Geodesic
Construction of exact solutions of nonlinear PDE via dressing chain in 3D
Ufa mathematical journal, Tome 16 (2024) no. 4, pp. 124-135 Cet article a éte moissonné depuis la source Math-Net.Ru

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The duality between a class of the Davey — Stewartson type coupled systems and a class of two–dimensional Toda type lattices is discussed. A new coupled system related to the recently found lattice is presented. A method for eliminating nonlocalities in coupled systems by virtue of special finite reductions of the lattices is suggested. An original algorithm for constructing explicit solutions of the coupled systems based on the finite reduction of the corresponding lattice is proposed. Some new solutions for coupled systems related to the Volterra lattice are presented as illustrative examples.
Keywords: 3D lattices, generalized symmetries, Darboux integrable reductions, Lax pairs, Davey — Stewartson type coupled system. 
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I. T. Habibullin; A. R. Khakimova. Construction of exact solutions of nonlinear PDE via dressing chain in 3D. Ufa mathematical journal, Tome 16 (2024) no. 4, pp. 124-135. http://geodesic.mathdoc.fr/item/UFA_2024_16_4_a9/

[1] A.N. Leznov, A.B. Shabat, R.I. Yamilov, “Canonical transformations generated by shifts in nonlinear lattices”, Phys. Lett., A, 174:5–6 (1993), 397–402 | DOI | MR

[2] A.B. Shabat, R.I. Yamilov, “To a transformation theory of two–dimensional integrable systems”, Phys. Lett., A,, 227:1–2 (1997), 15–23 | DOI | MR | Zbl

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

[4] M.N. Poptsova, I.T. Habibullin, “Algebraic properties of quasilinear two–dimensional lattices connected with integrability”, Ufa Math. J., 10:3 (2018), 86–105 | DOI | MR | Zbl

[5] I.T. Habibullin, M.N. Kuznetsova, “A classification algorithm for integrable two–dimensional lattices via Lie — Rinehart algebras”, Theor. Math. Phys., 203:1 (2020), 569–581 | DOI | MR | Zbl

[6] M.N. Kuznetsova, “Classification of a subclass of quasilinear two–dimensional lattices by means of characteristic algebras”, Ufa Math. J., 11:3 (2019), 10–131 | DOI | MR

[7] 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

[8] I.T. Habibullin, A.R. Khakimova, Symmetries of Toda type 3D lattices, 2024, arXiv: 2409.07017 [nlin.SI] | MR | Zbl

[9] B. Gürel, I. Habibullin, “Boundary conditions for two–dimensional integrable chains”, Phys. Lett., A, 233:1 (1997), 68–72 | DOI | MR | Zbl

[10] A.V. Zhiber, V.V. Sokolov, “Exactly integrable hyperbolic equations of Liouville type”, Russ. Math. Surv., 56:1 (2001), 61–101 | DOI | MR | Zbl

[11] I.T. Habibullin, A.U. Sakieva, “On integrable reductions of two–dimensional Toda–type lattices”, Part. Diff. Eq. Appl. Math., 11 (2024), 100854

[12] A.B. Shabat, R.I. Yamilov, “Symmetries of nonlinear chains”, Leningr. Math. J., 2:2 (1991), 377–400 | MR | Zbl

[13] V.B. Matveev, A.O. Smirnov, “Dubrovin method and Toda lattice”, St. Petersbg. Math. J., 34:6 (2023), 1019–1037 | DOI | MR | Zbl

[14] P.G. Grinevich, P.M. Santini, “The finite–gap method and the periodic Cauchy problem for (2+1)–dimensional anomalous waves for the focusing Davey — Stewartson 2 equation”, Russ. Math. Surv., 77:6 (2022), 1029–1059 | DOI | MR | Zbl

[15] I.A. Taimanov, “The Moutard transformation for the Davey — Stewartson II equation and its geometrical meaning”, Math. Notes, 110:5 (2021), 754–766 | DOI | MR | Zbl

[16] V.E. Zakharov, S.V. Manakov, “Construction of higher–dimensional nonlinear integrable systems and of their solutions”, Funct. Anal. Appl., 19:2 (1985), 89–101 | DOI | MR | Zbl

[17] R.Ch. Kulaev, A.B. Shabat, “Darboux system and separation of variables in the Goursat problem for a third order equation in $R^3$”, Russ. Math., 64:4 (2020), 35–43 | DOI | MR | Zbl

[18] B.G. Konopelchenko, B.T. Matkarimov, “Inverse spectral transform for the nonlinear evolution equation generating the Davey — Stewartson and Ishimori equations”, Stud. Appl. Math., 82:4 (1990), 319–359 | DOI | MR | Zbl

[19] M.N. Kuznetsova, “Construction of localized particular solutions of chains with three independent variables”, Theor. Math. Phys., 216:2 (2023), 1158–1167 | DOI | MR | Zbl

[20] M.N. Kuznetsova, I.T. Habibullin, A.R. Khakimova, “On the problem of classifying integrable chains with three independent variables”, Theor. Math. Phys., 215:2 (2023), 667–690 | DOI | MR | Zbl