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Journal of Soviet Mathematics

, Volume 17, Issue 1, pp 1624–1649 | Cite as

Geometry of nonlinear differential equations

  • A. M. Vinogradov
Article

Abstract

The paper contains a survey of certain contemporary concepts and results connected with the geometric foundations of the theory of nonlinear partial differential equations. At the base of the account is situated the geometry and analysis on jet spaces, finite and infinite.

Keywords

Differential Equation Partial Differential Equation Nonlinear Differential Equation Nonlinear Partial Differential Equation Contemporary Concept 
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Literature cited

  1. 1.
    V. I. Arnol'd, “Contact manifolds, Legendre mappings, and singularities of wave fronts,” Lisp. Mat. Nauk,29, No. 4, 153–154 (1974).Google Scholar
  2. 2.
    V. I. Bliznikas and Z. Yu. Lupeikis, “The geometry of differential equations,” in: Algebra. Topology. Geometry [in Russian], Vol. 11, Itogi Nauki i Tekhniki VINITI Akad. Nauk SSSR, Moscow (1974), pp. 209–259.Google Scholar
  3. 3.
    J. M. Boardman, “Singularities of differentiable mappings,” in: Singularities of Differentiate Mappings [Russian translation], Mir, Moscow (1968), pp. 102–152.Google Scholar
  4. 4.
    A. M. Vinogradov, “The algebra of the logic of the theory of linear differential operators,” Dokl. Akad. Nauk SSSR,205, No. 5, 1025–1028 (1972).Google Scholar
  5. 5.
    A. M. Vinogradov, “Multivalued solutions and the principle of classification of nonlinear differential equations,” Dokl. Akad. Nauk SSSR,210, No. 1, 11–14 (1973).Google Scholar
  6. 6.
    A. M. Vinogradov, “Theory of symmetry for nonlinear partial differential equations,” Moscow University (1974).Google Scholar
  7. 7.
    A. M. Vinogradov, “On the algebro-geometric foundations of the Lagrangian theory of fields,” Dokl. Akad. Nauk SSSR,236, 284–287 (1977).Google Scholar
  8. 8.
    A. M. Vinogradov, “A spectral sequence connected with nonlinear differential equations, and the algebro-geometric foundations of the Lagrangian theory of fields with constraints,” Dokl. Akad. Nauk SSSR,238, No. 5, 1028–1031 (1978).Google Scholar
  9. 9.
    A. M. Vinogradov, “Hamiltonian structures in the theory of fields,” Dokl. Akad. Nauk SSSR,241, No. 1, 18–21 (1978).Google Scholar
  10. 10.
    A. M. Vinogradov and E. M. Vorob'ev, “Application of symmetries in finding exact solutions of the Zabolotskaya-Khokhlov equation,” Akust. Zh.,22, No. 1, 23–27 (1976).Google Scholar
  11. 11.
    A. M. Vinogradov, I. S. Krasil'shchik, and V. V. Lychagin, Application of Nonlinear Equations to Civil Aviation [in Russian], Mosk. Inst. Inzh. Grazhd. Aviatsii, Moscow (1977).Google Scholar
  12. 12.
    A. M. Vinogradov, I. S. Krasil'shchik, and V. V. Lychagin, Geometry of Jet-Spaces and Nonlinear Differential Equations [in Russian], Mosk. In-t Elektr. Mash., Moscow (1979).Google Scholar
  13. 13.
    A. M. Vinogradov and B. A. Kupershmidt, “Structure of Hamiltonian mechanics,” Usp. Mat. Nauk,32, No. 4, 175–236 (1977).Google Scholar
  14. 14.
    M. M. Vinogradov, “On Spencer and de Rham algebraic cohomology,” Dokl. Akad. Nauk SSSR,242, No. 5, 989–992 (1978).Google Scholar
  15. 15.
