SYMMETRIES OF PARTIAL DIFFERENTIAL EQUATIONS

Title & Authors
SYMMETRIES OF PARTIAL DIFFERENTIAL EQUATIONS
Gaussier, Herve; Merker, Joel;

Abstract
We establish a link between the study of completely integrable systems of partial differential equations and the study of generic submanifolds in $\small{\mathbb{C}}$. 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;
Language
English
Cited by
1.
New extension phenomena for solutions of tangential Cauchy–Riemann equations, Communications in Partial Differential Equations, 2016, 41, 6, 925
2.
Characterization of the Newtonian Free Particle System in \$m\geqslant 2\$ Dependent Variables, Acta Applicandae Mathematicae, 2006, 92, 2, 125
3.
Lie symmetries and CR geometry, Journal of Mathematical Sciences, 2008, 154, 6, 817
References
1.
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

2.
G. W. Bluman and S. Kumei, Symmetries and differential equations, Springer- Verlag, Berlin, 1989

3.
E. Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes, I, Annali di Mat. 11 (1932), 17–90.

4.
S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), no. 2, 219–271.

5.
F. Engel and S. Lie, Theorie der Transformationsgruppen, I, II, II, Teubner, Leipzig, 1889, 1891, 1893

6.
M. Fels, The equivalence problem for systems of second-order ordinary differential equations, Proc. London Math. Soc. 71 (1995), 221–240.

7.
H. Gaussier and J. Merker, A new example of uniformly Levi degenerate hyper-surface in \$\mathbb{C}^3\$, Ark. Mat., to appear

8.
H. Gaussier and J. Merker, Nonalgebraizable real analytic tubes in \$\mathbb{C}^n\$, Math. Z., to appear

9.
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.

10.
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

11.
F. Gonzalez-Gascon and A. Gonzalez-Lopez, Symmetries of differential equations, IV. J. Math. Phys. 24 (1983), 2006–2021.

12.
A. Gonzalez-Lopez, Symmetries of linear systems of second order differential equations, J. Math. Phys. 29 (1988), 1097–1105.

13.
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.

14.
S. Lie, Theorie der Transformationsgruppen, Math. Ann. 16 (1880), 441–528.

15.
J. Merker, Vector field construction of Segre sets, preprint 1998, augmented in 2000. Downloadable at arXiv.org/abs/math.CV/9901010

16.
J. Merker, On the partial algebraicity of holomorphic mappings between two real algebraic sets, Bull. Soc. Math. France 129 (2001), no. 3, 547–591

17.
J. Merker, On the local geometry of generic submanifolds of \$\mathbb{C}^n\$ and the analytic reflection principle, Viniti, to appear

18.
P. J. Olver, Applications of Lie groups to differential equations. Springer-Verlag, Heidelberg, 1986

19.
P. J. Olver, Equivalence, Invariance and Symmetries, Cambridge University Press, Cambridge, 1995, pp. xvi+525

20.
H. Poincare, Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Mat. Palermo, II, Ser. 23 (1932), 185–220.

21.
B. Segre, Intorno al problema di Poincaré della rappresentazione pseudocon-forme, Rend. Acc. Lincei, VI, Ser. 13 (1931), 676–683

22.
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

23.
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

24.
A. Sukhov, Segre varieties and Lie symmetries, Math. Z. 238 (2001), no. 3, 483–492

25.
A. Sukhov, On transformations of analytic CR structures, Pub. Irma, Lille 2001, Vol. 56, no. II

26.
A. Sukhov, CR maps and point Lie transformations, Michigan Math. J. 50 (2002), 369–379

27.
H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171–188

28.
A. Tresse, Determination des invariants ponctuels de l'equation differentielle du second ordre y''= !(x, y, y'), Hirzel, Leipzig, 1896