Home On the internal approach to differential equations 3. Infinitesimal symmetries
Article
Licensed
Unlicensed Requires Authentication

On the internal approach to differential equations 3. Infinitesimal symmetries

  • Veronika Chrastinová EMAIL logo and Václav Tryhuk
Published/Copyright: December 30, 2016
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 6

Abstract

The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order of derivatives can be changed and the article is devoted to the higher-order infinitesimal symmetries which provide a simplifying “linear approximation” of general groups of higher-order symmetries. The classical Lie’s approach is appropriately adapted.

MSC 2010: Primary 54A17; 58J99; 35A30
Keywords: higher-order transformations; variation; infinitesimal symmetry; diffiety; standard basis

This paper was elaborated with the financial support of the European Union’s “Operational Programme Research and Development for Innovations”, No. CZ.1.05/2.1.00/03.0097, as an activity of the regional Centre AdMaS “Advanced Materials, Structures and Technologies”.

(Communicated by: Michal Fečkan)

References

[1] Cartan, E.: Sur la structure des groupes infinis de transformations, Ann. Sci. Éc. Norm. Supér. (3) 21 (1904), 153– 206.10.24033/asens.538Search in Google Scholar

[2] Chrastinová, V.: Report on the higher order contact transformations. 7-th Conference on Math. and Physics on Technical Universities. Proceedings part 1 – Mathematics, University of Defence, Brno, 2011.Search in Google Scholar

[3] Chrastinová, V.— Tryhuk, V.: Automorphisms of submanifolds, Adv. Difference Equ. 2010 (2010), Article ID 202731, 26 pp.10.1155/2010/202731Search in Google Scholar

[4] Chrastinová, V.— Tryhuk, V.: On the internal approach to differential equations 1. The involutiveness and standard basis Math. Slovaca 66 (2016), 999– 1018.10.1515/ms-2015-0198Search in Google Scholar

[5] Chrastinová, V.— Tryhuk, V.: On the internal approach to differential equations 2. The controllability structure Math. Slovaca 67 (2017).10.1515/ms-2017-0029Search in Google Scholar

[6] Chrastinová, V.— Tryhuk, V.: Generalized contact transformations, J. Appl. Math. Stat. Inform. (JAMSI) 3 (2007), 47– 62.Search in Google Scholar

[7] Newell, A. C.: Solitons in mathematics and physics.CBMS-NSF Regional Conf. Ser. in Appl. Math., SIAM, Philadelphia, PA, 1985.10.1137/1.9781611970227Search in Google Scholar

[8] Tryhuk, V.— Chrastinová, V.: Automorphisms of curves, J. Nonlinear Math. Phys. 16 (2009), 259– 281.10.1142/S1402925109000224Search in Google Scholar

[9] Tryhuk, V.— Chrastinová, V.: On the mapping of jet spaces, J. Nonlinear Math. Phys. 17 (2010), 293– 310.10.1142/S140292511000091XSearch in Google Scholar

[10] Tryhuk, V.— Chrastinová, V.: Automorphisms of ordinary differential equations, Abstr. Appl. Anal. 2014 (2014), Article ID 482963, 32 pp. http://dx.doi.org/10.1155/2014/48296310.1155/2014/482963Search in Google Scholar

[11] Tryhuk, V.— Chrastinová, V.— Dlouhý, O.: The Lie Group in Infinite Dimension. Abstr. Appl. Anal. 2011 (2011), Article ID 919538, 35 pp.10.1155/2011/919538Search in Google Scholar

[12] Vinogradov, A. M.: Cohomological Analysis of Partial Differential Equations and Secondary Calculus. Transl. Math. Monogr. 204, Amer. Math. Soc., Providence, RI, 2001.10.1090/mmono/204Search in Google Scholar

Received: 2014-3-4
Accepted: 2014-7-3
Published Online: 2016-12-30
Published in Print: 2016-12-1

© 2016 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  2. A triple representation of lattice effect algebras
  4. Periods of morgan-voyce sequences and elliptic curves
  6. Multiplicative generalized derivations on ideals in semiprime rings
  8. A few remarks on Poincaré-Perron solutions and regularly varying solutions
  10. Applications of extremal theorem and radius equation for a class of analytic functions
  12. On the “bang-bang” principle for a class of Riemann-Liouville fractional semilinear evolution inclusions
  14. New criteria for global exponential stability of linear time-varying volterra difference equations
  16. On approximation of functions by some hump matrix means of Fourier series
  18. Pseudo-amenability and pseudo-contractibility for certain products of Banach algebras
  20. Poisson kernels on semi-direct products of abelian groups
  22. Locally convex projective limit cones
  24. On the properties (wL) and (wV)
  26. On the existence of solutions for quadratic integral equations in Orlicz spaces
  28. Existence and uniqueness of best proximity points under rational contractivity conditions
  30. Almost Weyl structures on null geometry in indefinite Kenmotsu manifolds
  32. On the internal approach to differential equations 3. Infinitesimal symmetries
  34. A Characterization of the discontinuity point set of strongly separately continuous functions on products
  36. Wick differential and Poisson equations associated to the 𝚀𝚆𝙽-Euler operator acting on generalized operators
  38. Multivariate EIV models
  40. On codes over 𝓡k, m and constructions for new binary self-dual codes
  42. Domination number of total graphs
https://doi.org/10.1515/ms-2016-0236
Keywords for this article
higher-order transformations; variation; infinitesimal symmetry; diffiety; standard basis

Articles in the same Issue

  2. A triple representation of lattice effect algebras
  4. Periods of morgan-voyce sequences and elliptic curves
  6. Multiplicative generalized derivations on ideals in semiprime rings
  8. A few remarks on Poincaré-Perron solutions and regularly varying solutions
  10. Applications of extremal theorem and radius equation for a class of analytic functions
  12. On the “bang-bang” principle for a class of Riemann-Liouville fractional semilinear evolution inclusions
  14. New criteria for global exponential stability of linear time-varying volterra difference equations
  16. On approximation of functions by some hump matrix means of Fourier series
  18. Pseudo-amenability and pseudo-contractibility for certain products of Banach algebras
  20. Poisson kernels on semi-direct products of abelian groups
  22. Locally convex projective limit cones
  24. On the properties (wL) and (wV)
  26. On the existence of solutions for quadratic integral equations in Orlicz spaces
  28. Existence and uniqueness of best proximity points under rational contractivity conditions
  30. Almost Weyl structures on null geometry in indefinite Kenmotsu manifolds
  32. On the internal approach to differential equations 3. Infinitesimal symmetries
  34. A Characterization of the discontinuity point set of strongly separately continuous functions on products
  36. Wick differential and Poisson equations associated to the 𝚀𝚆𝙽-Euler operator acting on generalized operators
  38. Multivariate EIV models
  40. On codes over 𝓡k, m and constructions for new binary self-dual codes
  42. Domination number of total graphs
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 17.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2016-0236/html
Scroll to top button