Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions. (English. French summary) Zbl 1253.53075
Multidimensional second-order PDEs (i.e., PDEs with \(n\) independent variables) are considered as hypersurfaces \(\varepsilon\) in the Lagrangian Grassmann bundle \(M^{(1)}\) over a \((2n+1)\)-dimensional contact manifold \((M,C)\).
The main result of the paper is the following theorem. Let \(\varepsilon\subset M^{(1)}\) be a second-order PDE.
Then \(\varepsilon\) is locally of the form \(\varepsilon_D\) for some \(n\)-dimensional distribution \(D\subset C\) if the following properties are satisfied.
(1) Its conformal metric is decomposable, i.e., \((g\varepsilon)_{m^1}= \ell_{m^1}\vee \ell_{m^1}'\), where \(\ell_{m^1}, \ell_{m^1}'\subset L_{m^1}\) are lines.
(2) If we let the point \(m^1\) vary along the fibre \(\varepsilon_m\), then the lines \(\ell_{m^1}\), \(\ell_{m^1}'\) fill two \(n\)-dimensional spaces \(D_{1m}\), \(D_{2m}\) of \(C_m\).
Furthermore, \(D_1\) and \(D_2\) are mutually orthogonal \(\omega= d\theta\) and \(\varepsilon= \varepsilon_{D_1}= \varepsilon_{D_2}\).
The main result of the paper is the following theorem. Let \(\varepsilon\subset M^{(1)}\) be a second-order PDE.
Then \(\varepsilon\) is locally of the form \(\varepsilon_D\) for some \(n\)-dimensional distribution \(D\subset C\) if the following properties are satisfied.
(1) Its conformal metric is decomposable, i.e., \((g\varepsilon)_{m^1}= \ell_{m^1}\vee \ell_{m^1}'\), where \(\ell_{m^1}, \ell_{m^1}'\subset L_{m^1}\) are lines.
(2) If we let the point \(m^1\) vary along the fibre \(\varepsilon_m\), then the lines \(\ell_{m^1}\), \(\ell_{m^1}'\) fill two \(n\)-dimensional spaces \(D_{1m}\), \(D_{2m}\) of \(C_m\).
Furthermore, \(D_1\) and \(D_2\) are mutually orthogonal \(\omega= d\theta\) and \(\varepsilon= \varepsilon_{D_1}= \varepsilon_{D_2}\).
Reviewer: Mihail Banaru (Smolensk)
MSC:
| 53D10 | Contact manifolds (general theory) |
| 35A30 | Geometric theory, characteristics, transformations in context of PDEs |
| 58A30 | Vector distributions (subbundles of the tangent bundles) |
| 58A17 | Pfaffian systems |
Keywords:
hypersurfaces of Lagrangian Grassmannians; contact geometry; subdistribution of a contact distribution; Monge-Ampère equations; characteristics; intermediate integralsReferences:
| [1] | Akivis, Maks; Goldberg, Vladislav, Conformal differential geometry and its generalizations (1996) · Zbl 0863.53002 |
| [2] | Alonso-Blanco, R. J.; Manno, G.; Pugliese, F., Normal forms for Lagrangian distributions on 5-dimensional contact manifolds, Differential Geom. Appl., 27, 2, 212-229 (2009) · Zbl 1162.53325 · doi:10.1016/j.difgeo.2008.06.019 |
| [3] | Alonso Blanco, R. J.; Manno, Gianni; Pugliese, Fabrizio, Contact relative differential invariants for non generic parabolic Monge-Ampère equations, Acta Appl. Math., 101, 1-3, 5-19 (2008) · Zbl 1154.35004 · doi:10.1007/s10440-008-9204-8 |
| [4] | Boillat, Guy, Le champ scalaire de Monge-Ampère, Norske Vid. Selsk. Forh. (Trondheim), 41, 78-81 (1968) · Zbl 0183.10402 |
| [5] | Boillat, Guy, Sur l’équation générale de Monge-Ampère à plusieurs variables, C. R. Acad. Sci. Paris Sér. I Math., 313, 11, 805-808 (1991) · Zbl 0753.35107 |
| [6] | Doubrov, B.; Ferapontov, E. V., On the integrability of symplectic Monge-Ampère equations, J. Geom. Phys., 60, 10, 1604-1616 (2010) · Zbl 1195.35109 · doi:10.1016/j.geomphys.2010.05.009 |
| [7] | Ferapontov, Evgeny Vladimirovich; Hadjikos, Lenos; Khusnutdinova, Karima Robertovna, Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian, Int. Math. Res. Not. IMRN, 3, 496-535 (2010) · Zbl 1203.35215 |
| [8] | Forsyth, Andrew Russell, Theory of differential equations. 1. Exact equations and Pfaff’s problem; 2, 3. Ordinary equations, not linear; 4. Ordinary linear equations; 5, 6. Partial differential equations (1959) · Zbl 0088.05802 |
| [9] | Goursat, E., Leçons sur l’intégration des équations aux dérivées partielles du second ordre, 1 (1890) · JFM 22.0348.01 |
| [10] | Goursat, E., Sur les équations du second ordre à \(n\) variables analogues à l’équation de Monge-Ampère, Bull. Soc. Math. France, 27, 1-34 (1899) · JFM 30.0326.01 |
| [11] | Griffiths, Phillip; Harris, Joseph, Principles of algebraic geometry (1978) · Zbl 0836.14001 |
| [12] | Kushner, Alexei, Differential equations: geometry, symmetries and integrability, 5, 223-256 (2009) · Zbl 1233.35005 |
| [13] | Kushner, Alexei; Lychagin, Valentin; Rubtsov, Vladimir, Contact geometry and non-linear differential equations, 101 (2007) · Zbl 1122.53044 |
| [14] | Lax, P. D.; Milgram, A. N., Contributions to the theory of partial differential equations, 167-190 (1954) · Zbl 0128.09302 |
| [15] | Lyčagin, V., Contact geometry and second-order nonlinear differential equations, Uspekhi Mat. Nauk, 34, 1-205, 137-165 (1979) · Zbl 0405.58003 |
| [16] | Machida, Y.; Morimoto, T., On decomposable Monge-Ampère equations, Lobachevskii J. Math., 3, 185-196 (electronic) (1999) · Zbl 0957.35010 |
| [17] | Morimoto, Tohru, Symplectic singularities and geometry of gauge fields (Warsaw, 1995), 39, 105-121 (1997) · Zbl 0879.35008 |
| [18] | Muñoz Díaz, J., Ecuaciones diferenciales I (1982) |
| [19] | Petrovski, I. G., Lectures on partial differential equations (1991) |
| [20] | Ruggeri, Tommaso, Su una naturale estensione a tre variabili dell’equazione di Monge-Ampère, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 55, 445-449 (1974) (1973) · Zbl 0294.35013 |
| [21] | Valiron, Georges, The classical differential geometry of curves and surfaces (1986) · Zbl 0611.53001 |
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