Geodesic
On some categories of Monge-Ampère systems of equations
Sbornik. Mathematics, Tome 200 (2009) no. 11, pp. 1681-1714 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The paper looks at differential-geometric structures associated with Monge-Ampère systems of equations on manifolds and how they can be applied in the reduction of these equations. The category of Monge-Ampère systems of equations is investigated; its morphisms are changes of independent and dependent variables. Some subcategories of this category are also studied. The main emphasis is on subcategories of equations of locally equivalent triangular and semitriangular systems, systems that are linear with respect to derivatives (semilinear systems), systems with constant coefficients, and also complete differential systems. Tests, which can be verified effectively, are proved; these make it possible to establish whether a given system of Monge-Ampère equations belongs to the subcategories listed above. As corollaries, conditions for a Monge-Ampère system to be locally reducible to a single first- or second-order equation are obtained. Bibliography: 14 titles.
Keywords: Monge-Ampere systems on manifolds, equivalence of Monge-Ampere systems, linearization of Monge-Ampere systems. 
@article{SM_2009_200_11_a4,
     author = {D. V. Tunitsky},
     title = {On some categories of {Monge-Amp\`ere} systems of equations},
     journal = {Sbornik. Mathematics},
     pages = {1681--1714},
     year = {2009},
     volume = {200},
     number = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2009_200_11_a4/}
}
TY  - JOUR
AU  - D. V. Tunitsky
TI  - On some categories of Monge-Ampère systems of equations
JO  - Sbornik. Mathematics
PY  - 2009
SP  - 1681
EP  - 1714
VL  - 200
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/SM_2009_200_11_a4/
LA  - en
ID  - SM_2009_200_11_a4
ER  - 
%0 Journal Article
%A D. V. Tunitsky
%T On some categories of Monge-Ampère systems of equations
%J Sbornik. Mathematics
%D 2009
%P 1681-1714
%V 200
%N 11
%U http://geodesic.mathdoc.fr/item/SM_2009_200_11_a4/
%G en
%F SM_2009_200_11_a4
D. V. Tunitsky. On some categories of Monge-Ampère systems of equations. Sbornik. Mathematics, Tome 200 (2009) no. 11, pp. 1681-1714. http://geodesic.mathdoc.fr/item/SM_2009_200_11_a4/

[1] R. O. Wells, jr., Differential analysis on complex manifolds, Prentice-Hall, Englewood Cliffs, NJ, 1973 | MR | MR | Zbl

[2] D. V. Tunitskii, “Hyperbolic Monge–Ampère systems”, Sb. Math., 197:8 (2006), 1223–1258 | DOI | MR | Zbl

[3] I. Bucur, A. Deleanu, Introduction to the theory of categories and functors, Wiley, London–New York–Sydney, 1968 | MR | MR | Zbl | Zbl

[4] S. I. Bilchev, “Sistemy iz dvukh differentsialnykh uravnenii s chastnymi proizvodnymi pervogo poryadka (lokalnaya teoriya)”, Izv. vuzov. Matem., 1970, no. 3, 14–21 | MR | Zbl

[5] A. M. Vasilev, Teoriya differentsialno-geometricheskikh struktur, Izd-vo Mosk. un-ta, M., 1986 | MR | Zbl

[6] T. Morimoto, “La géométrie des équations de Monge–Ampère”, C. R. Acad. Sci. Paris Sér. A-B, 289:1 (1979), 25–28 | MR | Zbl

[7] D. V. Tunitskii, “On the contact linearization of Monge–Ampere equations”, Izv. Math., 60:2 (1996), 425–451 | DOI | MR | Zbl

[8] D. V. Tunitskii, “Equivalence and characteristic connections of the Monge–Ampere equations”, Sb. Math., 188:5 (1997), 771–797 | DOI | MR | Zbl

[9] R. Courant, Methods of mathematical physics. Vol. II: Partial differential equations, Intersci. Publ., New York–London, 1962 | MR | MR | Zbl | Zbl

[10] O. K. Fossum, On classification of a nonlinear first order system of Jacobi type, Preprints of the Sophus Lie seminar, University of Tromso, 2002; http://www.math.uit.no/seminar/preprints/02-12-of.pdf

[11] V. Lychagin, Lectures on geometry of differential equations, Part II. Roma, 1992

[12] A. Kushner, V. Lychagin, V. Rubtsov, Contact geometry and nonlinear differential equations, Encyclopedia Math. Appl., 101, Cambridge Univ. Press, Cambridge, 2007 | MR | Zbl

[13] N. E. Hurt, Geometric quantization in action. Applications of harmonic analysis in quantum statistical mechanics and quantum field theory, Math. Appl. (East European Ser.), 8, Reidel Publ., Dordrecht–Boston, MA, 1983 | MR | MR | Zbl

[14] É. Cartan, Les systèmes différentiels extérieurs et leurs applications géométriques, Hermann, Paris, 1945 | MR | Zbl