    I. M. Gel'fand and L. A. Dikii, “Asymptotics of the resolvent of Sturm-Liouville equations and the algebra of Korteweg-de Vries equations,” Usp. Matem. Nauk,30, No. 5, 67–100 (1975).Google Scholar
  16. 16.
    V. M. Zakalyukin, “On Lagrangian and Legendre singularities,” Funkts. Anal. Prilozhen.,10, No. 1, 26–36 (1976).Google Scholar
  17. 17.
    N. Kh. Ibragimov and R. D. Anderson, “Groups of Lie -Baecklund tangent transformations,” Dokl. Akad. Nauk SSSR,227, No. 3, 539–542 (1976).Google Scholar
  18. 18.
    E. Cartan, Exterior Differential Systems and Their Geometric Applications [Russian translation], Moscow University (1962).Google Scholar
  19. 19.
    A. P. Krishchenko, “On the structure of singularities of solutions of quasilinear equations,” Usp. Mat. Nauk,31, No. 3, 219–220 (1976).Google Scholar
  20. 20.
    A. P. Krishchenko, “On deviations of R-manifolds,” Vestn. Mosk. Univ. Mat., Mekh., No. 1, 17–20 (1977).Google Scholar
  21. 21.
    B. A. Kupershmidt, “On the geometry of jet manifolds,” Usp. Mat. Nauk,30, No. 5, 211–212 (1975).Google Scholar
  22. 22.
    B. A. Kupershmidt, “Lagrangian formalism in the calculus of variations,” Funkts. Anal. Prilozhen.,10, No. 2, 77–78 (1976).Google Scholar
  23. 23.
    V. V. Lygachin, “Local classification of first order nonlinear partial differential equations,” Usp. Mat. Nauk,30, No. 1, 101–171 (1975).Google Scholar
  24. 24.
    V. V. Lychagin, “On sufficient orbits of the group of contact diffeomorphisms,” Mat. Sb.,104, No. 2, 248–270 (1977).Google Scholar
  25. 25.
    V. V. Lychagin, “Contact geometry and second order linear differential equations,” Usp. Mat. Nauk,34, No. 1, 137–165 (1979).Google Scholar
  26. 26.
    Yu. I. Manin, “Algebraic aspects of nonlinear differential equations,” in: Contemporary Problems of Mathematics [in Russian], Vol. 11, Itogi Nauki i Tekhniki VINITI Akad. Nauk SSSR, Moscow (1978), pp. 5–152.Google Scholar
  27. 27.
    L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).Google Scholar
  28. 28.
    L. V. Ovsyannikov and N. Kh. Ibragimov, “Group analysis of the differential equations of mechanics,” in: General Mechanics [in Russian], Vol. 2, Itogi Nauki i Tekhniki VINITI Akad. Nauk SSSR, Moscow (1975), pp. 5–52.Google Scholar
  29. 29.
    P. K. Rashevskii, Geometric Theory of Partial Differential Equations [in Russian], Gostekhizdat, Moscow-Leningrad (1947).Google Scholar
  30. 30.
    D. Spencer, “Overdetermined systems of linear partial differential equations,” Matematika,14, No. 2, 66–90; No. 3, 99–126 (1970).Google Scholar
  31. 31.
    A. Andreotti, C. D. Hill, S. Lojasiewicz, and B. MacKichan, “Complexes of differential operators. The Mayer-Vietoris sequence,” Invent. Math.,35, No. 1, 43–86 (1976).Google Scholar
  32. 32.
    E. Cartan, “Sur certaines expressions differentielles et le probleme de Pfaff,” Ann. Sci. Ecole Norm. Super., Ser. 3,16, 239–332 (1899).Google Scholar
  33. 33.
    E. Cartan, “Sur l'integration des systemes d'equations aux differentielles totales,” Ann. Sci. Ecole Norm. Super., Ser. 3,18, 241–311 (1901).Google Scholar
  34. 34.
    P. Dedecker, “On the generalisation of symplectic geometry to multiple integrals in the calculus of variations,” Lect. Notes Math.,570, 395–456 (1977).Google Scholar
  35. 35.
