On the theory of nonlocal symmetries of nonlinear partial differential equations. (English. Russian original) Zbl 0604.58053

Vinogradov, A. M.; Krasil’shchik, I. S.

Sov. Math., Dokl. 29, 337-341 (1984); translation from Dokl. Akad. Nauk SSSR 275, 1044-1049 (1984).

Consider a partial differential equation, which is regarded as a subset \(y\subset {\mathcal J}^ k(\pi)\), where \(\pi\) : \(E\to M\) is a smooth bundle and \({\mathcal J}^ k(\pi)\to M\) is the bundle of k-jets. To this one associates \(y_{\infty}\), called ”infinite extension” of \(y\) which corresponds to the set of formal solutions to the equation and on which one can consider the Cartan distribution C(\(y_{\infty})\). Also consider some infinite-dimensional manifold \(\tilde y_{\infty}\) with an n- dimensional integrable distribution \(\tilde C:\) \(y\to \tilde C_ y\subset T_ y(\tilde y_{\infty})\), \(y\in \tilde y_{\infty}\) on it. A regular mapping \(\tau\) : \(\tilde y_{\infty}\to y_{\infty}\) is called a covering over \(y\) if the map \(\tau_{*,y}: \tilde C_{\nu}\to C_{\tau (\nu)}(y_{\infty})\) is an isomorphism for every \(y\in \tilde y_{\infty}\). The nonlocal symmetries of a differential equation are defined then to be the symmetries of its coverings. The main results from the paper show how one can compute these nonlocal symmetries for the Burgers equation.

Reviewer: O.Liess

MSC:

58J70	Invariance and symmetry properties for PDEs on manifolds
57R50	Differential topological aspects of diffeomorphisms
35Q99	Partial differential equations of mathematical physics and other areas of application
58A20	Jets in global analysis

Keywords:

Cartan distribution; nonlocal symmetries; Burgers equation