Geodesic
Construction of localized particular solutions of chains with three independent variables
Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 2, pp. 291-301 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider differential–difference chains with three independent variables of the form $u^j_{n+1,x} = F(u^j_{n,x}, u^{j+1}_n, u^j_n, u^j_{n+1}, u^{j-1}_{n+1})$. An effective approach to the study and classification of equations with three independent variables is the method based on Darboux-integrable reductions. Using the Darboux-integrable reductions, we construct localized particular solutions of chains with three independent variables.
Keywords: three-dimensional chains, characteristic algebras, Darboux integrability, characteristic integrals, integrable reductions. 
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M. N. Kuznetsova. Construction of localized particular solutions of chains with three independent variables. Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 2, pp. 291-301. http://geodesic.mathdoc.fr/item/TMF_2023_216_2_a6/

[1] I. Habibullin, “Characteristic Lie rings, finitely-generated modules and integrability conditions for $(2+ 1)$-dimensional lattices”, Phys. Scr., 87:6 (2013), 065005, 5 pp. | DOI | Zbl

[2] I. Habibullin, M. Poptsova, “Classification of a subclass of two-dimensional lattices via characteristic Lie rings”, SIGMA, 13 (2017), 073, 26 pp., arXiv: 1703.09963 | DOI | MR | Zbl

[3] M. N. Poptsova, I. T. Khabibullin, “Algebraicheskie svoistva kvazilineinykh dvumerizovannykh tsepochek, svyazannye s integriruemostyu”, Ufimsk. matem. zhurn., 10:3 (2018), 89–109 | DOI | MR

[4] I. T. Khabibullin, M. N. Kuznetsova, “O klassifikatsionnom algoritme integriruemykh dvumerizovannykh tsepochek na osnove algebr Li–Rainkharta”, TMF, 203:1 (2020), 161–173 | DOI | DOI | MR

[5] E. V. Ferapontov, I. T. Habibullin, M. N. Kuznetsova, V. S. Novikov, “On a class of 2D integrable lattice equations”, J. Math. Phys., 61:7 (2020), 073505, 15 pp. | DOI | MR | Zbl

[6] I. T. Habibullin, A. R. Khakimova, “Characteristic Lie algebras of integrable differential-difference equations in 3D”, J. Phys. A: Math. Theor., 54:29 (2021), 295202, 34 pp. | DOI | MR | Zbl

[7] I. T. Habibullin, M. N. Kuznetsova, “An algebraic criterion of the Darboux integrability of differential-difference equations and systems”, J. Phys. A: Math. Theor., 54:50 (2021), 505201, 20 pp. | DOI | MR

[8] M. N. Kuznetsova, I. T. Khabibullin, A. R. Khakimova, “K zadache o klassifikatsii tsepochek s tremya nezavisimymi peremennymi”, TMF, 215:2 (2023), 242–268 | DOI | DOI

[9] E. V. Ferapontov, V. S. Novikov, I. Roustemoglou, “On the classification of discrete Hirota-type equations in 3D”, Int. Math. Res. Not. IMRN, 2015:13 (2015), 4933–4974 | DOI | MR

[10] E. Goursat, “Recherches sur quelques équations aux dérivées partielles du second ordre”, Annales de la faculté des Sciences de l'Université de Toulouse: Mathématiques, Serie 2, 1:1 (1899), 31–78 | DOI

[11] A. N. Leznov, A. B. Shabat, “Usloviya obryva ryadov teorii vozmuschenii”, Integriruemye sistemy, BFAN SSSR, Ufa, 1982, 34–44

[12] A. N. Leznov, M. V. Savelev, Gruppovye metody integrirovaniya nelineinykh dinamicheskikh sistem, Nauka, M., 1985 | MR | Zbl

[13] A. V. Zhiber, V. V. Sokolov, “Tochno integriruemye giperbolicheskie uravneniya liuvillevskogo tipa”, UMN, 56:1(337) (2001), 63–106 | DOI | DOI | MR | Zbl

[14] O. S. Kostrigina, A. V. Zhiber, “Darboux-integrable two-component nonlinear hyperbolic systems of equations”, J. Math. Phys., 52:3 (2011), 033503, 32 pp. | DOI | MR | Zbl

[15] I. Habibullin, M. Zheltukhina, A. Sakieva, “On the Darboux-integrable semi-discrete chains”, J. Phys. A: Math. Theor., 43:43 (2010), 434017, 14 pp. | DOI | MR | Zbl

[16] V. E. Adler, S. Ya. Startsev, “O diskretnykh analogakh uravneniya Liuvillya”, TMF, 121:2 (1999), 271–284 | DOI | DOI | MR | Zbl

[17] A. B. Shabat, R. I. Yamilov, Eksponentsialnye sistemy tipa I i matritsy Kartana, Preprint BFAN SSSR, Ufa, 1981

[18] A. N. Leznov, V. G. Smirnov, A. B. Shabat, “Gruppa vnutrennikh simmetrii i usloviya integriruemosti dvumernykh dinamicheskikh sistem”, TMF, 51:1 (1982), 10–21 | DOI | MR | Zbl

[19] A. V. Zhiber, R. D. Murtazina, I. T. Khabibullin, A. B. Shabat, Kharakteristicheskie koltsa Li i nelineinye integriruemye uravneniya, In-t kompyuternykh issledovanii, M.–Izhevsk, 2012

[20] S. V. Smirnov, “Integriruemost po Darbu diskretnykh dvumerizovannykh tsepochek Tody”, TMF, 182:2 (2015), 231–255 | DOI | DOI | MR