Skip to main content
SpringerLink
Log in

Local algebraic analysis of differential systems

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

Abstract

We propose a new approach for studying the compatibility of partial differential equations. This approach is a synthesis of the Riquier method, Gröbner basis theory, and elements of algebraic geometry. As applications, we consider systems including the wave equation and the sine-Gordon equation.

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. J. Molk, ed., Encyclopédie des sciences mathématiques II, Vol. 4, Équations aux dérivées partielles, Jacques Gabay, Paris (1916).

    Google Scholar 

  2. M. E. Goursat, Leçons sur l’intégration des équations aux dérivées partielles du second order à deux variables indépendantes, Vol. 2, Librairie scientifique A. Hermann, Paris (1898).

    Google Scholar 

  3. M. Kuranishi, Lectures on Involutive Systems of Partial Differential Equations, Publ. Soc. Mat., São Paulo (1967).

    MATH  Google Scholar 

  4. D. Spencer, Bull. Amer. Math. Soc., 75, 179–239 (1969).

    Article  MATH  MathSciNet  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, J. Soviet Math., 17, 1624–1649 (1981).

    Article  MATH  Google Scholar 

  7. A. D. Wittkopf, “Algorithms and implementations for differential elimination,” Doctoral dissertation, Simon Fraser University, Burnaby, British Columbia, Canada (2004).

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. V. P. Gerdt and Yu. A. Blinkov, “Janet-like monomial division,” in: Computer Algebra in Scientific Computing/CASC 2005 (Lect. Notes Computer Sci., Vol. 3718), Springer, Berlin (2005), pp. 174–183; “Janet-like Gröbner bases,” in: Op. cit., pp. 184–195.

    Chapter  Google Scholar 

  10. B. Buchberger, “Gröbner bases: An algorithmic method in polynomial ideal theory,” in: Computer Algebra: Symbolic and Algebraic Computation (B. Buchberger, G. E. Collins, R. Loos, and R. Albrecht, eds.), Springer, New York (1983).

    Chapter  Google Scholar 

  11. D. A. Cox, J. B. Little, and D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, New York (1992).

    Book  MATH  Google Scholar 

  12. L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978); English transl., Acad. Press, New York (1982).

    Google Scholar 

  13. N. H. Ibragimov, Transformation Groups in Mathematical Physics [in Russian], Nauka, Moscow (1983); English transl.: Transformation Groups Applied to Mathematical Physics (Math.Its Appl. (Sov. Ser.), Vol. 3), D. Reidel, Dordrecht (1985).

    Google Scholar 

  14. A. F. Sidorov, V. P. Shapeev, and N. N. Yanenko, Method of Differential Constraints and Its Applications in Gas Dynamics [in Russian], Nauka, Novosibirsk (1984).

    Google Scholar 

  15. V. I. Fushchich and A. G. Nikitin, Symmetry of Equations of Quantum Mechanics [in Russian], Nauka, Moscow (1990); English transl., Allerton, New York (1994).

    Google Scholar 

  16. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM Stud. Appl. Math., Vol. 4), SIAM, Philadelphia (1981).

    Book  MATH  Google Scholar 

  17. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Mass. (1969).

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  19. J. Ritt, Differential Algebra, Amer. Math. Soc., New York (1950).

    MATH  Google Scholar 

  20. O. V. Kaptsov, Program. Comput. Soft., 40, No. 2, 63–70 (2014).

    Article  MATH  MathSciNet  Google Scholar 

  21. O. Zariski and P. Samuel, Commutative Algebra, Vol. 2, Van Nostrand, Princeton, N. J. (1960).

    Book  Google Scholar 

  22. L. M. Brehovskikh, Waves in Layered Media [in Russian], Nauka, Moscow (1973); English transl. (Appl. Math. Mech., Vol. 16), Acad. Press, New York (1980).

    Google Scholar 

  23. O. V. Kaptsov, Integration Methods for Partial Differential Equations [in Russian], Fizmatlit, Moscow (2009).

    Google Scholar 

  24. V. B. Matveev and M. A. Salle, Darboux Trasformations and Solitons, Springer, New York (1991).

    Book  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. 183, No. 3, pp. 342–358, June, 2015.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kaptsov, O.V. Local algebraic analysis of differential systems. Theor Math Phys 183, 740–755 (2015). https://doi.org/10.1007/s11232-015-0293-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-015-0293-z

Keywords

Search

Navigation