Petrera, Christian Scimiterna, Ravil Yamilov, “On Miura Transformations and Volterra-Type Equations Associated with the Adler–Bobenko–Suris Equations”, SIGMA, 4 (2008), 077, 14 pp.

Citation in format AMSBIB
\by Decio Levi, Matteo Petrera, Christian Scimiterna, Ravil Yamilov
\paper On Miura Transformations and Volterra-Type Equations Associated with the Adler--Bobenko--Suris Equations
\jour SIGMA
\yr 2008
\vol 4
\papernumber 077
\totalpages 14

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    This publication is cited in the following articles:
    1. D. Levi, R. I. Yamilov, “On a nonlinear integrable difference equation on the square”, Ufimsk. matem. zhurn., 1:2 (2009), 101–105  mathnet  zmath  elib
    2. Levi D., Yamilov R.I., “The generalized symmetry method for discrete equations”, J. Phys. A, 42:45 (2009), 454012, 18 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Levi D., Winternitz P., Yamilov R.I., “Lie point symmetries of differential-difference equations”, J. Phys. A, 43:29 (2010), 292002, 14 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    4. Levi D., Yamilov R.I., “Integrability test for discrete equations via generalized symmetries”, Symmetries in Nature, AIP Conference Proceedings, 1323, 2010, 203–214  crossref  mathscinet  adsnasa  isi  scopus
    5. Levi D., Yamilov R.I., “Generalized symmetry integrability test for discrete equations on the square lattice”, J. Phys. A, 44:14 (2011), 145207, 22 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Mikhailov A.V., Wang J.P., Xenitidis P., “Cosymmetries and Nijenhuis recursion operators for difference equations”, Nonlinearity, 24:7 (2011), 2079–2097  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    7. Decio Levi, Pavel Winternitz, Ravil I. Yamilov, “Symmetries of the Continuous and Discrete Krichever–Novikov Equation”, SIGMA, 7 (2011), 097, 16 pp.  mathnet  crossref  mathscinet
    8. Svinin A.K., “On some integrable lattice related by the Miura-type transformation to the Itoh-Narita-Bogoyavlenskii lattice”, Journal of Physics A-Mathematical and Theoretical, 44:46 (2011), 465210  crossref  mathscinet  zmath  adsnasa  isi  scopus
    9. Xenitidis P., “Symmetries and conservation laws of the ABS equations and corresponding differential-difference equations of Volterra type”, Journal of Physics A-Mathematical and Theoretical, 44:43 (2011), 435201  crossref  mathscinet  zmath  adsnasa  isi  scopus
    10. Kassotakis P., Nieszporski M., “On Non-Multiaffine Consistent-Around-the-Cube Lattice Equations”, Phys. Lett. A, 376:45 (2012), 3135–3140  crossref  zmath  adsnasa  isi  elib  scopus
    11. Garifullin R.N. Yamilov R.I., “Generalized Symmetry Classification of Discrete Equations of a Class Depending on Twelve Parameters”, J. Phys. A-Math. Theor., 45:34 (2012), 345205  crossref  mathscinet  zmath  isi  elib  scopus
    12. Zhang, DJ; Cheng, JW; Sun, YY, “Deriving conservation laws for ABS lattice equations from Lax pairs”, Journal of Physics A: Mathematical and Theoretical, 46:26 (2013), 265202  crossref  mathscinet  zmath  adsnasa  isi  scopus
    13. R. N. Garifullin, A. V. Mikhailov, R. I. Yamilov, “Discrete equation on a square lattice with a nonstandard structure of generalized symmetries”, Theoret. and Math. Phys., 180:1 (2014), 765–780  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    14. Scimiterna Ch., Hay M., Levi D., “on the Integrability of a New Lattice Equation Found By Multiple Scale Analysis”, J. Phys. A-Math. Theor., 47:26 (2014), 265204  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    15. Adler V.E., Postnikov V.V., “on Discrete 2D Integrable Equations of Higher Order”, J. Phys. A-Math. Theor., 47:4 (2014), 045206  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    16. Garifullin R.N. Habibullin I.T. Yamilov R.I., “Peculiar Symmetry Structure of Some Known Discrete Nonautonomous Equations”, J. Phys. A-Math. Theor., 48:23 (2015), 235201  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    17. Garifullin R.N. Yamilov R.I., “Integrable Discrete Nonautonomous Quad-Equations as Backlund Auto-Transformations For Known Volterra and Toda Type Semidiscrete Equations”, Seventh International Workshop: Group Analysis of Differential Equations and Integrable Systems (Gadeisvii), Journal of Physics Conference Series, 621, IOP Publishing Ltd, 2015, UNSP 012005  crossref  isi  scopus
    18. Garifullin R.N., Yamilov R.I., Levi D., “Classification of five-point differential-difference equations”, J. Phys. A-Math. Theor., 50:12 (2017), 125201  crossref  mathscinet  zmath  isi  scopus
    19. Lou S. Shi Y. Zhang D.-j., “Spectrum transformation and conservation laws of lattice potential KdV equation”, Front. Math. China, 12:2 (2017), 403–416  crossref  mathscinet  zmath  isi  scopus
    20. Gubbiotti G. Scimiterna C. Levi D., “The Non-Autonomous Ydkn Equation and Generalized Symmetries of Boll Equations”, J. Math. Phys., 58:5 (2017), 053507  crossref  mathscinet  zmath  isi  scopus
    21. Garifullin R.N., Yamilov R.I., Levi D., “Classification of Five-Point Differential-Difference Equations II”, J. Phys. A-Math. Theor., 51:6 (2018), 065204  crossref  mathscinet  zmath  isi  scopus
    22. V. E. Adler, “Integrable seven-point discrete equations and second-order evolution chains”, Theoret. and Math. Phys., 195:1 (2018), 513–528  mathnet  crossref  crossref  adsnasa  isi  elib
    23. Garifullin R.N., Gubbiotti G., Yamilov I R., “Integrable Discrete Autonomous Quad-Equations Admitting, as Generalized Symmetries, Known Five-Point Differential-Difference Equations”, J. Nonlinear Math. Phys., 26:3 (2019), 333–357  crossref  isi
  • Symmetry, Integrability and Geometry: Methods and Applications
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