Geodesic
Meta-Symplectic Geometry of $3^{\mathrm{rd}}$ Order Monge–Ampère Equations and their Characteristics
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper is a natural companion of [Alekseevsky D. V., Alonso Blanco R., Manno G., Pugliese F., Ann. Inst. Fourier (Grenoble) 62 (2012), 497–524, arXiv:1003.5177], generalising its perspectives and results to the context of third-order (2D) Monge–Ampère equations, by using the so-called “meta-symplectic structure” associated with the 8D prolongation $M^{(1)}$ of a 5D contact manifold $M$. We write down a geometric definition of a third-order Monge–Ampère equation in terms of a (class of) differential two-form on $M^{(1)}$. In particular, the equations corresponding to decomposable forms admit a simple description in terms of certain three-dimensional distributions, which are made from the characteristics of the original equations. We conclude the paper with a study of the intermediate integrals of these special Monge–Ampère equations, herewith called of Goursat type.
Keywords: Monge–Ampère equations; prolongations of contact manifolds; characteristics of PDEs; distributions on manifolds; third-order PDEs. 
@article{SIGMA_2016_12_a31,
     author = {Gianni Manno and Giovanni Moreno},
     title = {Meta-Symplectic {Geometry} of $3^{\mathrm{rd}}$ {Order} {Monge{\textendash}Amp\`ere} {Equations} and their {Characteristics}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2016},
     volume = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a31/}
}
TY  - JOUR
AU  - Gianni Manno
AU  - Giovanni Moreno
TI  - Meta-Symplectic Geometry of $3^{\mathrm{rd}}$ Order Monge–Ampère Equations and their Characteristics
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2016
VL  - 12
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a31/
LA  - en
ID  - SIGMA_2016_12_a31
ER  - 
%0 Journal Article
%A Gianni Manno
%A Giovanni Moreno
%T Meta-Symplectic Geometry of $3^{\mathrm{rd}}$ Order Monge–Ampère Equations and their Characteristics
%J Symmetry, integrability and geometry: methods and applications
%D 2016
%V 12
%U http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a31/
%G en
%F SIGMA_2016_12_a31
Gianni Manno; Giovanni Moreno. Meta-Symplectic Geometry of $3^{\mathrm{rd}}$ Order Monge–Ampère Equations and their Characteristics. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a31/

[1] Agafonov S. I., Ferapontov E. V., “Systems of conservation laws in the setting of the projective theory of congruences: reducible and linearly degenerate systems”, Differential Geom. Appl., 17 (2002), 153–173 | DOI | MR | Zbl

[2] Alekseevsky D. V., Alonso Blanco R., Manno G., Pugliese F., “Contact geometry of multidimensional Monge–Ampère equations: characteristics, intermediate integrals and solutions”, Ann. Inst. Fourier (Grenoble), 62 (2012), 497–524, arXiv: 1003.5177 | DOI | MR | Zbl

[3] Alekseevsky D. V., Alonso Blanco R., Manno G., Pugliese F., “Finding solutions of parabolic Monge–Ampère equations by using the geometry of sections of the contact distribution”, Differential Geom. Appl., 33, suppl. (2014), 144–161 | DOI | MR | Zbl

[4] Alonso Blanco R., Manno G., Pugliese F., “Contact relative differential invariants for non generic parabolic Monge–Ampère equations”, Acta Appl. Math., 101 (2008), 5–19 | DOI | MR | Zbl

[5] Alonso Blanco R., Manno G., Pugliese F., “Normal forms for Lagrangian distributions on 5-dimensional contact manifolds”, Differential Geom. Appl., 27 (2009), 212–229, arXiv: 0707.0683 | DOI | MR | Zbl

[6] Bächtold M. J., Fold-type solution singularities and charachteristic varieties of non-linear PDEs, Ph.D. Thesis, Universität Zürich, 2009 | DOI

[7] Bächtold M. J., Moreno G., “Remarks on non-maximal integral elements of the Cartan plane in jet spaces”, J. Geom. Phys., 85 (2014), 185–195, arXiv: 1208.5880 | DOI | MR

[8] 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, Translations of Mathematical Monographs, 182, Amer. Math. Soc., Providence, RI, 1999 | MR

[9] Boillat G., “Sur l'équation générale de Monge–Ampère à plusieurs variables”, C. R. Acad. Sci. Paris Sér. I Math., 313 (1991), 805–808 | MR | Zbl

