Browse > Article
http://dx.doi.org/10.4134/JKMS.2003.40.3.517

SYMMETRIES OF PARTIAL DIFFERENTIAL EQUATIONS  

Gaussier, Herve (CNRS, Universite de Provence)
Merker, Joel (CNRS, Universite de Provence)
Publication Information
Journal of the Korean Mathematical Society / v.40, no.3, 2003 , pp. 517-561 More about this Journal
Abstract
We establish a link between the study of completely integrable systems of partial differential equations and the study of generic submanifolds in $\mathbb{C}$. Using the recent developments of Cauchy-Riemann geometry we provide the set of symmetries of such a system with a Lie group structure. Finally we determine the precise upper bound of the dimension of this Lie group for some specific systems of partial differential equations.
Keywords
Lie symmetry; completely integrable system; prologation; CR geometry; local holomorphic automorphism;
Citations & Related Records

Times Cited By Web Of Science : 2  (Related Records In Web of Science)
Times Cited By SCOPUS : 3
연도 인용수 순위
1 B. Segre, Intorno al problema di Poincaré della rappresentazione pseudocon-forme, Rend. Acc. Lincei, VI, Ser. 13 (1931), 676–683
2 B. Segre, Questioni geometriche legate colla teoria delle funzioni di due variabili complesse, Rendiconti del Seminario di Matematici di Roma, II, Ser. 7 (1932), no. 2, 59–107
3 O. Stormark, Lie's structural approach to PDE systems, Encyclopaedia of math ematics and its applications, vol. 80, Cambridge University Press, Cambridge, 2000, pp. xv+572
4 A. Sukhov, Segre varieties and Lie symmetries, Math. Z. 238 (2001), no. 3, 483–492   DOI
5 A. Sukhov, On transformations of analytic CR structures, Pub. Irma, Lille 2001, Vol. 56, no. II
6 A. Sukhov, CR maps and point Lie transformations, Michigan Math. J. 50 (2002), 369–379   DOI
7 H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171–188   DOI
8 A. Tresse, Determination des invariants ponctuels de l'equation differentielle du second ordre y''= !(x, y, y'), Hirzel, Leipzig, 1896
9 H. Gaussier and J. Merker, Göomötrie des sous-variötös analytiques röelles de $\mathbb{C}^n$ et symötries de Lie des öquations aux dörivöes partielles, Bull. Soc. Math. Tunisie, to appear
10 F. Gonzalez-Gascon and A. Gonzalez-Lopez, Symmetries of differential equations, IV. J. Math. Phys. 24 (1983), 2006–2021.   DOI
11 A. Gonzalez-Lopez, Symmetries of linear systems of second order differential equations, J. Math. Phys. 29 (1988), 1097–1105.   DOI
12 N. H. Ibragimov, Group analysis of ordinary differential equations and the in-variance principle in mathematical physics, Russian Math. Surveys 47:4 (1992), 89–156.   DOI   ScienceOn
13 S. Lie, Theorie der Transformationsgruppen, Math. Ann. 16 (1880), 441–528.   DOI
14 J. Merker, Vector field construction of Segre sets, preprint 1998, augmented in 2000. Downloadable at arXiv.org/abs/math.CV/9901010
15 J. Merker, On the partial algebraicity of holomorphic mappings between two real algebraic sets, Bull. Soc. Math. France 129 (2001), no. 3, 547–591
16 J. Merker, On the local geometry of generic submanifolds of $\mathbb{C}^n$ and the analytic reflection principle, Viniti, to appear
17 P. J. Olver, Applications of Lie groups to differential equations. Springer-Verlag, Heidelberg, 1986
18 P. J. Olver, Equivalence, Invariance and Symmetries, Cambridge University Press, Cambridge, 1995, pp. xvi+525
19 H. Poincare, Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Mat. Palermo, II, Ser. 23 (1932), 185–220.
20 M. S. Baouendi, P. Ebenfelt and L. P. Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series 47, Princeton University Press, Princeton, NJ, 1999, pp. xii+404
21 G. W. Bluman and S. Kumei, Symmetries and differential equations, Springer- Verlag, Berlin, 1989
22 E. Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes, I, Annali di Mat. 11 (1932), 17–90.   DOI
23 S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), no. 2, 219–271.   DOI
24 F. Engel and S. Lie, Theorie der Transformationsgruppen, I, II, II, Teubner, Leipzig, 1889, 1891, 1893
25 M. Fels, The equivalence problem for systems of second-order ordinary differential equations, Proc. London Math. Soc. 71 (1995), 221–240.   DOI
26 H. Gaussier and J. Merker, A new example of uniformly Levi degenerate hyper-surface in $\mathbb{C}^3$, Ark. Mat., to appear   DOI
27 H. Gaussier and J. Merker, Nonalgebraizable real analytic tubes in $\mathbb{C}^n$, Math. Z., to appear
28 H. Gaussier and J. Merker, Sur l'algöbrisabilitö locale de sous-variötös analytiques röelles gönöriques de $\mathbb{C}^n$, C. R. Acad. Sci. Paris Sör. I 336 (2003), 125–128.   DOI   ScienceOn