Advances in Analysis

Download PDF (690.1 KB) PP. 41 - 61 Pub. Date: January 15, 2017

DOI: 10.22606/aan.2017.11007

• Veronika Chrastinová^*
  Brno University of Technology, Faculty of Civil Engineering, Department of Mathematics, Veverí 331/95, 602 00 Brno, Czech Republic
• Václav Tryhuk
  Brno University of Technology, Faculty of Civil Engineering, AdMaS centre, Purkynova 139, 612 00 Brno, Czech Republic

The article provides a survey of the absolute theory of general systems of (partial) differential equations. The equations are relieved of all additional structures and subject to quite arbitrary change of the variables. An abstract mathematical theory in the Bourbaki sense with its own concepts and technical tools follows. In particular the external, internal, generalized and higher–order symmetries and infinitesimal symmetries together with the E. Cartan’s prolongations, various characteristics, the involutivity and the controllability structures are clarified in genuinely coordinate–free terms without any use of the common jet mechanisms.

Higher–order transformations, symmetry, diffiety, involutivity, controllability, characteristics.

[1] Veronika Chrastinová and Václav Tryhuk. Automorphisms of submanifolds. Adv. Difference Equ., pages Art. ID 202731, 26, 2010.

[2] Václav Tryhuk and Veronika Chrastinová. Automorphisms of curves. J. Nonlinear Math. Phys., 16(3):259–281, 2009.

[3] Veronika Chrastinová. Parallel curves in three–dimensional space. In 5-th Conference on Mathematics and Physics on Technical Universities, pages 120–124. University of Defence, Brno, 2007.

[4] Veronika Chrastinová and Václav Tryhuk. Generalized contact transformations. Journal of Applied Mathematics, Statistics and Informatics, 3(1):47–62, 2007.

[5] Jan Chrastina. The formal theory of differential equations, volume 6 of Folia Facultatis Scientiarium Naturalium Universitatis Masarykianae Brunensis. Mathematica. Masaryk University, Brno, 1998.

[6] V. Tryhuk, V. Chrastinová, and O. Dlouhy. The Lie group in infinite dimension. Abstr. Appl. Anal., pages Art. ID 919538, 35, 2011.

[7] Václav Tryhuk and Veronika Chrastinová. Automorphisms of ordinary differential equations. Abstr. Appl. Anal., pages Art. ID 482963, 32, 2014.

[8] Veronika Chrastinová and Václav Tryhuk. On the internal approach to differential equations 1. The involutiveness and standard basis. Math. Slovaca, 66(4):999–1018, 2016. DOI: 10.1515/ms-2015-0198.

[9] Veronika Chrastinová and Václav Tryhuk. On the internal approach to differential equations 2. The controllability structure. Math. Slovaca, 2014. (arXiv:1409.5904) accepted.

[10] Veronika Chrastinová and Václav Tryhuk. On the internal approach to differential equations 3. Infinitesimal symmetries. Math. Slovaca, 2014. (arXiv:1403.0781) accepted.

[11] R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths. Exterior differential systems, volume 18 of Mathematical Sciences Research Institute Publications. Springer-Verlag, New York, 1991.

[12] I. S. Krasil0shchik, V. V. Lychagin, and A. M. Vinogradov. Geometry of jet spaces and nonlinear partial differential equations, volume 1 of Advanced Studies in Contemporary Mathematics. Gordon and Breach Science Publishers, New York, 1986. Translated from the Russian by A. B. Sosinski??.

[13] A. M. Vinogradov. Cohomological analysis of partial differential equations and secondary calculus, volume 204 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 2001. Translated from the Russian manuscript by Joseph Krasil0shchik.

[14] Ch. Riquier. Les systèmes d’équations aux dérivées partielles. Gauthier–Villars, 1910.

[15] élie Cartan. L’intégration des systèmes d’équations aux différentielles totales. Ann. Sci. école Norm. Sup. (3), 18:241–311, 1901.

[16] élie Cartan. Les sous-groupes des groupes continus de transformations. Ann. Sci. éc. Norm. Supér. (3), 25:57–194, 1908.

[17] élie Cartan. Les systèmes différentiels exterieurs et leurs applications geométriques. Paris: Hermann & Cie. 214 p. (1945)., 1945.

[18] Bernard Malgrange. Cartan involutiveness = Mumford regularity. In Commutative algebra (Grenoble/Lyon, 2001), volume 331 of Contemp. Math., pages 193–205. Amer. Math. Soc., Providence, RI, 2003.

[19] élie Cartan. Sur l’équivalence absolue de certains systèmes d’équations différentielles et sur certaines familles de courbes. Bull. Soc. Math. France, 42:12–48, 1914.

[20] David Hilbert. über den begriff der klasse von differentialgleichungen. Math. Ann., 73(1):95–108, 1912.

[21] Elie Cartan. Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre. Ann. Sci. école Norm. Sup. (3), 27:109–192, 1910.

[22] Shlomo Sternberg. Lectures on differential geometry. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.

[23] Boris A. Kupershmidt. KP or mKP, volume 78 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000. Noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems.

[24] Václav Tryhuk and Veronika Chrastinová. On the mapping of jet spaces. J. Nonlinear Math. Phys., 17(3):293–310, 2010.

[25] Jan Chrastina. Examples from the calculus of variations. I. Nondegenerate problems. Math. Bohem., 125(1):55–76, 2000.

[26] Jan Chrastina. Examples from the calculus of variations. II. A degenerate problem. Math. Bohem., 125(2):187– 197, 2000.

[27] Jan Chrastina. Examples from the calculus of variations. III. Legendre and Jacobi conditions. Math. Bohem., 126(1):93–111, 2001.

[28] Jan Chrastina. Examples from the calculus of variations. IV. Concluding review. Math. Bohem., 126(4):691–710, 2001.