@article{SIGMA_2005_1_a5,
author = {O. I. Morozov},
title = {Structure of {Symmetry} {Groups} via {Cartan's} {Method:} {Survey} of {Four} {Approaches}},
journal = {Symmetry, integrability and geometry: methods and applications},
year = {2005},
volume = {1},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SIGMA_2005_1_a5/}
}
O. I. Morozov. Structure of Symmetry Groups via Cartan's Method: Survey of Four Approaches. Symmetry, integrability and geometry: methods and applications, Tome 1 (2005). http://geodesic.mathdoc.fr/item/SIGMA_2005_1_a5/
[1] Bryant R. L., Griffiths Ph. A., “Characteristic cohomology of differential systems. II. Conservation laws for a class of parabolic equations”, Duke Math. J., 78 (1995), 531–676 | DOI | MR | Zbl
[2] Bryant R., Griffiths Ph., Hsu L., “Hyperbolic exterior differential systems and their conservation laws. I, II”, Selecta Math., New Ser., 1:1 (1995), 21–112 ; 265–323 | DOI | MR | Zbl | DOI | MR | Zbl
[3] Bryant R., Griffiths Ph., Hsu L., “Toward a geometry of differential equations”, Geometry, Topology Physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Cambridge, Internat. Press, 1995, 1–76 | MR
[4] Bluman G. W., Kumei S., Symmetries and differential equations, Springer, New York, 1989 | MR | Zbl
[5] Cartan É., “Sur la structure des groupes infinis de transformations”, {ØE}uvres Complètes, Partie II, Vol. 2, Gauthier-Villars, Paris, 1953, 571–714
[6] Cartan É., “Les sous-groupes des groupes continus de transformations”, {ØE}uvres Complètes, Partie II, Vol. 2, Gauthier-Villars, Paris, 1953, 719–856
[7] Cartan É., “La structure des groupes infinis”, {ØE}uvres Complètes, Partie II, Vol. 2, Gauthier-Villars, Paris, 1953, 1335–1384 | MR
[8] Cartan É., “Les problèmes d'équivalence”, {ØE}uvres Complètes, Partie II, Vol. 2, Gauthier-Villars, Paris, 1953, 1311–1334
[9] Cheh J., Olver P. J., Pohjanpelto J., “Maurer–Cartan equations for Lie symmetry pseudo-groups of differential equations”, J. Math. Phys., 46 (2005), 023504, 11 pp., ages | DOI | MR
[10] Clelland J. N., “Geometry of conservation laws for a class of parabolic partial differential equations”, Selecta Math., New Ser., 3:1 (1997), 1–77 | DOI | MR | Zbl
[11] Fels M., “The equivalence problem for systems of second order ordinary differential equations”, Proc. London Math. Soc., 71 (1995), 221–240 | DOI | MR | Zbl
[12] Fels M., Olver P. J., “Moving coframes. I. A practical algorithm”, Acta. Appl. Math., 51 (1998), 161–213 | DOI | MR
[13] Foltinek K., Quasilinear third-order scalar evolution equations and their conservation laws, Ph.D. Thesis, Duke University, 1996
[14] Flanders H., Differential forms with applications to the physical sciences, Academic Press, New York–London, 1963 | MR | Zbl
[15] Gardner R. B., The method of equivalence and its applications, CBMS–NSF Regional Conference Series in Applied Math., 58, SIAM, Philadelphia, 1989 | MR | Zbl
[16] Golovin S. V., “Group foliation of Euler equations in nonstationary rotationally symmetrical case”, Proceedings of Fifth International Conference “Symmetry in Nonlinear Mathematical Physics”, Part 1 (June 23–29, 2003, Kyiv), Proceedings of Institute of Mathematics, 50, eds. A. G. Nikitin, V. M. Boyko, R. O. Popovych and I. A. Yehorchenko, Kyiv, 2004, 110–117 | MR | Zbl
[17] Grissom C., Thompson G., Wilkens G., “Linearization of second order odes via Cartan's equivalence method”, J. Differential Equations, 77:1 (1989), 1–15 | DOI | MR | Zbl
[18] Dryuma V., On the Riemannian and Einstein–Weyl Geometry in Theory of the Second Order Ordinary Differential Equations, arXiv:gr-qc/0104095
[19] Hsu L., Kamran N., “Classification of second order ordinary differential equations admitting Lie groups of fiber-preserving symmetries”, Proc. London Math. Soc., Ser. 3, 58:2 (1989), 387–416 | DOI | MR | Zbl
[20] Hunter J. K., Saxton R., “Dynamics of director fields”, SIAM J. Appl. Math., 51:6 (1991), 1498–1521 | DOI | MR | Zbl
[21] Ibragimov N. H., Transformation groups applied to mathematical physics, Reidel, Dordrecht, 1985 | MR | Zbl
[22] Ibragimov N. H., “Invariants of hyperbolic equations: solution to Laplace's problem”, J. Appl. Mech. Tech. Phys., 45:2 (2004), 11–21 | DOI | MR | Zbl
