Contact integrable extensions and differential coverings for the generalized (2 + 1)-dimensional dispersionless Dym equation

• Autorzy: Oleg Morozov
• Języki publikacji: EN

Abstrakty

EN

The method of contact integrable extensions is used to find new differential coverings for the generalized (2 + 1)-dimensional dispersionless Dym equation and corresponding Bäcklund transformations.

Słowa kluczowe

EN

Lie pseudo-groups; Maurer-Cartan forms; Symmetries of differential equations; Differential coverings

Kategorie tematyczne

• 58H05: Pseudogroups and differentiable groupoids
• 35A30: Geometric theory, characteristics, transformations
• 58J70: Invariance and symmetry properties

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 5
• Strony: 1688-1697

Daty

wydano

2012-10-01

online

2012-07-24

Twórcy

autor

Oleg Morozov

• University of Tromsø

Bibliografia

• [1] Alfinito E., Profilo G., Soliani G., Properties of equations of the continuous Toda type, J. Phys. A, 1997, 30(5), 1527–1547 http://dx.doi.org/10.1088/0305-4470/30/5/019
• [2] Błaszak M., Classical R-matrices on Poisson algebras and related dispersionless systems, Phys. Lett. A, 2002, 297(3–4), 191–195 http://dx.doi.org/10.1016/S0375-9601(02)00421-8
• [3] Bryant R.L., Griffiths Ph.A., Characteristic cohomology of differential systems II: Conservation laws for a class of parabolic equations, Duke Math. J., 1995, 78(3), 531–676 http://dx.doi.org/10.1215/S0012-7094-95-07824-7
• [4] Cartan É., Sur la structure des groupes infinis de transformations, In: Œuvres complétes, 2(2), Gauthier-Villars, Paris, 1953, 571–714
• [5] Cartan É., Les sous-groupes des groupes continus de transformations, In: Œuvres Complètes, 2(2), Gauthier-Villars, Paris, 1953, 719–856
• [6] Cartan É., Les problèmes d’équivalence, In: Œuvres Complètes, 2(2), Gauthier-Villars, Paris, 1953, 1311–1334
• [7] Cartan É., La structure des groupes infinis, In: Œuvres Complètes, 2(2), Gauthier-Villars, Paris, 1953, 1335–1384
• [8] Dodd R., Fordy A., The prolongation structures of quasi-polynomial flows, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 1983, 385(1789), 389–429 http://dx.doi.org/10.1098/rspa.1983.0020
• [9] Dryuma V.S., Non-linear multi-dimensional equations related to commuting vector fields, and their integration, In: 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, Moscow State University, 2007 (book of abstracts, in Russian)
• [10] Estabrook F.B., Moving frames and prolongation algebras, J. Math. Phys., 1982, 23(11), 2071–2076 http://dx.doi.org/10.1063/1.525248
• [11] Fels M., Olver P.J., Moving coframes: I. A practical algorithm, Acta Appl. Math., 1998, 51(2), 161–213 http://dx.doi.org/10.1023/A:1005878210297
• [12] Ferapontov E.V., Khusnutdinova K.R., On integrability of (2 + 1)-dimensional quasilinear systems, Comm. Math. Phys., 2004, 248(1), 187–206 http://dx.doi.org/10.1007/s00220-004-1079-6
• [13] Ferapontov E.V., Odesskii A.V., Stoilov N.M., Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions, preprint available at http://arxiv.org/abs/1007.3782
• [14] Gardner R.B., The Method of Equivalence and its Applications, CBMS-NSF Regional Conf. Ser. in Appl. Math., 58, SIAM, Philadelphia, 1989
• [15] Harrison B.K., On methods of finding Bäcklund transformations in systems with more than two independent variables, J. Nonlinear Math. Phys., 1995, 2(3–4), 201–215 http://dx.doi.org/10.2991/jnmp.1995.2.3-4.1
• [16] Harrison B.K., Matrix methods of searching for Lax pairs and a paper by Estévez, In: Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 30(1–2), Natsïonal. Akad. Nauk Ukraïni, Īnst. Mat., Kiev, 2000, 17–24
• [17] Hoenselaers C., More prolongation structures, Progr. Theoret. Phys., 1986, 75(5), 1014–1029 http://dx.doi.org/10.1143/PTP.75.1014
• [18] Igonin S., Coverings and fundamental algebras for partial differential equations, J. Geom. Phys., 2006, 56(6), 939–998 http://dx.doi.org/10.1016/j.geomphys.2005.06.001
• [19] Kamran N., Contributions to the Study of the Equivalence Problem of Élie Cartan and its Applications to Partial and Ordinary Differential Equations, Acad. Roy. Belg. Cl. Sci. Mém. Collect. 8, 45(7), Brussels, 1989
• [20] Krasil’shchik I.S., Vinogradov A.M., Nonlocal symmetries and the theory of coverings, Acta Appl. Math., 1984, 2(1), 79–86 http://dx.doi.org/10.1007/BF01405492
• [21] Krasil’shchik I.S., Vinogradov A.M., Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations, Acta Appl. Math., 1989, 15(1–2), 161–209 http://dx.doi.org/10.1007/BF00131935
