zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Nonlocal symmetries and the theory of coverings: An addendum to A. M. Vinogradov’s ’Local Symmetries and Conservation Laws’. (English) Zbl 0547.58043
For a system 𝒴 of (nonlinear partial) differential equations, the notion of a covering 𝒴 ˜ 𝒴 is introduced, where 𝒴 is the infinite prolongation of 𝒴. It is shown how the local infinitesimal symmetries of 𝒴 ˜ generate nonlocal (involving integro-differential operators) symmetries of 𝒴 . This provides an effective procedure for calculating the nonlocal symmetries of 𝒴: 1) find the coverings 𝒴 ˜ of 𝒴 , 2) compute the local symmetries of 𝒴 ˜ using, for example, the technique of the second author [ibid. 2, 21-78 (1984; Zbl 0534.58005)].
Reviewer: S.V.Duzhin

MSC:
58J70Invariance and symmetry properties
57M10Covering spaces (manifolds)
35A30Geometric theory for PDE, characteristics, transformations
35Q99PDE of mathematical physics and other areas
References:
[1]VinogradovA. M.: ?Local Symmetries and Conservation Laws,?Acta Appl. Math. 2 (1984), 21-78 (this issue). · Zbl 0534.58005 · doi:10.1007/BF01405491
[2]VinogradovA. M. and KrasilchchikI. S.: ?A Method of Computing Higher Symmetries of Nonlinear Evolution Equations and Nonlocal Symmetries.Doklady AN SSSR,253 (1980), 1289-1293 (in Russian).
[3]KaptsovO. V.: ?An Extension of Symmetries of Evolution Equations?,Doklady AN SSSR,265 (1982), 1056-1059 (in Russian).
[4]OlverP. J.: ?Evolution Equations Possessing Infinitely Many Symmetries?,J. Math. Phys.,18 (1977), 1212-1215. · Zbl 0348.35024 · doi:10.1063/1.523393
[5]FushchichV. I.: ?On Additional Invariance of Vector Fields?,Doklady AN SSSR,257 (1981), 1105-1109 (in Russian).
[6]IbragimovN. Kh. and ShabatA. B.: On Infinite Algebras of Lie-Bäcklund’,Funct. Anal. Appl. 14 (1980), 79-80 (in Russian). · Zbl 0447.52011 · doi:10.1007/BF01086547
[7]KonopelchenkoB. G. and MokhnakovV. G.: ?On the Group Theoretical Analysis of Differential Equations?,J. Phys. A: Math. Gen.,13 (1980), 3113-3124. · Zbl 0448.35005 · doi:10.1088/0305-4470/13/10/009
[8]Vinogradov, A. M.: ?The Category of Nonlinear Differential Equations?, inEquations on manifolds, 1982, pp. 26-51 (in Russian).
[9]WahlquistH. D. and EstabrookF. B.: ?Prolongation Structures of Nonlinear Evolution Equations?,J. Math. Phys. 16 (1975), 1-7. · Zbl 0298.35012 · doi:10.1063/1.522396