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Compatibility, Multi-brackets and Integrability of Systems of PDEs

Acta Applicandae Mathematicae volume 109, pages 151–196 (2010)Cite this article

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.

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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

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Authors and Affiliations

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

    Boris Kruglikov & Valentin Lychagin

Authors
  1. Boris Kruglikov
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  2. Valentin Lychagin
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Correspondence to Boris Kruglikov.

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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

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

Keywords

  • Multi-brackets
  • Jacobi-Mayer bracket
  • Spencer cohomology
  • Koszul homology
  • Buchsbaum-Rim complex
  • Integral
  • Characteristics
  • System of PDEs
  • Symbols
  • Compatibility

Mathematics Subject Classification (2000)

  • 35N10
  • 58A20
  • 58H10
  • 35A30

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