Krasil’shchik, I. S.; Vinogradov, A. M. Nonlocal symmetries and the theory of coverings: An addendum to A. M. Vinogradov’s ’Local Symmetries and Conservation Laws’. (English) Zbl 0547.58043 Acta Appl. Math. 2, 79-96 (1984). 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 Cited in 2 ReviewsCited in 58 Documents 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 Keywords:(nonlinear partial) differential equations Citations:Zbl 0534.58005 PDF BibTeX XML Cite \textit{I. S. Krasil'shchik} and \textit{A. M. Vinogradov}, Acta Appl. Math. 2, 79--96 (1984; Zbl 0547.58043) Full Text: DOI OpenURL 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.