    C. Ehresmann, “Introduction a la theorie des structures infinitesimales et des pseudo-groupes de Lie,” in: Coloq.Internat. Centre Nat. Rech. Scient. 52 Strasbourg, 1953, Paris (1953), pp. 97–110.Google Scholar
  36. 36.
    A. R. Forsyth, Theory of Differential Equations, Dover, New York (1960).Google Scholar
  37. 37.
    H. Goldschmidt, “Existence theorems for analytic partial differential equations,” Ann. Math.,86, No. 2, 246–270 (1967).Google Scholar
  38. 38.
    H. Goldschmidt, “Integrability criteria for systems of nonlinear partial differential equations,” J. Different. Geom.,1, No. 3, 269–307 (1967).Google Scholar
  39. 39.
    H. Goldschmidt, “Prolongations of linear partial differential equations. I. A conjecture of E. Cartan,” Ann. Sci. Ecole Norm. Super.,1, No. 2, 417–444 (1968).Google Scholar
  40. 40.
    H. Goldschmidt, “Prolongations of linear partial differential equations. II. Inhomogeneous equations,” Ann. Sci. Ecole Norm. Super.,1, No. 4, 617–625 (1968).Google Scholar
  41. 41.
    H. Goldschmidt, “Prolongementes d'equations differentielles lineaire. III. La suite exacte de cohomologie de Spencer,” Ann. Sci. Ecole Norm. Super.,7, No. 1, 5–27 (1974).Google Scholar
  42. 42.
    H. Goldschmidt and D. Spencer, “On the nonlinear cohomology of Lie equations. I, II,” Acta Math.,136, Nos. 1–2, 103–239 (1976); Nos. 3–4, 171–239 (1976).Google Scholar
  43. 43.
    H. Goldschmidt and S. Sternberg, “The Hamilton-Cartan formalism in the calculus of variations,” Ann. Inst. Fourier, Grenoble,23, No. 1, 203–267 (1973).Google Scholar
  44. 44.
    A. Grothendieck, Elements de Geometrie Algebrique. IV. Etude Locale des Schemas et des Morphismes des Schemas, Publs. Math. Inst. Hautes Etudes Scient.,32 (1967).Google Scholar
  45. 45.
    R. Hermann, “Cartan's geometric theory of partial differential equations” Adv. Math.,1, No. 3, 265–317 (1965).Google Scholar
  46. 46.
    E. Kähler, Einführung in die Theorie der Systeme von Differentialgleichungen, Hamburger Math. Einzelschribte,16 (1934).Google Scholar
  47. 47.
    A. Kumpera, “Invariants differentiels d'un pseudogroupe de Lie,” J. Different. Geom.,10, No. 2, 289–417 (1974).Google Scholar
  48. 48.
    M. Kuranishi, “On E. Cartan's prolongations theorem of exterior differential systems,” Am. J. Math.,79, No. 1, 1–47 (1957).Google Scholar
  49. 49.
    M. Kuranishi, Lectures on Exterior Differential Systems, Tata Inst. of Fundamental Research, Bombay (1962).Google Scholar
  50. 50.
    M. Kuranishi, Lectures on Involutive Systems of Partial Differential Equations, Publ. Soc. Mat. San Paulo, San Paulo (1967).Google Scholar
  51. 51.
    S. Lie, Gesamelte Abhandlungen, Bd. 4, Teubner, Leipzig (1929).Google Scholar
  52. 52.
    S. Lie and F. Engel, Theorie der Transformationsgruppen, Bd. 1–3, Teubner, Leipzig, 1886, 1890, 1893.Google Scholar
  53. 53.
    S. Sternberg, Notes on Partial Differential Equations, Harvard University (1968).Google Scholar
  54. 54.
    F. Takens, “A global version of the inverse problem of the calculus of variations,” Math. Inst. Rijksuniversiteit Groningen, Preprint No. ZW-7701 (1977).Google Scholar

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© Plenum Publishing Corporation 1981

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  • A. M. Vinogradov

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