PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2014 | 12 | 2 | 271-283
Tytuł artykułu

A recursion operator for the universal hierarchy equation via Cartan’s method of equivalence

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We apply Cartan’s method of equivalence to find a Bäcklund autotransformation for the tangent covering of the universal hierarchy equation. The transformation provides a recursion operator for symmetries of this equation.
Twórcy
autor
Bibliografia
  • [1] Bluman G.W., Cole J.D., Similarity Methods for Differential Equations, Appl. Math. Sci., 13, Springer, New York-Heidelberg, 1974 http://dx.doi.org/10.1007/978-1-4612-6394-4
  • [2] Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor’kova N.G., Krasil’shchik I.S., Samokhin A.V., Torkhov Yu.N., Verbovetsky A.M., Vinogradov A.M., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Transl. Math. Monogr., 182, American Mathematical Society, Providence, 1999
  • [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: OEuvres Complètes, 2(2), Gauthier-Villars, Paris, 1953, 571–714
  • [5] Cartan É., Les sous-groupes des groupes continus de transformations, In: OEuvres Complètes, 2(2), Gauthier-Villars, Paris, 1953, 719–856
  • [6] Cartan É., Les problèmes d’équivalence, In: OEuvres Complètes, 2(2), Gauthier-Villars, Paris, 1953, 1311–1334
  • [7] Cartan É., La structure des groupes infinis, In: OEuvres Complètes, 2(2), Gauthier-Villars, Paris, 1953, 1335–1384
  • [8] 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
  • [9] Fokas A.S., Symmetries and integrability, Stud. Appl. Math., 1987, 77(3), 253–299
  • [10] Fokas A.S., Santini P.M., The recursion operator of the Kadomtsev-Petviashvili equation and the squared eigenfunctions of the Schrödinger operator, Stud. Appl. Math., 1986, 75(2), 179–185
  • [11] Fokas A.S., Santini P.M., Recursion operators and bi-Hamiltonian structures in multidimensions. II, Comm. Math. Phys., 1988, 116(3), 449–474 http://dx.doi.org/10.1007/BF01229203
  • [12] Fuchssteiner B., Application of hereditary symmetries to nonlinear evolution equations, Nonlinear Anal., 1979, 3(6), 849–862 http://dx.doi.org/10.1016/0362-546X(79)90052-X
  • [13] Gardner R.B., The Method of Equivalence and its Applications, CBMS-NSF Regional Conf. Ser. in Appl. Math., 58, SIAM, Philadelphia, 1989
  • [14] Gürses M., Karasu A., Sokolov V.V., On construction of recursion operators from Lax representation, J. Math. Phys., 1999, 40(12), 6473–6490 http://dx.doi.org/10.1063/1.533102
  • [15] Guthrie G.A., Recursion operators and non-local symmetries, Proc. Roy. Soc. London Ser. A, 1994, 446(1926), 107–114 http://dx.doi.org/10.1098/rspa.1994.0094
  • [16] Ibragimov N.H., Transformation Groups Applied to Mathematical Physics, Math. Appl. (Soviet Ser.), Reidel, Dordrecht, 1985
  • [17] 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), Brussls, 1989
  • [18] Krasil’shchik I.S., Kersten P.H.M., Deformations and recursion operators for evolution equations, In: Geometry in Partial Differential Equations, World Scientific, River Edge, 1994, 114–154 http://dx.doi.org/10.1142/9789814354394_0008
  • [19] Krasil’shchik I.S., Kersten P.H.M., Graded differential equations and their deformations: a computational theory for recursion operators, Acta Appl. Math., 1995, 41(1–3), 167–191 http://dx.doi.org/10.1007/BF00996112
  • [20] Krasil’shchik I.S., Lychagin V.V., Vinogradov A.M., Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Adv. Stud. Contemp. Math., 1, Gordon and Breach, New York, 1986
  • [21] Krasil’shchik J., Verbovetsky A., Geometry of jet spaces and integrable systems, J. Geom. Phys., 2011, 61(9), 1633–1674 http://dx.doi.org/10.1016/j.geomphys.2010.10.012
  • [22] Krasil’shchik I.S., Verbovetsky A.M., Vitolo R., A unified approach to computation of integrable structures, Acta Appl. Math., 2012, 120, 199–218 http://dx.doi.org/10.1007/s10440-012-9699-x
  • [23] 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
  • [24] Martínez Alonso L., Shabat A.B., Energy-dependent potentials revisited: a universal hierarchy of hydrodynamic type, Phys. Lett. A, 2002, 299(4), 359–365 http://dx.doi.org/10.1016/S0375-9601(02)00662-X