[10] Boillat G., “Sur l'équation générale de Monge–Ampère d'ordre supérieur”, C. R. Acad. Sci. Paris Sér. I Math., 315 (1992), 1211–1214 | MR | Zbl

[11] Bryant R. L., Chern S. S., Gardner R. B., Goldschmidt H. L., Griffiths P. A., Exterior differential systems, Mathematical Sciences Research Institute Publications, 18, Springer-Verlag, New York, 1991 | DOI | MR | Zbl

[12] De Paris A., Vinogradov A. M., “Scalar differential invariants of symplectic Monge–Ampère equations”, Cent. Eur. J. Math., 9 (2011), 731–751, arXiv: 1102.0426 | DOI | MR | Zbl

[13] De Poi P., Mezzetti E., “Congruences of lines in ${\mathbb P}^5$, quadratic normality, and completely exceptional Monge–Ampère equations”, Geom. Dedicata, 131 (2008), 213–230, arXiv: 0710.5110 | DOI | MR | Zbl

[14] Dubrovin B., “Geometry of $2$D topological field theories”, Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., 1620, Springer, Berlin, 1996, 120–348, arXiv: hep-th/9407018 | DOI | MR | Zbl

[15] Ferapontov E. V., “Decomposition of higher-order equations of Monge–Ampère type”, Lett. Math. Phys., 62 (2002), 193–198, arXiv: nlin.SI/0205056 | DOI | MR | Zbl

[16] Goursat E., “Sur les équations du second ordre à $n$ variables analogues à l'équation de Monge–Ampère”, Bull. Soc. Math. France, 27 (1899), 1–34 | MR | Zbl

[17] Kolář I., Michor P. W., Slovák J., Natural operations in differential geometry, Springer-Verlag, Berlin, 1993 | DOI | MR | Zbl

[18] Krasil'shchik J., Verbovetsky A., “Geometry of jet spaces and integrable systems”, J. Geom. Phys., 61 (2011), 1633–1674, arXiv: 1002.0077 | DOI | MR | Zbl

[19] Kruglikov B., Lychagin V., “Geometry of differential equations”, Handbook of Global Analysis, Elsevier Sci. B. V., Amsterdam, 2008, 725–771 | DOI | MR | Zbl

[20] Kushner A., Lychagin V., Rubtsov V., Contact geometry and non-linear differential equations, Encyclopedia of Mathematics and its Applications, 101, Cambridge University Press, Cambridge, 2007 | MR

[21] Lax P. D., Milgram A. N., Contributions to the Theory of Partial Differential Equations, Annals of Mathematics Studies, 33, Princeton University Press, Princeton, N. J., 1954, Parabolic equations | MR

[22] Lychagin V. V., “Singularities of multivalued solutions of nonlinear differential equations, and nonlinear phenomena”, Acta Appl. Math., 3 (1985), 135–173 | DOI | MR | Zbl

[23] Lychagin V. V., J. Sov. Math., 51 (1990), Geometric theory of singularities of solutions of nonlinear differential equations | DOI

[24] Moreno G., Submanifolds in the Grassmannian of $n$-dimensional subspaces determined by a submanifold in the Grassmannian of $l$-dimensional subspaces, MathOverflow, , 2013 http://mathoverflow.net/q/138544

[25] Moreno G., “The geometry of the space of Cauchy data of nonlinear PDEs”, Cent. Eur. J. Math., 11 (2013), 1960–1981, arXiv: 1207.6290 | DOI | MR | Zbl

[26] Morimoto T., “Monge–Ampère equations viewed from contact geometry”, Symplectic Singularities and Geometry of Gauge Fields (Warsaw, 1995), Banach Center Publ., 39, Polish Acad. Sci., Warsaw, 1997, 105–121 | MR | Zbl

[27] Piccione P., Tausk D. V., “The single-leaf Frobenius theorem with applications”, Resenhas, 6 (2005), 337–381, arXiv: math.DG/0510555 | MR | Zbl

[28] Strachan I. A. B., “On the integrability of a third-order Monge–Ampère type equation”, Phys. Lett. A, 210 (1996), 267–272 | DOI | MR | Zbl

[29] Vitagliano L., “Characteristics, bicharacteristics and geometric singularities of solutions of PDEs”, Int. J. Geom. Methods Mod. Phys., 11 (2014), 1460039, 35 pp., arXiv: 1311.3477 | DOI | MR | Zbl