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Hirota difference equation and Darboux system: mutual symmetry. (English) Zbl 1423.39013

Summary: We considered the relation between two famous integrable equations: The Hirota difference equation (HDE) and the Darboux system that describes conjugate curvilinear systems of coordinates in \(\mathbb{R}^3\). We demonstrated that specific properties of solutions of the HDE with respect to independent variables enabled introduction of an infinite set of discrete symmetries. We showed that degeneracy of the HDE with respect to parameters of these discrete symmetries led to the introduction of continuous symmetries by means of a specific limiting procedure. This enabled consideration of these symmetries on equal terms with the original HDE independent variables. In particular, the Darboux system appeared as an integrable equation where continuous symmetries of the HDE served as independent variables. We considered some cases of intermediate choice of independent variables, as well as the relation of these results with direct and inverse problems.

MSC:

39A14 Partial difference equations
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
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[1] Hirota, R.; Nonlinear partial difference equations II; Discrete time Toda equations; J. Phys. Soc. Jpn.: 1977; Volume 43 ,2074-2078. · Zbl 1334.39014
[2] Hirota, R.; Discrete analogue of a generalized Toda equation; J. Phys. Soc. Jpn.: 1981; Volume 50 ,3785-3791.
[3] Miwa, T.; On Hirota’s difference equation; Proc. Jpn. Acad. A Math. Sci.: 1982; Volume 58 ,9-12. · Zbl 0508.39009
[4] Zakharov, V.E.; Manakov, S.V.; Construction of higher-dimensional nonlinear integrable systems and of their solutions; Funct. Anal. Appl.: 1985; Volume 19 ,89-101. · Zbl 0597.35115
[5] Bogdanov, L.V.; Konopelchenko, B.G.; Generalized KP hierarchy: Möbius symmetry, symmetry constraints and Calogero—Moser system; Physica D: 2001; Volume 152-153 ,85-96. · Zbl 0988.37085
[6] Zabrodin, A.V.; Hirota’s difference equations; Theor. Math. Phys.: 1997; Volume 113 ,1347-1392.
[7] Zabrodin, A.V.; Bäcklund transformations for the difference Hirota equation and the supersymmetric Bethe ansatz; Theor. Math. Phys.: 2008; Volume 155 ,567-584. · Zbl 1157.82313
[8] Saito, S.; Octahedral structure of the Hirota-Miwa equation; J. Nonlinear Math. Phys.: 2012; Volume 10 ,1250032. · Zbl 1303.37027
[9] Krichever, I.; Wiegmann, P.; Zabrodin, A.; Elliptic solutions to difference non-linear equations and related many-body problems; Commun. Math. Phys.: 1998; Volume 193 ,373-396. · Zbl 0907.35124
[10] Hone, A.N.W.; Kouloukas, T.E.; Ward, C.; On Reductions of the Hirota-Miwa Equation; Symmetry Integrab. Geom. Methods Appl.: 2017; Volume 13 ,057. · Zbl 1425.70032
[11] Doliwa, A.; Lin, R.; Discrete KP equation with self-consistent sources; Phys. Lett. A: 2014; Volume 378 ,1925-1931. · Zbl 1342.37067
[12] Pogrebkov, A.K.; Hirota difference equation: Inverse scattering transform, Darboux transformation, and solitons; Theor. Math. Phys.: 2014; Volume 181 ,1585-1598. · Zbl 1317.81252
[13] Pogrebkov, A.K.; Hirota difference equation and a commutator identity on an associative algebra; St. Petersburg Math. J.: 2011; Volume 22 ,473-483. · Zbl 1222.37073
[14] Pogrebkov, A.K.; Commutator identities on associative algebras. The non-Abelian Hirota difference equation and its reductions; Theor. Math. Phys.: 2016; Volume 187 ,823-834. · Zbl 1346.37056
[15] Pogrebkov, A.K.; Symmetries of the Hirota difference equation; Symmetry Integrab. Geom. Methods Appl.: 2017; Volume 13 ,053. · Zbl 1372.35267
[16] Darboux, G.; ; Lecons sur les Systèmes Orthogonaux et les Coordonnées Curvilignes: Paris, France 1898; . · JFM 29.0515.03
[17] Eisenhart, L.P.; ; A Treatise on the Differential Geometry of Curves and Surfaces: Whitefish, MT, USA 2010; .
[18] Bianchi, L.; Opere; Sisteme Tripli Orthogonali: Roma, Italy 1956; .
[19] Rogers, C.; Schief, W.K.; ; BDT, Geometry and Modern Appliation in Soliton Theory: Cambridge, UK 2002; . · Zbl 1019.53002
[20] Ferapontov, E.V.; Systems of three differential equations of hydrodynamic type with hexagonal 3-web of characteristics on the solutions; Funct. Anal. Appl.: 1989; Volume 23 ,151-153. · Zbl 0714.35070
[21] Tsarev, S.P.; The geometry of Hamiltonian systems of hydrodymanic type. The generalized hodograph method; Math. USSR-Izvestiya: 1991; Volume 37 ,397-419. · Zbl 0796.76014
[22] Zakharov, V.E.; Description of the n-orthogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type. I: Integration of the Lamé equations; Duke Math. J.: 1998; Volume 94 ,103-139. · Zbl 0963.37068
[23] Bogdanov, L.V.; Konopelchenko, B.G.; Generalized integrable hierarchies and Combescure symmetry transformations; J. Phys. A Math. Gen.: 1997; Volume 30 ,1591-1603. · Zbl 1001.37501
[24] Garagash, T.I.; Pogrebkov, A.K.; Scattering problem for the differential operator ∂x∂y + 1 + a(x,y)∂y + b(x,y); Theor. Math. Phys.: 1995; Volume 102 ,117-132. · Zbl 0856.35096
[25] Pogrebkov, A.K.; Higher Hirota difference equations and their reductions; Theor. Math. Phys.: 2018; Volume 197 ,1779-1796. · Zbl 1429.39011
[26] Kulaev, R.C.; Pogrebkov, A.K.; Shabat, A.B.; Darboux system: Liouville reduction and an explicit solution; Proc. Steklov Inst. Math.: 2018; Volume 302 ,250-269. · Zbl 1439.35401
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