×

zbMATH — the first resource for mathematics

Differential invariants of generic hyperbolic Monge-Ampère equations. (English) Zbl 1129.58015
A hyperbolic Monge-Ampére equation is interpreted as a pair of 2-dimensional, skew-orthogonal, non-Lagrangian subdistributions of the contact distribution on a 5-dimensional contact manifold, see also [V. V. Lychagin, Russ. Math. Surv. 34, No. 1, 149–180 (1979); translation from Usp. Mat. Nauk 34, No.1(205), 137-165 (1979; Zbl 0405.58003); A. G. Kushner, Dokl. Math. 58, No. 1, 103–104 (1998); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 361, No. 5, 595–596 (1998; Zbl 0958.58002); O. P. Tchii, Lobachevskii J. Math. 4, 109–162, electronic only (1999; Zbl 0938.35011); V. B. Levenshtam, Dokl. Math. 72, No. 3, 872–875 (2005); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 405, No. 2, 169–172 (2005; Zbl 1133.34344)]. Differential invariants are easily visible due to the existence of bicharacteristics. The equivalence problem is solved.

MSC:
58J70 Invariance and symmetry properties for PDEs on manifolds
58J45 Hyperbolic equations on manifolds
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] D.V. Alekseevsky, A.M. Vinogradov and V.V. Lychagin: “Basic ideas and concepts of differential geometry”, In: Geometry, I, Encyclopaedia Math. Sci., Vol. 28, Springer, Berlin, 1991, pp. 1-264.
[2] A. Frölicher and A. Nijenhuis: “Theory of vector valued differential forms. Part I: Derivations in the graded ring of differential forms,” Indag. Math., Vol. 18, (1956), pp. 338-359.
[3] P. Hartman and A. Wintner: “On hyperbolic partial differential equations”, Am. J. Math., Vol. 74, (1952), pp. 834-864. http://dx.doi.org/10.2307/2372229 · Zbl 0048.33302
[4] I.S. Krasil’shchik, V.V. Lychagin and A.M. Vinogradov: Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach, New York, 1986.
[5] I.S. Krasil’shchik and A.M. Vinogradov (Ed.): Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Translations of Mathematical Monographs, Vol. 182, American Mathematical Society, Providence RI, 1999.
[6] B.S. Kruglikov: “Some classificational problems in four-dimensional geometry: distributions, almost complex structures and the generalized Monge-Ampère equations”, Math. Sbornik, Vol. 189(11), (1998), pp. 61-74 (in Russian); English translation in Sb. Math., Vol. 186(11-12), (1998), pp. 1643-1656; e-print: http://xxx.lanl.gov/abs/dg-ga/9611005.
[7] B.S. Kruglikov: “Symplectic and contact Lie algebras with application to the Monge-Ampère equations”, Trudy Mat. Inst. Steklova, Vol. 221, (1998), pp. 232-246 (in Russian); English translation in Proc. Steklov Math. Inst., Vol. 221(2), (1998), pp. 221-235; e-print: http://xxx.lanl.gov/abs/dg-ga/9709004 · Zbl 1032.53067
[8] B.S. Kruglikov: “Classification of Monge-Ampère equations with two variables”, In: Geometry and Topology of Caustics — CAUSTICS’ 98 (Warsaw), Banach Center Publications, Vol. 50, Polish Acad. Sci., Warsaw, 1999, pp. 179-194.
[9] A. Kushner: “Monge-Ampère equations and e-structures”, Dokl. Akad. Nauk, Vol. 361(5), (1998), pp. 595-596. · Zbl 0958.58002
[10] H. Lewy: “Über das Anfangswertproblem bei einer hyperbolischen nichtlinearen partiellen Differentialgleichung zweiter Ordnung mit zwei unabhängigen Verànderlichen”, Math. Annalen, Vol. 98, (1928), pp. 179-191. http://dx.doi.org/10.1007/BF01451588 · JFM 53.0473.15
[11] V.V. Lychagin: “Contact geometry and non-linear second order differential equations”, Russian Math. Surveys, Vol. 34, (1979), pp. 149-180. http://dx.doi.org/10.1070/RM1979v034n01ABEH002873 · Zbl 0427.58002
[12] V.V. Lychagin: Lectures on Geometry of Differential Equations, Universita “La Sapienza”, Roma, 1992.
[13] V.V. Lychagin and V.N. Rubtsov: “Local classification of Monge-Ampere equations”, Soviet Math. Doklady, Vol. 272(1), (1983), pp. 34-38. · Zbl 0555.35016
[14] V.V. Lychagin and V.N. Rubtsov: “On the Sophus Lie theorems for Monge-Ampere equations”, Belorussian Acad. Sci. Doklady, Vol. 27(5, (1983), pp. 396-398 · Zbl 0526.58002
[15] V.V. Lychagin, V.N. Rubtsov and I.V. Chekalov: “A classification of Monge-Ampere equations”, Ann. Sc. Ecole Norm. Sup., Vol. 4(26), (1993), pp. 281-308. · Zbl 0789.58078
[16] M. Marvan, A.M. Vinogradov and V.A. Yumaguzhin: “Differential invariants of generic hyperbolic Monge-Ampère equations”, Russian Acad. Sci. Dokl. Math., Vol. 405, (2005), pp. 299-301 (in Russian); English translation in: Doklady Mathematics, Vol. 72, (2005), pp. 883-885.
[17] M. Matsuda: “Two methods of integrating Monge-Ampère’s equations”, Trans. Amer. Math. Soc., Vol. 150, (1970), pp. 327-343. http://dx.doi.org/10.2307/1995496 · Zbl 0201.42101
[18] M. Matsuda: “Two methods of integrating Monge-Ampère’s equations. II”, Trans. Amer. Math. Soc., Vol. 166, (1972), pp. 371-386. http://dx.doi.org/10.2307/1996056 · Zbl 0235.35002
[19] T. Morimoto: “La géométrie des équations de Monge-Ampère”, C. R. Acad. Sci., Paris, Vol. 289, (1979), pp. A-25-A-28.
[20] T. Morimoto: “Monge-Ampère equations viewed from contact geometry”, In: Symplectic Singularities and Geometry of Gauge Fields (Warsaw, 1995), Banach Center Publ., Vol. 39, Polish Acad. Sci., Warsaw, 1997, pp. 105-121.
[21] O.P. Tchij: “Contact geometry of hyperbolic Monge-Ampère eqquations”, Lobachevskii Journal of Mathematics, Vol. 4, (1999), pp. 109-162. · Zbl 0938.35011
[22] D.V. Tunitsky: “On the global solvability of hyperbolic Monge-Ampère equations”, Izv. Ross. Akad. Nauk Ser. Mat., Vol. 61(5), (1997), pp. 177-224 (in Russian); English translation in: Izv. Math, Vol. 61(5), (1997), pp. 1069-1111.
[23] D.V. Tunitsky: “Monge-Ampère equations and functors of characteristic connection”, Izv. RAN, Ser. Math., Vol. 65(6), (2001), pp. 173-222.
[24] A.M. Vinogradov: “Scalar differential invariants, diffieties and characteristic classes”, In: Mechanics, Analysis and Geometry: 200 Years after Lagrange, M. Francaviglia, North-Holland, 1991, pp. 379-414. · Zbl 0735.57012
[25] A.M. Vinogradov and V.A. Yumaguzhin: “Differential invariants of webs on 2-dimensional manifolds”, Mat. Zametki, Vol. 48(1), (1990), pp. 46-68 (in Russian). · Zbl 0714.53019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.