Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions

Dmitri V. Alekseevsky[1]; Ricardo Alonso-Blanco[2]; Gianni Manno[3]; Fabrizio Pugliese[4]

  • [1] University of Edinburgh School of Mathematics and Maxwell Institute for Mathematical Sciences The Kings Buildings, JCMB Mayfield Road Edinburgh, EH9 3JZ (UK)
  • [2] Universidad de Salamanca Departamento de Matemáticas plaza de la Merced 1-4 37008 Salamanca (Spain)
  • [3] Università di Milano-Bicocca Dipartimento di Matematica e Applicazioni via Cozzi 53 20125 Milano (Italy)
  • [4] Università di Salerno Dipartimento di Matematica via Ponte don Melillo 84084 Fisciano (Italy)

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 2, page 497-524
  • ISSN: 0373-0956

Abstract

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We study the geometry of multidimensional scalar 2 n d order PDEs (i.e. PDEs with n independent variables), viewed as hypersurfaces ℰ in the Lagrangian Grassmann bundle M ( 1 ) over a ( 2 n + 1 ) -dimensional contact manifold ( M , 𝒞 ) . We develop the theory of characteristics of ℰ in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of ℰ . After specializing such results to general Monge-Ampère equations (MAEs), we focus our attention to MAEs of type introduced by Goursat in 1899: det ∂ 2 f ∂ x i ∂ x j - b i j x , f , ∇ f = 0 . We show that any MAE of this class is associated with an n -dimensional subdistribution 𝒟 of the contact distribution 𝒞 , and viceversa. We characterize these Goursat-type equations together with their intermediate integrals in terms of their characteristics and give a criterion of local contact equivalence. Finally, we develop a method to solve Cauchy problems for this kind of equations.

How to cite

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Alekseevsky, Dmitri V., et al. "Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions." Annales de l’institut Fourier 62.2 (2012): 497-524. <http://eudml.org/doc/251116>.

@article{Alekseevsky2012,
abstract = {We study the geometry of multidimensional scalar $2^\{nd\}$ order PDEs (i.e. PDEs with $n$ independent variables), viewed as hypersurfaces $\mathcal\{E\}$ in the Lagrangian Grassmann bundle $M^\{(1)\}$ over a $(2n+1)$-dimensional contact manifold $(M,\mathcal\{C\})$. We develop the theory of characteristics of $\mathcal\{E\}$ in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of $\mathcal\{E\}$. After specializing such results to general Monge-Ampère equations (MAEs), we focus our attention to MAEs of type introduced by Goursat in 1899:\[ \det \Bigl \Vert \frac\{\partial ^\{2\} f\}\{\partial x^\{i\}\partial x^\{j\}\}-b\_\{ij\}\left( x,f,\nabla f\right) \Bigr \Vert =0. \]We show that any MAE of this class is associated with an $n$-dimensional subdistribution $\mathcal\{D\}$ of the contact distribution $\mathcal\{C\}$, and viceversa. We characterize these Goursat-type equations together with their intermediate integrals in terms of their characteristics and give a criterion of local contact equivalence. Finally, we develop a method to solve Cauchy problems for this kind of equations.},
affiliation = {University of Edinburgh School of Mathematics and Maxwell Institute for Mathematical Sciences The Kings Buildings, JCMB Mayfield Road Edinburgh, EH9 3JZ (UK); Universidad de Salamanca Departamento de Matemáticas plaza de la Merced 1-4 37008 Salamanca (Spain); Università di Milano-Bicocca Dipartimento di Matematica e Applicazioni via Cozzi 53 20125 Milano (Italy); Università di Salerno Dipartimento di Matematica via Ponte don Melillo 84084 Fisciano (Italy)},
author = {Alekseevsky, Dmitri V., Alonso-Blanco, Ricardo, Manno, Gianni, Pugliese, Fabrizio},
journal = {Annales de l’institut Fourier},
keywords = {Hypersurfaces of Lagrangian Grassmannians; contact geometry; subdistributions of a contact distribution; Monge-Ampère equations; characteristics; intermediate integrals; hypersurfaces of Lagrangian Grassmannians; subdistribution of a contact distribution},
language = {eng},
number = {2},
pages = {497-524},
publisher = {Association des Annales de l’institut Fourier},
title = {Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions},
url = {http://eudml.org/doc/251116},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Alekseevsky, Dmitri V.
AU - Alonso-Blanco, Ricardo
AU - Manno, Gianni
AU - Pugliese, Fabrizio
TI - Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 2
SP - 497
EP - 524
AB - We study the geometry of multidimensional scalar $2^{nd}$ order PDEs (i.e. PDEs with $n$ independent variables), viewed as hypersurfaces $\mathcal{E}$ in the Lagrangian Grassmann bundle $M^{(1)}$ over a $(2n+1)$-dimensional contact manifold $(M,\mathcal{C})$. We develop the theory of characteristics of $\mathcal{E}$ in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of $\mathcal{E}$. After specializing such results to general Monge-Ampère equations (MAEs), we focus our attention to MAEs of type introduced by Goursat in 1899:\[ \det \Bigl \Vert \frac{\partial ^{2} f}{\partial x^{i}\partial x^{j}}-b_{ij}\left( x,f,\nabla f\right) \Bigr \Vert =0. \]We show that any MAE of this class is associated with an $n$-dimensional subdistribution $\mathcal{D}$ of the contact distribution $\mathcal{C}$, and viceversa. We characterize these Goursat-type equations together with their intermediate integrals in terms of their characteristics and give a criterion of local contact equivalence. Finally, we develop a method to solve Cauchy problems for this kind of equations.
LA - eng
KW - Hypersurfaces of Lagrangian Grassmannians; contact geometry; subdistributions of a contact distribution; Monge-Ampère equations; characteristics; intermediate integrals; hypersurfaces of Lagrangian Grassmannians; subdistribution of a contact distribution
UR - http://eudml.org/doc/251116
ER -

