Advertisement

Theoretical and Mathematical Physics

, Volume 122, Issue 1, pp 121–133 | Cite as

Factoring linear partial differential operators and the Darboux method for integrating nonlinear partial differential equations

  • S. P. Tsarev
Article

Abstract

Using a new definition of the generalized factorization of linear partial differential operators, we discuss possible generalizations of the Darboux integrability of nonlinear partial differential equations.

Keywords

Left Ideal Nonlinear Partial Differential Equation Principal Ideal Modular Lattice Linear Partial Differential Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. Darboux,Ann. École Normale Sup.,7, 163–173 (1870).MathSciNetGoogle Scholar
  2. 2.
    I. M. Anderson and N. Kamran,Duke Math. J.,87, 265–319 (1997).CrossRefMathSciNetGoogle Scholar
  3. 3.
    M. Juras, “Generalized Laplace invariants and classical integration methods for second-order scalar hyperbolic partial differential equations in the plane,” in:Proc. 6th Intl. Conf. “Differential Geometry and Applications” (Brno, Czech Republik, August 28–September 1, 1995) (J. Janyske, ed.), Masaryk Univ., Brno (1996), pp. 275–284.Google Scholar
  4. 4.
    A. V. Zhiber, V. V. Sokolov, and S. Ya. Startsev,Dokl. Math.,52, 128–130 (1995); V. V. Sokolov and A. V. Zhiber,Phys. Lett. A,208, 303–308 (1995).Google Scholar
  5. 5.
    G. Darboux,Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, Vols. 1–4, Gautier-Villars, Paris (1887–1896).MATHGoogle Scholar
  6. 6.
    E. Landau,J. Reine Angew. Math.,124, 115–120 (1901/1902).Google Scholar
  7. 7.
    S. P. Tsarev, “An algorithm for complete enumeration of all factorizations of a linear ordinary differential operator,” in:Proc. 1996 Intl. Symp. Symbolic and Algebraic Computation, ISSAC'96 (Zuerich, Switzerland, July 24–26, 1996) (Y. N. Lakshman, ed.), ACM Press, New York (1996), pp. 226–231.Google Scholar
  8. 8.
    A. Loewy,Math. Annal.,56, 549–584 (1903).CrossRefMathSciNetGoogle Scholar
  9. 9.
    A. Lewy,Math. Annal.,62, 89–117 (1906).CrossRefGoogle Scholar
  10. 10.
    O. Ore,Ann. Math.,34, 480–508 (1933).CrossRefMathSciNetGoogle Scholar
  11. 11.
    E. Beke,Math. Annal.,45, 278–300 (1894).CrossRefMathSciNetGoogle Scholar
  12. 12.
    H. Blumberg, “Über algebraische Eigenschaften von linearen homogenen Differentialausdrücken,” Diss., Univ. Göttingen, Göttingen (1912).Google Scholar
  13. 13.
    K. Ireland and M. Rosen,Classical Introduction to Modern Number Theory (2nd ed.), Springer, New York (1990).MATHGoogle Scholar
  14. 14.
    O. Ore,Ann. Math.,32, 463–477 (1931).CrossRefMathSciNetGoogle Scholar
  15. 15.
    G. Birkhoff,Lattice Theory, Am. Math. Soc., Providence, RI (1967).MATHGoogle Scholar
  16. 16.
    G. Grätzer,General Lattice Theory, Akademie, Berlin (1978).Google Scholar
  17. 17.
    N. Jacobson,Lectures in Abstract Algebra, Vol. 1,Basic Concepts, Van Nostrand, Toronto (1951).Google Scholar
  18. 18.
    E. Goursat,Am. J. Math.,18, 347–385 (1896).CrossRefMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 2000

Authors and Affiliations

  • S. P. Tsarev
    • 1
  1. 1.Krasnoyarsk State Pedagogical UniversityKrasnoyarskRussia

Personalised recommendations