Skip to main content
SpringerLink
Log in

Compatibility, Multi-brackets and Integrability of Systems of PDEs

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

We establish an efficient compatibility criterion for a system of generalized complete intersection type in terms of certain multi-brackets of differential operators. These multi-brackets generalize the higher Jacobi-Mayer brackets, important in the study of evolutionary equations and the integrability problem. We also calculate Spencer δ-cohomology of generalized complete intersections and evaluate the formal functional dimension of the solutions space. The results are used to establish new integration methods.

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. Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  2. Bruns, W., Vetter, U.: Determinental Rings. Lect. Notes in Math., vol. 1327. Springer, Berlin (1988)

    Google Scholar 

  3. Bryant, R.L., Chern, S.S., Gardner, R.B., Goldschmidt, H.L., Griffiths, P.A.: Exterior Differential Systems. MSRI Publications, vol. 18. Springer, Berlin (1991)

    MATH  Google Scholar 

  4. Buchsbaum, D.A., Eisenbud, D.: What annihilates a module?. J. Algebra 47, 231–243 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  5. Buchsbaum, D.A., Rim, D.S.: A generalized Koszul complex, II. Depth and multiplicity. Trans. Am. Math. Soc. 111, 197–224 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  6. Burch, L.: On ideals of finite homological dimension in local rings. Proc. Camb. Philos. Soc. 64, 941–948 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cartan, E.: Les systèmes différentiels extérieurs et leurs applications géométriques. Actualités Sci. Ind., vol. 994. Hermann, Paris (1945) (French)

    MATH  Google Scholar 

  8. Darboux, G.: Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, III Partie. Gauthier-Villars, Paris (1894)

    Google Scholar 

  9. de Azcárraga, J.A., Perelomov, A.M., Pérez Bueno, J.C.: The Schouten-Nijenhuis bracket, cohomology and generalized Poisson structures. J. Phys. A 29(24), 7993–8009 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311(1), 167–183 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, Berlin (1995)

    MATH  Google Scholar 

  12. Gelfand, I., Gelfand, S., Retakh, V., Wilson, R.L.: Quasideterminants. Adv. Math. 193(1), 56–141 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Goldberg, V., Lychagin, V.: On the Blaschke conjecture for 3-webs. J. Geom. Anal. 16(1), 69–115 (2006)

    MATH  MathSciNet  Google Scholar 

  14. Goldberg, V., Lychagin, V.: Abelian equations and rank problems for planar webs. Russ. Math. (Iz. VUZ) 51(10), 39–75 (2007)

    MathSciNet  Google Scholar 

  15. Goldschmidt, H.: Integrability criteria for systems of nonlinear partial differential equations. J. Differ. Geom. 1(3), 269–307 (1967)

    MATH  MathSciNet  Google Scholar 

  16. Goursat, E.: Lecons sur l’intégration des équations aux dérivées partielles du premier ordere. Hermann, Paris (1891)

    Google Scholar 

  17. Guillemin, V., Sternberg, S.: An algebraic model of transitive differential geometry. Bull. Am. Math. Soc. 70, 16–47 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  18. Guillemin, V., Quillen, D., Sternberg, S.: The integrability of characteristics. Commun. Pure Appl. Math. 23(1), 39–77 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gunter, N.M.: Integration of PDEs of the First Order. ONTI, Leningrad (1934) (Russian)

    Google Scholar 

  20. Heckman, G.J.: Quantum integrability for the Kovalevskaya top. Indag. Math. (N.S.) 9(3), 359–365 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hörmander, L.: Linear Partial Differential Operators, 3rd edn. Springer, Berlin (1969)

    MATH  Google Scholar 

  22. Hubert, E.: Notes on triangular sets and triangulation-decomposition algorithms: I. Polynomial systems; II. Differential systems. In: Symbolic and Numerical Scientific Computation (Hagenberg, 2001). Lecture Notes in Comput. Sci., vol. 2630, pp. 1–39, 40–87. Springer, Berlin (2003)

  23. Janet, M.: Leçons sur les systèms d’équations. Gauthier-Villers, Paris (1929)

    Google Scholar 

  24. Krasilschik, I.S., Lychagin, V.V., Vinogradov, A.M.: Geometry of Jet Spaces and Differential Equations. Gordon and Breach, New York (1986)

    Google Scholar 

  25. Kruglikov, B.S.: Note on two compatibility criteria: Jacobi-Mayer bracket vs. differential Gröbner basis. Lobachevskii J. Math. 23, 57–70 (2006)

    MATH  MathSciNet  Google Scholar 

  26. Kruglikov, B.S.: Invariant characterization of Liouville metrics and polynomial integrals. J. Geom. Phys. 58(8), 979–995 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kruglikov, B.S.: Anomaly of linearization and auxiliary integrals. In: SPT 2007—Symmetry and Perturbation Theory, pp. 108–115. World Scientific, Singapore (2008)

