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


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


Zbl 0534.58005
Full Text: DOI


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