×

zbMATH — the first resource for mathematics

Rank two prolongations of second-order PDE and geometric singular solutions. (English) Zbl 1301.58002
The authors prove that the fibers of the first prolongation of a second order hyperbolic, parabolic or elliptic (respectively) 2nd order PDE are tori, pinched tori (with one singular point) or spheres (respectively). They compute out the tangent nilpotent graded Lie algebras that arise as symbol algebras in Tanaka’s theory for any equation of each of the three classes. They give examples of singular solutions of an equation of each type, so that the solution is a smooth surface in a suitable prolongation. The authors point out that the torus fibers in the hyperbolic case were described in a paper of R. Bryant, P. Griffiths and L. Hsu [Sel. Math., New Ser. 1, No. 1, 21–112 (1995; Zbl 0853.58102)]; the reader might also enjoy reading Eendebak’s work on tori in first order hyperbolic systems for two functions of two variables.
MSC:
58A15 Exterior differential systems (Cartan theory)
58A17 Pfaffian systems
58J05 Elliptic equations on manifolds, general theory
35R01 PDEs on manifolds
35J15 Second-order elliptic equations
35K10 Second-order parabolic equations
35L10 Second-order hyperbolic equations
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] R. Bryant, S. S. Chern, R. Gardner, H. Goldscmidt and P. Griffiths, Exterior Differential Systems , MSRI Publ. vol. 18 , Springer Verlag, Berlin (1991). · Zbl 0726.58002
[2] R. Bryant, P. Griffiths and L. Hsu, Hyperbolic Exterior Differential Systems and their Conservation Laws, Part I, Selecta Math , New Series, Vol. 1 , No. 1 (1995). · Zbl 0853.58102
[3] B. Kruglikov and V. Lychagin, Geometry of differential equations, Handbook of global analysis , 1214 , Elsevier Sci. B. V., Amsterdam, (2008), 725-771. · Zbl 1236.58039
[4] V. Lychagin, Geometric theory of singularities of solutions of nonlinear differential equations (Russian), Translated in J. Soviet Math. 51 (1990), no 6, 2735-2757. · Zbl 1267.35056
[5] V. Lychagin, Differential equations on two-dimensional manifolds (Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 1992, no. 5, 25-35.
[6] T. Morimoto, Geometric structures on filtered manifolds, Hokkaido Math. J. 22 (1993), 263-347. · Zbl 0801.53019
[7] R. Montgomery and M. Zhitomirskii, Geometric approach to Goursat flags, Ann.Inst. H.PoincarĂ©-AN 18 (2001), 459-493. · Zbl 1013.58004
[8] T. Noda and K. Shibuya, Second order type-changing PDE for a scalar function on a plane, Osaka J. Math. 49 (2012), 101-124. · Zbl 1246.35137
[9] K. Shibuya, On the prolongation of 2-jet space of 2 independent and 1 dependent variables, Hokkaido Math. J. 38 (2009), 587-626. · Zbl 1178.53028
[10] K. Shibuya, Rank 4 distributions of type hyperbolic, parabolic and elliptic, in preparation.
[11] K. Shibuya and K. Yamaguchi, Drapeau theorem for differential systems, Diff Geom and its Appl 27 (2009), 793-808. · Zbl 1182.58003
[12] N. Tanaka, On differential systems, graded Lie algebras and pseudo-groups, J. Math. Kyoto. Univ. 10 (1970), 1-82. · Zbl 0206.50503
[13] N. Tanaka, On generalized graded Lie algebras and geometric structures I, J. Math. Soc. Japan 19 (1967), 215-254. · Zbl 0165.56002
[14] N. Tanaka On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J. 8 (1979), no. 1, 23-84. · Zbl 0409.17013
[15] K. Yamaguchi, Contact geometry of higher order, Japan. J. Math., 8 (1982), 109-176. · Zbl 0548.58002
[16] K. Yamaguchi, Geometrization of jet bundles, Hokkaido Math. J. 12 (1983), 27-40. · Zbl 0561.58002
[17] K. Yamaguchi, Differential systems associated with simple graded Lie algebras, Advanced Studies in Pure Math. 22 (1993), 413-494. · Zbl 0812.17018
[18] K. Yamaguchi, Contact geometry of second order I, Differential Equations -Geometry, Symmetries and Integrability . The Abel symposium 2008, Abel symposia 5 , 2009, 335-386. · Zbl 1187.58003
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.