[23] Johnpillai I. K., Mahomed F. M., Wafo Soh C., “Basis of joint invariants for (1+1)-linear hyperbolic equations”, J. Nonlinear Math. Phys., 9, Suppl. 2 (2002), 49–59 | DOI | MR
[24] Kamran N., Lamb K. G., Shadwick W. F., “The local equivalence problem for $d^2y/dx^2=F(x,y,dy/dx)$ and the Painlevé transcendents”, J. Diff. Geometry, 22:2 (1985), 139–150 | MR | Zbl
[25] Kamran N., Shadwick W. F., “A differential geometric characterization of the first Painlevé transcendents”, Mathematische Annalen, 279:1 (1987), 117–123 | DOI | MR | Zbl
[26] Kamran N., Shadwick W. F., “Équivalence locale des équations aux dérivées partielles quasi lineaires du deixième ordre et pseudo-groupes infinis”, Comptes Rendus Acad. Sci. (Paris). Ser. I, 303 (1986), 555–558 | MR | Zbl
[27] 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$^\circ$ (2), 45:7 (1989), 121 | MR | Zbl
[28] Krasil'shchik I. S., Lychagin V. V., Vinogradov A. M., Geometry of jet spaces and nonlinear partial differential equations, Gordon and Breach, New York, 1986 | MR
[29] Laplace P. S., “Recherches sur le calcul intégral aux différences partielles”, Mémoires de l'Academie Royale de Sciences de Paris, 1777, 341–401; Reprinted in: {ØE}uvres Complètes, Vol. 9, Gauthier-Villars, Paris, 1893, 3–68; English translation: New York, 1966
[30] Lie S., Gesammelte Abhandlungen, V. 1–6, Teubner, Leipzig, 1922–1937 | Zbl
[31] Lisle I. G., Reid G. J., Boulton A., “Algorithmic determination of structure of infinite Lie pseudogroups of symmetries of PDEs”, Proc. ISSAC'95, ACM Press, New York, 1995, 1–6 | Zbl
[32] Lisle I. G., Reid G. J., “Geometry and structure of Lie pseudogroups from infinitesimal defining equations”, J. Symb. Comp., 26 (1998), 355–379 | DOI | MR | Zbl
[33] Morozov O. I., “Moving coframes and symmetries of differential equations”, J. Phys. A: Math. Gen., 35:12, 2965–2977 | DOI | MR | Zbl
[34] Morozov O. I., “Symmetries of differential equations and Cartan's equivalence method”, Proceedings of Fifth International Conference “Symmetry in Nonlinear Mathematical Physics”, Part 1 (June 23–29, 2003, Kyiv), Proceedings of Institute of Mathematics, 50, eds. A. G. Nikitin, V. M. Boyko, R. O. Popovych and I. A. Yehorchenko, Kyiv, 2004, 196–203 | MR | Zbl
[35] Morozov O. I., “The contact equivalence problem for linear hyperbolic equations”, Proceedings of I. G. Petrovsky's Seminar, 25, 2006, 119–142 ; arXiv:math-ph/0406004 | MR | Zbl
[36] Olver P. J., Applications of Lie groups to differential equations, Springer, New York, 1986 | MR
[37] Olver P. J., Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995 | MR | Zbl
[38] Olver P. J., Pohjanpelto J., Moving frames for pseudo-groups. I. The Maurer–Cartan forms, Preprint, University of Minnesota, 2003 | MR
[39] Olver P. J., Pohjanpelto J., Moving frames for pseudo-groups. II. Differential invariants for submanifolds, Preprint, University of Minnesota, 2003
[40] Olver P. J., Rosenau Ph., “Tri-Hamiltonian duality between solitons and solitary wave solutions having compact support”, Phys. Rev. E, 53 (1996), 1900–1906 | DOI | MR
[41] Ovsiannikov L. V., “Group properties of the Chaplygin equation”, J. Appl. Mech. Tech. Phys., 3 (1960), 126–145
[42] Ovsiannikov L. V., Group analysis of differential equations, Academic Press, New York, 1982 | MR | Zbl
[43] Pavlov M. V., “The Calogero equation and Liouville type equations”, Theor. and Math. Phys., 128:1 (2001), 927–932 ; arXiv:nlin.SI/0101034 | DOI | MR | Zbl
[44] Reyes E. G., “The soliton content of the Camassa–Holm and Hunter–Saxton equations”, Proceedings of Fourth International Conference “Symmetry in Nonlinear Mathematical Physics”, Part 1 (July 9–15, 2001, Kyiv), Proceedings of Institute of Mathematics, 43, eds. A. G. Nikitin, V. M. Boyko and R. O. Popovych, Kyiv, 2002, 201–208 | MR | Zbl
[45] Surovikhin K. P., “Cartan's exterior forms and computation of the basic group admitted by a given system of differential equations”, Moscow Univ. Bulletin, Ser. Math., Mech., 6 (1965), 70–81 (in Russian) | MR
[46] Tod K. P., “Einstein–Weil spaces and third order differential equations”, J. Math. Phys., 41 (2000), 5572–5581 | DOI | MR | Zbl