Skip to main content
SpringerLink
Log in

Algebraic and geometric structures of analytic partial differential equations

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the problem of the compatibility of nonlinear partial differential equations. We introduce the algebra of convergent power series, the module of derivations of this algebra, and the module of Pfaffian forms. Systems of differential equations are given by power series in the space of infinite jets. We develop a technique for studying the compatibility of differential systems analogous to the Gröbner bases. Using certain assumptions, we prove that compatible systems generate infinite manifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, New York (1997).

    MATH  Google Scholar 

  2. B. Buchberger, “Gröbner bases: An algorithmic method in polynomial ideal theory,” in: Progress, Directions, and Open Problems in Multidimensional Systems Theory (N. K. Bose, ed.), D. Reidel, Dordrecht (1985), pp. 184–232.

    Chapter  Google Scholar 

  3. C. Riquier, Les systèmes d’équations aux dérivées partielles, Gauthier-Villars, Paris (1909).

    MATH  Google Scholar 

  4. M. Janet, Leçons sur les systémes des équations aux dérivées partielles, Gauthier-Villars, Paris (1929).

    MATH  Google Scholar 

  5. J. F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach, New York (1978).

    MATH  Google Scholar 

  6. A. M. Vinogradov, I. S. Krasil’schik, and V. V. Lychagin, Introduction to the Geometry of Nonlinear Differential Equations [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  7. V. V. Zharinov, Lecture Notes on Geometrical Aspects of Partial Differential Equations (Series Sov. East Eur. Math., Vol. 9), World Scientific, Singapore (1992).

    Book  MATH  Google Scholar 

  8. W. M. Seiler, Involution: The Formal Theory of Differential Equations and Its Applications in Computer Algebra (Algor. Comput. Math., Vol. 24), Springer, New York (2010).

    Book  MATH  Google Scholar 

  9. M. Marvan, Found. Comput. Math., 9, 651–674 (2009).

    Article  MathSciNet  Google Scholar 

  10. Maple, http://www.maplesoft.com/products/Maple/index.aspx (2015).

  11. O. V. Kaptsov, Program. Comput. Softw., 40, 63–70 (2014).

    Article  MathSciNet  Google Scholar 

  12. O. V. Kaptsov, Theor. Math. Phys., 183, 740–755 (2015).

    Article  MathSciNet  Google Scholar 

  13. H. Grauert and R. Remmert, Analytische Stellenalgebren (Grundlehren Math. Wiss., Vol. 176), Springer, Berlin (1971).

    Book  MATH  Google Scholar 

  14. R. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, New York (1965).

    MATH  Google Scholar 

  15. J.-P. Serre, Lie Algebras and Lie Groups (Lect. Notes Math., Vol. 1500), Springer, Berlin (1992).

    MATH  Google Scholar 

  16. N. Bourbaki, Algèbre, Hermann, Paris (1962).

    MATH  Google Scholar 

  17. L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).

    MATH  Google Scholar 

  18. A. M. Vinogradov and I. S. Krasil’schik, eds., Symmetries and Conservation Laws of Equations of Mathematical Physics [in Russian], Faktorial, Moscow (1997).

    Google Scholar 

  19. V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, and A. A. Rodionov, Applications of Group-Theoretical Methods in Hydrodynamics, Springer, Amsterdam (2010).

    MATH  Google Scholar 

  20. N. Bourbaki, General Topology, Springer, Berlin (1998).

    MATH  Google Scholar 

  21. A. I. Belousov and S. B. Tkachev, Discrete Mathematics [in Russian], Bauman Moscow State Technical Univ., Moscow (2004).

    Google Scholar 

  22. J. F. Ritt, Differential Algebra, Dover, New York (1966).

    Google Scholar 

  23. D. Mumford, The Red Book of Varieties and Schemes (Lect. Notes Math., Vol. 1358), Springer, Berlin (1999).

    Book  MATH  Google Scholar 

  24. C.-S. Yih, Stratified Flows, Acad. Press, New York (1980).

    MATH  Google Scholar 

  25. Yu. V. Shan’ko, Vychisl. Tekhnol., 6, No. 5, 106–117 (2001).

    MathSciNet  Google Scholar 

  26. M. Golubitsky and V. Guillemin, StableMappings and Their Singularities (Grad. TextsMath., Vol. 14), Springer, New York (1973).

    Book  MATH  Google Scholar 

  27. V. A. Dorodnitsyn, J. Soviet Math., 55, 1490–1517 (1991).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computational Modeling, Siberian Branch, RAS, Krasnoyarsk, Russia

    O. V. Kaptsov

  2. Siberian Federal University, Krasnoyarsk, Russia

    O. V. Kaptsov

Authors
  1. O. V. Kaptsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. V. Kaptsov.

Additional information

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 189, No. 2, pp. 219–238, November, 2016.

This research was performed under the financial support of a grant from the Russian government for the conduct of research under the direction of leading scientists at the Siberian Federal University (Contract No. 14.U26.31.006) and the Program for Supporting Leading Scientific Schools (Grant Nos. NSh-544.2012.1 and NSh-6293.2012.9).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kaptsov, O.V. Algebraic and geometric structures of analytic partial differential equations. Theor Math Phys 189, 1592–1608 (2016). https://doi.org/10.1134/S0040577916110052

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577916110052

Keywords

Search

Navigation