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SYMMETRIES OF PARTIAL DIFFERENTIAL EQUATIONS

  • 발행 : 2003.05.06

초록

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.

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참고문헌

  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. https://doi.org/10.1007/BF02417822
  4. S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), no. 2, 219–271. https://doi.org/10.1007/BF02392146
  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. https://doi.org/10.1112/plms/s3-71.1.221
  7. H. Gaussier and J. Merker, A new example of uniformly Levi degenerate hyper-surface in $\mathbb{C}^3$, Ark. Mat., to appear https://doi.org/10.1007/BF02384568
  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. https://doi.org/10.1016/S1631-073X(02)00020-1
  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. https://doi.org/10.1063/1.525960
  12. A. Gonzalez-Lopez, Symmetries of linear systems of second order differential equations, J. Math. Phys. 29 (1988), 1097–1105. https://doi.org/10.1063/1.527948
  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. https://doi.org/10.1070/RM1992v047n04ABEH000916
  14. S. Lie, Theorie der Transformationsgruppen, Math. Ann. 16 (1880), 441–528. https://doi.org/10.1007/BF01446218
  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 https://doi.org/10.1007/s002090100262
  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 https://doi.org/10.1307/mmj/1028575739
  27. H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171–188 https://doi.org/10.1090/S0002-9947-1973-0321133-2
  28. A. Tresse, Determination des invariants ponctuels de l'equation differentielle du second ordre y''= !(x, y, y'), Hirzel, Leipzig, 1896

피인용 문헌

  1. New extension phenomena for solutions of tangential Cauchy–Riemann equations vol.41, pp.6, 2016, https://doi.org/10.1080/03605302.2016.1180536
  2. Lie symmetries and CR geometry vol.154, pp.6, 2008, https://doi.org/10.1007/s10958-008-9201-5
  3. Characterization of the Newtonian Free Particle System in $m\geqslant 2$ Dependent Variables vol.92, pp.2, 2006, https://doi.org/10.1007/s10440-006-9064-z