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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 \({\mathcal Y}\) of (nonlinear partial) differential equations, the notion of a covering \(\tilde {\mathcal Y}_{\infty}\to {\mathcal Y}_{\infty}\) is introduced, where \({\mathcal Y}_{\infty}\) is the infinite prolongation of \({\mathcal Y}\). It is shown how the local infinitesimal symmetries of \(\tilde {\mathcal Y}_{\infty}\) generate nonlocal (involving integro-differential operators) symmetries of \({\mathcal Y}_{\infty}\). This provides an effective procedure for calculating the nonlocal symmetries of \({\mathcal Y}:\) 1) find the coverings \(\tilde {\mathcal Y}_{\infty}\) of \({\mathcal Y}_{\infty}\), 2) compute the local symmetries of \(\tilde {\mathcal Y}_{\infty}\) 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 for PDEs on manifolds
57M10 Covering spaces and low-dimensional topology
35A30 Geometric theory, characteristics, transformations in context of PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application

Citations:

Zbl 0534.58005
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References:

[1] VinogradovA. M.: ?Local Symmetries and Conservation Laws,?Acta Appl. Math. 2 (1984), 21-78 (this issue). · Zbl 0534.58005
[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
[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
[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
[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
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