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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 ${\cal Y}$ of (nonlinear partial) differential equations, the notion of a covering $\tilde {\cal Y}\sb{\infty}\to {\cal Y}\sb{\infty}$ is introduced, where ${\cal Y}\sb{\infty}$ is the infinite prolongation of ${\cal Y}$. It is shown how the local infinitesimal symmetries of $\tilde {\cal Y}\sb{\infty}$ generate nonlocal (involving integro-differential operators) symmetries of ${\cal Y}\sb{\infty}$. This provides an effective procedure for calculating the nonlocal symmetries of ${\cal Y}:$ 1) find the coverings $\tilde {\cal Y}\sb{\infty}$ of ${\cal Y}\sb{\infty}$, 2) compute the local symmetries of $\tilde {\cal Y}\sb{\infty}$ using, for example, the technique of the second author [ibid. 2, 21-78 (1984; Zbl 0534.58005)].
Reviewer: S.V.Duzhin

##### MSC:
 58J70 Invariance and symmetry properties 57M10 Covering spaces (manifolds) 35A30 Geometric theory for PDE, characteristics, transformations 35Q99 PDE of mathematical physics and other areas
##### Keywords:
(nonlinear partial) differential equations
Full Text:
##### 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