    Chapter  Google Scholar 

  28. Kruglikov, B.S., Lychagin, V.V.: Mayer brackets and solvability of PDEs—I. Differ. Geom. Appl. 17, 251–272 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  29. Kruglikov, B.S., Lychagin, V.V.: A compatibility criterion for systems of PDEs and generalized Lagrange-Charpit method. A.I.P. Conference Proceedings, Global Analysis and Applied Mathematics International Workshop on Global Analysis, vol. 729, no. 1, pp. 39–53 (2004)

  30. Kruglikov, B.S., Lychagin, V.V.: Mayer brackets and solvability of PDEs—II. Trans. Am. Math. Soc. 358(3), 1077–1103 (2005)

    Article  MathSciNet  Google Scholar 

  31. Kruglikov, B.S., Lychagin, V.V.: Multi-brackets of differential operators and compatibility of PDE systems. C. R. Math. 342(8), 557–561 (2006)

    MATH  MathSciNet  Google Scholar 

  32. Kruglikov, B.S., Lychagin, V.V.: Dimension of the solutions space of PDEs. In: J. Calmet, W. Seiler, R. Tucker (eds.) Global Integrability of Field Theories, Proc. of GIFT-2006, pp. 5–25 (2006). ArXive e-print: math.DG/0610789

  33. Kruglikov, B.S., Lychagin, V.V.: Compatibility, multi-brackets and integrability of systems of PDEs. Prepr. Univ. Tromsø 2006-49. ArXive:math.DG/0610930

  34. Lada, T., Stasheff, J.: Introduction to SH Lie algebras for physicists. Int. J. Theor. Phys. 32(7), 1087–1103 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  35. Lie, S.: Gesammelte Abhandlungen, Bd. 1–4. Teubner, Leipzig (1929)

    Google Scholar 

  36. Lie, S., Engel, F.: Theorie der Transformationsgruppen. Begründungstransformationen, vol. II. Teubner, Leipzig (1888–1893)

    MATH  Google Scholar 

  37. Lychagin, V.V.: Homogeneous geometric structures and homogeneous differential equations. In: V. Lychagin (ed.) The Interplay between Differential Geometry and Differential Equations, Am. Math. Soc. Transl., ser. 2, vol. 167, pp. 143–164 (1995)

  38. Lychagin, V.V.: Quantum mechanics on manifolds. Acta Appl. Math. 56(2–3), 231–251 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  39. Malgrange, B.: Equations differentielles sans solutions (d’apres Lars Hormander). In: Sem. Bourbaki. Exp. No. 213, vol. 6, pp. 119–125. Soc. Math. France, Paris (1995)

    Google Scholar 

  40. Mansfield, E.L.: A simple criterion for involutivity. J. Lond. Math. Soc. 54(2), 323–345 (1996)

    MATH  MathSciNet  Google Scholar 

  41. Matveev, V.S., Topalov, P.J.: Quantum integrability for the Beltrami-Laplace operator as geodesic equivalence. Math. Z. 238, 833–866 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  42. Michor, P.W., Vinogradov, A.M.: n-ary Lie and associative algebras. In: Geometrical Structures for Physical Theories, II (Vietri, 1996). Rend. Sem. Mat. Univ. Politec. Torino 54(4), pp. 373–392 (1996)

  43. Moyal, J.: Quantum mechanics as a statistical theory. Proc. Camb. Philos. Soc. 45, 99–124 (1945)

    Article  MathSciNet  Google Scholar 

  44. Nambu, Y.: Generalized Hamiltonian dynamics. Phys. Rev. D 7, 2405–2412 (1973)

    Article  MathSciNet  Google Scholar 

  45. Palais, R.: Seminar on the Atiyah-Singer Index Theorem. Annals of Mathematics Studies, vol. 57. Princeton Univ. Press, Princeton (1965)

    MATH  Google Scholar 

  46. Perelomov, A.M.: Integrable Systems of Classical Mechanics and Lie Algebras, vol. 1. Birkhäuser, Basel (1990) (transl. from Russian)

    Google Scholar 

  47. Spencer, D.C.: Overdetermined systems of linear partial differential equations. Bull. Am. Math. Soc. 75, 179–239 (1969)

    Article  MATH  Google Scholar 

  48. Sternberg, S.: Lectures on Differential Geometry. Prentice-Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

  49. Sweeney, W.J.: The δ-Poincaré estimate. Pac. J. Math. 20, 559–570 (1967)

    MATH  MathSciNet  Google Scholar 

  50. Verbovetsky, A.: On the cohomology of compatibility complex. Usp. Mat. Nauk 53(1) (1998) (319) 213–214; transl. Russ. Math. Surv. 53(1) (1998) 225–226

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Statistics, University of Tromsø, 90-37, Tromsø, Norway

    Boris Kruglikov & Valentin Lychagin

Authors
  1. Boris Kruglikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Valentin Lychagin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Boris Kruglikov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kruglikov, B., Lychagin, V. Compatibility, Multi-brackets and Integrability of Systems of PDEs. Acta Appl Math 109, 151–196 (2010). https://doi.org/10.1007/s10440-009-9446-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-009-9446-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation