Acta Applicandae Mathematica

, Volume 2, Issue 1, pp 21–78

Local symmetries and conservation laws

  • A. M. Vinogradov
Article

Abstract

Starting with Lie's classical theory, we carefully explain the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessary calculation procedures. Roughly speaking, we explain what analogs of ‘higher KdV equations’ are for an arbitrary system of partial differential equations and also how one can find and use them. The cohomological nature of conservation laws is shown and some basic results are exposed which allow one to calculate, in principle, all conservation laws for a given system of partial differential equations. In particular, it is shown that ‘symmetry’ and ‘conservation law’ are, in some sense, the ‘dual’ conceptions which coincides in the ‘self-dual’ case, namely, for Euler-Lagrange equations. Training examples are also given.

AMS (MOS) Subject classifications (1980)

35A30 58G05 58G35 58H05 

Key words

System of partial differential equations differential operator infinitely prolonged equation contact structure ofkth order (infinitesimal) symmetry transformation conservation law De Rham cohomology 

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References

  1. 1.
    DhoogheP.: ‘Les transformations de contact sur un espace fibré des jets d'application’,C.R. Acad. Sci. 287 (1978), A1125-A1128.Google Scholar
  2. 2.
    GardnerC. S.: ‘Korteweg-de Vries equation and generalisation. IV: The Korteweg-de Vries equation as a Hamiltonian system’,J. Math. Phys. 12 (1971), 1548–1551.Google Scholar
  3. 3.
    IbragimovN. H. and AndersonR. L.: ‘Lie-Bäcklund tangent transformations’,J. Math. Anal. Appl. 59 (1977), 145–162.Google Scholar
  4. 4.
    KumeiS.: ‘Invariance transformations, invariance group transformations and invariance groups of the sine-Gordon equation’,J. Math. Phys. 16 (1975), 2461–2468.Google Scholar
  5. 5.
    KupershmidtB.: ‘On geometry of jet manifolds’,Uspehi Mat. Nauk. 30 (1975), 211–212 (in Russian).Google Scholar
  6. 6.
    KupershmidtB.: ‘Geometry of jet bundles and the Structure of Lagrangian and Hamiltonian formalisms’,Lecture Notes in Math., Vol 775, Springer-Verlag, New York, 1980, pp. 162–217.Google Scholar
  7. 7.
    LaxP. D.: ‘Integrals of nonlinear equations of evolution and solitary waves’,Comm. Pure Appl. Math. 21 (1968), 467–490.Google Scholar
  8. 8.
    LychaginV. V.: ‘Local classification of nonlinear first-order partial differential equations’,Uspehi Mat. Nauk. 30 (1975), 101–171. (English translation inRussian Math. Surveys 3 (1975), 105–176).Google Scholar
  9. 9.
    LychaginV. V.: ‘Geometric singularities of solutions of nonlinear differential equations’,Dokl. Akad. Nauk. SSSR 261 (1981), 1299–1303 (in Russian); 680–685 (English).Google Scholar
  10. 10.
    LychaginV. V.: ‘Geometry and topology of shock waves’,Dokl. Akad. Nauk. SSSR 264 (1982), 551–555 (in Russian); 685–689 (English).Google Scholar
  11. 11.
    MarsdenJ. and WeinsteinA.: ‘Reduction of symplectic manifolds with symmetry’,Rep. Math. Phys. 5 (1974), 121–130.Google Scholar
  12. 12.
    MiuraR. M., GardnerC. S., and KruskalM. D.: ‘Korteweg-de Vries equation and generalisations. II: Existence of conservation laws and constants of motion’,J. Math. Phys. 9 (1968), 1204–1209.Google Scholar
  13. 13.
    OlverP.: ‘Symmetry groups and group invariant solutions of partial differential equations’,J. Diff. Geom. 14 (1979), 497–542.Google Scholar
  14. 14.
    OttersonP. and SvetlichnyG.: ‘On derivative-dependent infinitesimal deformations of differential maps’,J. Diff. Eq. 36 (1981), 270–294.Google Scholar
  15. 15.
    OvsiannikovL. V.:Group Analysis of Differential Equations, Nauka, Moscow, 1978 (English translation: Academic Press, 1982).Google Scholar
  16. 16.
    Pommaret, J.-F.:Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach, 1978.Google Scholar
  17. 17.
    NovikovS. P. (ed.):Soliton Theory, Nauka, Moscow, 1980.Google Scholar
  18. 18.
    TsujishitaT.: ‘On variation bicomplexes associated to differential equations’,Osaka J. Math. 19 (1982), 311–363.Google Scholar
  19. 19.
    VinogradovA. M.: ‘On the algebra-geometric foundations of Lagrangian field theory’,Dokl. Akad. Nauk. SSSR 236 (1977), 284–287 (English translation inSoviet Math. Dokl. 18 (1977), 1200–1204.Google Scholar
  20. 20.
    VinogradovA. M.: ‘A spectral sequence associated with a nonlinear differential equation and algebrageometric foundations of Lagrangian field theory with constraints’,Dokl. Akad. Nauk. SSSR 238 (1978), 1028–1031 (English translation inSoviet Math. Dokl. 19 (1978), 144–148).Google Scholar
  21. 21.
    VinogradoyA. M.: ‘The Theory of higher infinitesimal symmetries of nonlinear partial differential equations’,Dokl. Akad. Nauk. SSSR 248 (1979), 274–278 (English translation inSoviet Math. Dokl. 20 (1979), 985–990).Google Scholar
  22. 22.
    VinogradovA. M.: ‘Geometry of nonlinear differential equations’,Itogi Nauki i Tekniki, VINITI, Ser. ‘Problemy Geometrii’ 11 (1980), 89–134 (English translation inJ. Soviet Math. 17 (1981), 1624–1649).Google Scholar
  23. 23.
    VinogradovA. M.: ‘The category of nonlinear differential equations’, inEquations on manifolds, Izdat. Voronezh. Gosudarstv. Universitet, Voronezh, 1982, pp. 26–51 (in Russian).Google Scholar
  24. 24.
    Vinogradov, A. M.: ‘TheC-spectral sequence, Lagrangian formalism and conservation laws’,J. Math. Anal. Appl. (1984) to appear.Google Scholar
  25. 25.
    VinogradovA. M. and Krasil'scikI. S.: ‘A Method of computing higher symmetries of nonlinear evolution equations, and nonlocal symmetries’,Dokl. Akad. Nauk. SSSR 253 (1980), 1289–1293 (English translation inSoviet Math. Dokl. 22 (1980), 235–239).Google Scholar
  26. 26.
    WahlquistH. D. and EstabrookF. B.: ‘Prolongation structures on nonlinear evolution equations’,J. Math. Phys. 16 (1975), 1–7.Google Scholar
  27. 27.
    ZakharovV. E. and FaddeevL. D.: ‘Korteweg-de Vries equation: a completely integrable Hamiltonian system’,Funkt. Anal. Appl. 5 (1971), 18–27 (Russian); 280–287 (English).Google Scholar

Copyright information

© D. Reidel Publishing Company 1984

Authors and Affiliations

  • A. M. Vinogradov
    • 1
  1. 1.Department of Mechanics and MathematicsMoscow UniversityMoscowUSSR

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