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SIGMA 8 (2012), 051, 5 pages arXiv:1205.1149
https://doi.org/10.3842/SIGMA.2012.051
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”
A Two-Component Generalization of the Integrable rdDym Equation
Oleg I. Morozov
Institute of Mathematics and Statistics, University of Tromsø, Tromsø 90-37, Norway
Received May 26, 2012, in final form August 09, 2012; Published online August 11, 2012
Abstract
We find a two-component generalization of the integrable case of rdDym equation. The reductions of this system
include the general rdDym equation, the Boyer-Finley equation, and the deformed Boyer-Finley equation.
Also we find a Bäcklund transformation between our generalization and Bodganov's two-component
generalization of the universal hierarchy equation.
Key words:
coverings of differential equations; Bäcklund transformations.
pdf (251 kb)
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References
- Blaszak M., Classical R-matrices on Poisson algebras and related
dispersionless systems, Phys. Lett. A 297 (2002), 191-195.
- Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor'kova N.G., Krasil'shchik I.S., Samokhin A.V., Torkhov Y.N., Verbovetsky A.M., Vinogradov A.M.,
Symmetries and conservation laws for differential equations of mathematical
physics, Translations of Mathematical Monographs, Vol. 182, American
Mathematical Society, Providence, RI, 1999.
- Bogdanov L.V., Non-Hamiltonian generalizations of the dispersionless 2DTL
hierarchy, J. Phys. A: Math. Theor. 43 (2010), 434008,
8 pages, arXiv:1003.0287.
- Boyer C.P., Finley III J.D., Killing vectors in self-dual, Euclidean
Einstein spaces, J. Math. Phys. 23 (1982), 1126-1130.
- Dryuma V.S., Non-linear multi-dimensional equations related to commuting vector
fields, in Book of abstracts of International Conference "Differential
Equations and Related Topics" dedicated to I.G. Petrovskii, XXII Joint
Session of Moscow Mathematical Society and Petrovskii Seminar (Moscow, May
21-26, 2007), Moscow Lomonosov State University, Moscow, 2007, 78.
- Dunajski M., A class of Einstein-Weyl spaces associated to an integrable
system of hydrodynamic type, J. Geom. Phys. 51 (2004),
126-137, nlin.SI/0311024.
- Dunajski M., An interpolating dispersionless integrable system,
J. Phys. A: Math. Theor. 41 (2008), 315202, 9 pages,
arXiv:0804.1234.
- Dunajski M., Anti-self-dual four-manifolds with a parallel real spinor,
Proc. R. Soc. Lond. Ser. A 458 (2002), 1205-1222,
math.DG/0102225.
- Ferapontov E.V., Moro A., Sokolov V.V., Hamiltonian systems of hydrodynamic
type in 2+1 dimensions, Comm. Math. Phys. 285 (2009),
31-65, arXiv:0710.2012.
- Kashaev R.M., Saveliev M.V., Savelieva S.A., Vershik A.M., On nonlinear
equations associated with Lie algebras of diffeomorphism groups of
two-dimensional manifolds, in Ideas and Methods in Mathematical Analysis,
Stochastics, and Applications (Oslo, 1988), Cambridge University Press,
Cambridge, 1992, 295-307.
- Krasil'shchik I.S., Lychagin V.V., Vinogradov A.M., Geometry of jet spaces and
nonlinear partial differential equations, Advanced Studies in
Contemporary Mathematics, Vol. 1, Gordon and Breach Science Publishers, New
York, 1986.
- Krasil'shchik I.S., Vinogradov A.M., Nonlocal symmetries and the theory of
coverings: an addendum to Vinogradov's "Local symmetries and
conservation laws", Acta Appl. Math. 2 (1984), 79-96.
- Krasil'shchik I.S., Vinogradov A.M., Nonlocal trends in the geometry of
differential equations: symmetries, conservation laws, and Bäcklund
transformations, Acta Appl. Math. 15 (1989), 161-209.
- Krichever I.M., The τ-function of the universal Whitham hierarchy,
matrix models and topological field theories, Comm. Pure Appl. Math.
47 (1994), 437-475, hep-th/9205110.
- Malykh A.A., Nutku Y., Sheftel M.B., Winternitz P., Invariant solutions of
complex Monge-Ampère equation and gravitational instantons,
Phys. Atomic Nuclei 61 (1989), 1986-1989.
- Manakov S.V., Santini P.M., Cauchy problem on the plane for the dispersionless
Kadomtsev-Petviashvili equation, JETP Lett. 83 (2006),
462-466, nlin.SI/0604023.
- Mañas M., Medina E., Martínez Alonso L., On the Whitham hierarchy:
dressing scheme, string equations and additional symmetries,
J. Phys. A: Math. Gen. 39 (2006), 2349-2381,
nlin.SI/0509017.
- Martínez Alonso L., Shabat A.B., Energy-dependent potentials revisited:
a universal hierarchy of hydrodynamic type, Phys. Lett. A
299 (2002), 359-365, nlin.SI/0202008.
- Martínez Alonso L., Shabat A.B., Hydrodynamic reductions and solutions
of the universal hierarchy, Theoret. and Math. Phys. 140
(2004), 1073-1085, nlin.SI/0312043.
- Morozov O.I., Contact integrable extensions of symmetry pseudo-groups and
coverings of (2+1) dispersionless integrable equations, J. Geom.
Phys. 59 (2009), 1461-1475, arXiv:0809.1218.
- Pavlov M.V., Classification of integrable Egorov hydrodynamic chains,
Theoret. and Math. Phys. 138 (2004), 45-58,
nlin.SI/0603055.
- Pavlov M.V., Integrable hydrodynamic chains, J. Math. Phys.
44 (2003), 4134-4156, nlin.SI/0301010.
- Pavlov M.V., The Kupershmidt hydrodynamic chains and lattices, Int.
Math. Res. Not. (2006), Art. ID 46987, 43 pages, nlin.SI/0604049.
- Pavlov M.V., Chang J.H., Chen Y.T., Integrability of the Manakov-Santini
hierarchy, arXiv:0910.2400.
- Plebanski J.F., Some solutions of complex Einstein equations,
J. Math. Phys. 16 (1975), 2395-2402.
- Zakharov V.E., Integrable systems in multidimensional spaces, in Mathematical
Problems in Theoretical Physics (Berlin, 1981), Lecture Notes in
Phys., Vol. 153, Springer, Berlin, 1982, 190-216.
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