• [22] Kuzmina G.M., On geometry of a system of two partial differential equations, Moskov. Gos. Ped. Inst. Uchen. Zap., 1965, 243, 99–108 (in Russian)
• [23] Kuzmina G.M., On a possibility to reduce a system of two partial differential equations of the first order to one equation of the second order, Moskov. Gos. Ped. Inst. Uchen. Zap., 1967, 271, 67–76 (in Russian)
• [24] Marvan M., On zero-curvature representations of partial differential equations, In: Differential Geometry and its Applications, 1992, August 24–28, Opava, Math. Publ., 1, Silesian University in Opava, Opava, 1993, 103–122
• [25] Marvan M., A direct procedure to compute zero-curvature representations. The case sl2, In: Secondary Calculus and Cohomological Physics, Moscow, August 24–31, 1997, Diffiety Institute of the Russian Academy of Natural Sciences, Pereslavl’ Zalesskiy, 1997
• [26] Morozov O.I., Moving coframes and symmetries of differential equations, J. Phys. A, 2002, 35(12), 2965–2977 http://dx.doi.org/10.1088/0305-4470/35/12/317
• [27] Morozov O.I., The contact-equivalence problem for linear hyperbolic equations, J. Math. Sci. (N.Y.), 2006, 135(1), 2680–2694 http://dx.doi.org/10.1007/s10958-006-0138-2
• [28] Morozov O.I., Coverings of differential equations and Cartan’s structure theory of Lie pseudo-groups, Acta Appl. Math., 2007, 99(3), 309–319 http://dx.doi.org/10.1007/s10440-007-9167-1
• [29] Morozov O.I., Cartan’s structure theory of symmetry pseudo-groups, coverings and multi-valued solutions for the Khokhlov-Zabolotskaya equation, Acta Appl. Math., 2008, 101(1–3), 231–241 http://dx.doi.org/10.1007/s10440-008-9191-9
• [30] Morozov O.I., Contact integrable extensions of symmetry pseudo-groups and coverings of (2 + 1) dispersionless integrable equations, J. Geom. Phys., 2009, 59(11), 1461–1475 http://dx.doi.org/10.1016/j.geomphys.2009.07.009
• [31] Morozov O.I., Contact integrable extensions and zero-curvature representations for the second heavenly equation, Global and Stochastic Analysis, 2011, 1, 89–100
• [32] Morozov O.I., A covering for the dKP-hyper CR interpolating equation and multi-valued Einstein-Weyl structures, preprint available at http://arxiv.org/abs/0903.3788
• [33] Morozov O.I., Contact integrable extensions of symmetry pseudo-group and coverings for the r-th double modified dispersionless Kadomtsev-Petviashvili equation, preprint available at http://arxiv.org/abs/1010.1828
• [34] Morris H.C., Prolongation structures and nonlinear evolution equations in two spatial dimensions, J. Math. Phys., 1976, 17(10), 1870–1872 http://dx.doi.org/10.1063/1.522809
• [35] Morris H.C., Prolongation structures and nonlinear evolution equations in two spatial dimensions: a general class of equations, J. Phys. A, 1979, 12(3), 261–267 http://dx.doi.org/10.1088/0305-4470/12/3/003
• [36] Nucci M.C., Pseudopotentials for nonlinear evolution equations in 2+1 orders, Internat. J. Non-Linear Mech., 1988, 23(5–6), 361–367 http://dx.doi.org/10.1016/0020-7462(88)90033-9
• [37] Olver P.J., Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, 1995
• [38] Ovsienko V., Bi-Hamiltionian nature of the equation u tx = u xyu y − u yyu x, Adv. Pure Appl. Math., 2010, 1(1), 7–17
• [39] Palese M., Bäcklund loop algebras for compact and noncompact nonlinear spin models in 2+1 dimensions, Theoret. and Math. Phys., 2005, 144(1), 1014–1021 http://dx.doi.org/10.1007/s11232-005-0129-3
• [40] Palese M., Leo R.A., Soliani G., The prolongation problem for the heavenly equation, In: Recent Developments in General Relativity, Bari, September 21–25, 1998, Springer, Milan, 2000
• [41] Ritt J.F., Integration in Finite Terms. Liouville’s Theory of Elementary Methods, Columbia University Press, New York, 1948
• [42] Sakovich S.Yu., On zero-curvature representations of evolution equations, J. Phys. A, 1995, 28(10), 2861–2869 http://dx.doi.org/10.1088/0305-4470/28/10/016
• [43] Stormark O., Lie’s Structural Approach to PDE Systems, Encyclopedia Math. Appl., 80, Cambridge University Press, Cambridge, 2000
• [44] Tondo G.S., The eigenvalue problem for the three-wave resonant interaction in (2+1) dimensions via the prolongation structure, Lett. Nuovo Cimento, 1985, 44(5), 297–302 http://dx.doi.org/10.1007/BF02746684
• [45] Vasilieva M.V., Structure of Infinite Lie Groups of Transformations, Moskov. Gosudarstv. Ped. Inst., Moscow, 1972 (in Russian)
• [46] Wahlquist H.D., Estabrook F.B., Prolongation structures of nonlinear evolution equations, J. Math. Phys., 1975, 16, 1–7 http://dx.doi.org/10.1063/1.522396

Identyfikatory

DOI

10.2478/s11533-012-0039-1