  • [25] Martínez Alonso L., Shabat A.B., Hydrodynamic reductions and solutions of the universal hierarchy, Theoret. and Math. Phys., 2004, 140(2), 1073–1085 http://dx.doi.org/10.1023/B:TAMP.0000036538.41884.57
  • [26] Marvan M., Another look on recursion operators, In: Differential Geometry and Applications, Brno, August 28–September 1, 1995, Masaryk University, Brno, 1996, 393–402
  • [27] Marvan M., Recursion operator for vacuum Einstein equations with symmetries, Symmetry in Nonlinear Mathematical Physics, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 50(1,2,3), Natsīonal Akad. Nauk Ukraïni, Kiev, 2004, 179–183
  • [28] Marvan M., Reducibility of zero curvature representations with application to recursion operators, Acta Appl. Math., 2004, 83(1–2), 39–68 http://dx.doi.org/10.1023/B:ACAP.0000035588.67805.0b
  • [29] Marvan M., On the spectral parameter problem, Acta Appl. Math., 2010, 109(1), 239–255 http://dx.doi.org/10.1007/s10440-009-9450-4
  • [30] Marvan M., Pobořil M., A recursion operator for the intrinsic generalized sine-Gordon equation, J. Math. Sci. (N.Y.), 2008, 151(4), 3151–3158 http://dx.doi.org/10.1007/s10958-008-9024-4
  • [31] Marvan M., Sergyeyev A., Recursion operator for the stationary Nizhnik-Veselov-Novikov equation, J. Phys. A, 2003, 36(5), L87–L92 http://dx.doi.org/10.1088/0305-4470/36/5/102
  • [32] Marvan M., Sergyeyev A., Recursion operators for dispersionless integrable systems in any dimension, Inverse Problems, 2012, 28(2), #025011 http://dx.doi.org/10.1088/0266-5611/28/2/025011
  • [33] 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
  • [34] 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
  • [35] 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
  • [36] Olver P.J., Evolution equations possessing infinitely many symmetries, J. Math. Phys., 1977, 18(6), 1212–1215 http://dx.doi.org/10.1063/1.523393
  • [37] Olver P.J., Applications of Lie Groups to Differential Equations, 2nd ed., Grad. Texts in Math., 107, Springer, New York, 1993 http://dx.doi.org/10.1007/978-1-4612-4350-2
  • [38] Olver P.J., Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, 1995 http://dx.doi.org/10.1017/CBO9780511609565
  • [39] Papachristou C.J., Potential symmetries for self-dual gauge fields, Phys. Lett. A, 1990, 145(5), 250–254 http://dx.doi.org/10.1016/0375-9601(90)90359-V
  • [40] Papachristou C.J., Harrison B.K., Bäcklund-transformation-related recursion operators: application to self-dual Yang-Mills equation, J. Nonlinear Math. Phys., 2010, 17(1), 35–49 http://dx.doi.org/10.1142/S1402925110000581
  • [41] Pavlov M.V., Integrable hydrodynamic chains, J. Math. Phys., 2003, 44(9), 4134–4156 http://dx.doi.org/10.1063/1.1597946
  • [42] Sanders J.A., Wang J.P., On recursion operators, Phys. D, 2001, 149(1–2), 1–10 http://dx.doi.org/10.1016/S0167-2789(00)00188-3
  • [43] Santini P.M., Fokas A.S., Recursion operators and bi-Hamiltonian structures in multidimensions. I, Comm. Math. Phys., 1988, 115(3), 375–419 http://dx.doi.org/10.1007/BF01218017
  • [44] Sergyeyev A., On recursion operators and nonlocal symmetries of evolution equations, In: Proceedings of the Seminar on Differential Geometry, Opava, 2000, Math. Publ., 2, Silesian University at Opava, Opava, 2000, 159–173
  • [45] Stormark O., Lie’s Structural Approach to PDE Systems, Encyclopedia Math. Appl., 80, Cambridge University Press, Cambridge, 2000 http://dx.doi.org/10.1017/CBO9780511569456
  • [46] Vasilieva M.V., Structure of Infinite Lie Groups of Transformations, Moskov. Gosudartv. Ped. Inst., Moscow, 1972 (in Russian)
  • [47] Vinogradov A.M., Local symmetries and conservation laws, Acta Appl. Math., 1984, 2(1), 21–78 http://dx.doi.org/10.1007/BF01405491
  • [48] 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
  • [49] Wang J.P., A list of 1+1 dimensional integrable equations and their properties, J. Nonlinear Math. Phys., 2002, 9(Suppl. 1), 213–233 http://dx.doi.org/10.2991/jnmp.2002.9.s1.18
  • [50] Zakharov V.E., Konopel’chenko B.G., On the theory of recursion operator, Commun. Math. Phys., 1984, 94(4), 483–509 http://dx.doi.org/10.1007/BF01403883
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-013-0345-2
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.