Abstract
The theory of coverings over differential equations is exposed which is an adequate language for describing various nonlocal phenomena: nonlocal symmetries and conservation laws, Bäcklund transformations, prolongation structures, etc. A notion of a nonlocal cobweb is introduced which seems quite useful for dealing with nonlocal objects.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bäcklund. A. V., Einiges uber Curven und Flächentransformationen, Lund Universitëts Arsskrift 10 (1875),
Bäcklund. A. V. Om Ytor med konstant negativ Krokning, Lund Universitëts Arsskrift 19 (1883).
Dodd, R. K. and Fordy, A. P., Prolongation structures of complex quasi-polynomial evolution equations, J. Phys. A 17 (1984), 3249–3266.
Khor’kova, N. G., Conservation laws and nonlocal symmetries, Mat. Zametki 44 (1988), 134–144 (in Russian).
Kirnasov, E. G., On the coverings of Wahlquist-Estabrook type over the heat equation, Mat. Zametki 42 (1987), 422–433 (in Russian).
Kiso, K., Pseudopotentials and symmetries of evolution equations, Preprint (1986), 18 pp.
Krasil’shchik, I. S., Lychagin, V. V., and Vinogradov, A. M, Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach, 1986.
Krasil’shchik, I. S. and Vinogradov, A. M., Nonlocal symmetries and the theory of coverings, Acta Appl. Math. 2 (1984), 79–96.
Kupershmidt, B. A., On the nature of Gardner transformation, J. Math. Phys. 22 (1981), 449–451.
Lamb, G. L. Jr., Bäcklund transformations at the turn of the century, Springer Lecture Notes in Math. 515 (1975), 69–79.
Olver, P., Evolution equations possessing infinitely many symmetries, J. Math. Phys. 18 (1977), 1212–1215.
Ovsjannikov, L. V., Group Analysis of Differential Equations, Nauka, Moscow, 1978, English Transl.: Academic Press, N.Y., 1982.
Pirani, F. A. E., Robinson, D. C, and Shadwick, W. F., Local Jet Bundle Formulation of Bäcklund Transformations, D. Reidel, Dordrecht, 1979.
K. Lonngren and A. Scott (eds.), Solitons in Action, Academic Press, 1978.
Vinogradov, A. M., Category of nonlinear differential equations, Springer Lecture Notes in Math. 1108 (1984), 77–102.
Vinogradov, A. M., The C-spectral sequence, Lagrangian formalism and conservation laws. Part 1, J. Math. Anal. Appl, 100 (1984), 1–40.
Vinogradov, A. M., The C-spectral sequence, Lagrangian formalism and conservation laws. Part 2, J. Math. Anal. Appl, 100 , 41–129.
Vinogradov, A. M., Local symmetries and conservation laws, Acta Appl. Math. 3 (1984), 21–78.
Vinogradov, A. M., Symmetries and conservation laws of partial differential equations: Basic notations and results, Acta Appl. Math. 15 (1989), 3–21 (this issue).
Wahlquist, H. D. and Estabrook, F. B., Prolongation structures of nonlinear evolution equations, J. Math. Phys. 16(1975), 1–7.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Kluwer Academic Publishers
About this chapter
Cite this chapter
Krasil’shchik, I.S., Vinogradov, A.M. (1989). Nonlocal Trends in the Geometry of Differential Equations: Symmetries, Conservation Laws, and Bäcklund Transformations. In: Vinogradov, A.M. (eds) Symmetries of Partial Differential Equations. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1948-8_8
Download citation
DOI: https://doi.org/10.1007/978-94-009-1948-8_8
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7370-7
Online ISBN: 978-94-009-1948-8
eBook Packages: Springer Book Archive