References

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  1. Maks Akivis, Vladislav Goldberg, Conformal differential geometry and its generalizations, (1996), John Wiley & Sons Inc., New York Zbl0863.53002MR1406793
  2. R.J. Alonso-Blanco, G. Manno, F. Pugliese, Normal forms for Lagrangian distributions on 5-dimensional contact manifolds, Differential Geom. Appl. 27 (2009), 212-229 Zbl1162.53325MR2503974
  3. R.J. Alonso Blanco, Gianni Manno, Fabrizio Pugliese, Contact relative differential invariants for non generic parabolic Monge-Ampère equations, Acta Appl. Math. 101 (2008), 5-19 Zbl1154.35004MR2383541
  4. Guy Boillat, Le champ scalaire de Monge-Ampère, Norske Vid. Selsk. Forh. (Trondheim) 41 (1968), 78-81 Zbl0183.10402MR240472
  5. Guy Boillat, Sur l’équation générale de Monge-Ampère à plusieurs variables, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), 805-808 Zbl0753.35107MR1139843
  6. B. Doubrov, E. V. Ferapontov, On the integrability of symplectic Monge-Ampère equations, J. Geom. Phys. 60 (2010), 1604-1616 Zbl1195.35109MR2661158
  7. Evgeny Vladimirovich Ferapontov, Lenos Hadjikos, Karima Robertovna Khusnutdinova, Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian, Int. Math. Res. Not. IMRN (2010), 496-535 Zbl1203.35215MR2587572
  8. Andrew Russell Forsyth, 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), Dover Publications Inc., New York Zbl0088.05802MR123757
  9. E. Goursat, Leçons sur l’intégration des équations aux dérivées partielles du second ordre, 1 (1890), Gauthier-Villars, Paris 
  10. E. Goursat, Sur les équations du second ordre à n variables analogues à l’équation de Monge-Ampère, Bull. Soc. Math. France 27 (1899), 1-34 Zbl30.0326.01MR1504329
  11. Phillip Griffiths, Joseph Harris, Principles of algebraic geometry, (1978), Wiley-Interscience [John Wiley & Sons], New York Zbl0836.14001MR507725
  12. Alexei Kushner, Classification of Monge-Ampère equations, Differential equations: geometry, symmetries and integrability 5 (2009), 223-256, Springer-Verlag, Berlin Zbl1233.35005MR2562576
  13. Alexei Kushner, Valentin Lychagin, Vladimir Rubtsov, Contact geometry and non-linear differential equations, 101 (2007), Cambridge University Press, Cambridge Zbl1122.53044MR2352610
  14. P. D. Lax, A. N. Milgram, Parabolic equations, Contributions to the theory of partial differential equations (1954), 167-190, Princeton University Press, Princeton, N. J. Zbl0128.09302MR67317
  15. V. Lyčagin, Contact geometry and second-order nonlinear differential equations, Uspekhi Mat. Nauk 34 (1979), 137-165 Zbl0405.58003MR525652
  16. Y. Machida, T. Morimoto, On decomposable Monge-Ampère equations, Lobachevskii J. Math. 3 (1999), 185-196 (electronic) Zbl0957.35010MR1743137
  17. Tohru Morimoto, Monge-Ampère equations viewed from contact geometry, Symplectic singularities and geometry of gauge fields (Warsaw, 1995) 39 (1997), 105-121, Polish Acad. Sci., Warsaw Zbl0879.35008MR1458653
  18. J. Muñoz Díaz, Ecuaciones diferenciales I, (1982) 
  19. I. G. Petrovski, Lectures on partial differential equations, (1991) 
  20. Tommaso Ruggeri, Su una naturale estensione a tre variabili dell’equazione di Monge-Ampère, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 55 (1973), 445-449 (1974) Zbl0294.35013MR380066
  21. Georges Valiron, The classical differential geometry of curves and surfaces, (1986), Math Sci Press, Brookline, MA Zbl0611.53001MR869256

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