Geodesic
Modified series of integrable discrete equations on a quadratic lattice with a nonstandard symmetry structure
Teoretičeskaâ i matematičeskaâ fizika, Tome 205 (2020) no. 1, pp. 23-40 Cet article a éte moissonné depuis la source Math-Net.Ru

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We recently constructed a series of integrable discrete autonomous equations on a quadratic lattice with a nonstandard structure of higher symmetries. Here, we construct a modified series using discrete nonpoint transformations. We use both noninvertible linearizable transformations and nonpoint transformations that are invertible on the solutions of the discrete equation. As a result, we obtain several new examples of discrete equations together with their higher symmetries and master symmetries. The constructed higher symmetries give new integrable examples of five- and seven-point differential–difference equations together with their master symmetries. The method for constructing noninvertible linearizable transformations using conservation laws is considered for the first time in the case of discrete equations.
Keywords: discrete equation, higher symmetry, master symmetry, differential–difference equation
Mots-clés : noninvertible transformation. 
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R. N. Garifullin; R. I. Yamilov. Modified series of integrable discrete equations on a quadratic lattice with a nonstandard symmetry structure. Teoretičeskaâ i matematičeskaâ fizika, Tome 205 (2020) no. 1, pp. 23-40. http://geodesic.mathdoc.fr/item/TMF_2020_205_1_a1/

[1] R. N. Garifullin, R. I. Yamilov, “Neobychnaya seriya avtonomnykh diskretnykh integriruemykh uravnenii na kvadratnoi reshetke”, TMF, 200:1 (2019), 50–71 | DOI | DOI

[2] R. N. Garifullin, I. T. Habibullin, R. I. Yamilov, “Peculiar symmetry structure of some known discrete nonautonomous equations”, J. Phys. A: Math. Theor., 48:23 (2015), 235201, 27 pp., arXiv: 1501.05435 | DOI | MR

[3] R. N. Garifullin, R. I. Yamilov, “Examples of Darboux integrable discrete equations possessing first integrals of an arbitrarily high minimal order”, Ufimsk. matem. zhurn., 4:3 (2012), 177–183 | MR

[4] R. N. Garifullin, R. I. Yamilov, “On series of Darboux integrable discrete equations on square lattice”, Ufimsk. matem. zhurn., 11:3 (2019), 100–109 | DOI

[5] R. N. Garifullin, R. I. Yamilov, D. Levi, “Non-invertible transformations of differential-difference equations”, J. Phys. A: Math. Theor., 49:37 (2016), 37LT01, 12 pp. | DOI | MR

[6] S. Ya. Startsev, “On non-point invertible transformations of difference and differential-difference equations”, SIGMA, 6 (2010), 092, 14 pp. | MR

[7] R. I. Yamilov, “Obratimye zameny peremennykh, porozhdennye preobrazovaniyami Beklunda”, TMF, 85:3 (1990), 368–375 | DOI | MR | Zbl

[8] A. V. Mikhailov, “Formalnaya diagonalizatsiya skhem Laksa–Darbu”, Model. i analiz inform. sistem, 22:6 (2015), 795–817 | DOI | MR

[9] D. Levi, R. I. Yamilov, “Generalized symmetry integrability test for discrete equations on the square lattice”, J. Phys. A: Math. Theor., 44:14 (2011), 145207, 22 pp., arXiv: 1011.0070 | DOI | MR

[10] R. I. Yamilov, “O klassifikatsii diskretnykh evolyutsionnykh uravnenii”, UMN, 38:6(234) (1983), 155–156

[11] R. Yamilov, “Symmetries as integrability criteria for differential difference equations”, J. Phys. A: Math. Gen., 39:45 (2006), R541–R623 | DOI | MR

[12] V. E. Adler, A. B. Shabat, R. I. Yamilov, “Simmetriinyi podkhod k probleme integriruemosti”, TMF, 125:3 (2000), 355–424 | DOI | DOI | MR | Zbl

[13] D. Levi, M. Petrera, C. Scimiterna, R. Yamilov, “On Miura transformations and Volterra–type equations associated with the Adler–Bobenko–Suris equations”, SIGMA, 4 (2008), 077, 14 pp. | MR

[14] V. E. Adler, “Integriruemye Mebius-invariantnye evolyutsionnye tsepochki vtorogo poryadka”, Funkts. analiz i ego pril., 50:4 (2016), 13–25 | DOI | DOI | MR

[15] R. I. Yamilov, “On the construction of Miura type transformations by others of this kind”, Phys. Lett. A, 173 (1993), 53–57 | DOI | MR

[16] R. I. Yamilov, “Construction scheme for discrete Miura transformations”, J. Phys. A: Math. Gen., 27:20 (1994), 6839–6851 | DOI | MR

[17] H. Zhang, G. Tu, W. Oevel, B. Fuchssteiner, “Symmetries, conserved quantities and hierarchies for some lattice systems with soliton structure”, J. Math. Phys., 32:7 (1991), 1908–1918 | DOI | MR

[18] R. N. Garifullin, R. I. Yamilov, D. Levi, “Classification of five-point differential-difference equations”, J. Phys. A: Math. Theor., 50:12 (2017), 125201, 27 pp., arXiv: 1610.07342 | DOI | MR

[19] W. Oevel, H. Zhang, B. Fuchssteiner, “Mastersymmetries and multi-Hamiltonian formulations for some integrable lattice systems”, Progr. Theor. Phys., 81:2 (1989), 294–308 | DOI | MR

[20] V. E. Adler, “Integriruemye semitochechnye diskretnye uravneniya i evolyutsionnye tsepochki vtorogo poryadka”, TMF, 195:1 (2018), 27–43 | DOI | DOI