Nonlocal symmetries and ghosts. (English) Zbl 1072.35193

Olver, Peter J.

Shabat, A.B.(ed.) et al., New trends in integrability and partial solvability. Proceedings of the NATO Advanced Research Workshop, Cadiz, Spain, June 12–16, 2002. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1835-5/hbk). NATO Science Series II: Mathematics, Physics and Chemistry 132, 199-215 (2004).

The reviewed article has a problem character, because the theory of nonlocal symmetries of nonlocal differential equations remains incomplete. The reason is the absence of calculus for nonlocal vector fields. The author proposes a new theoretical and computational basis for nonlocal theory – nonlocal differential algebra, applicable to a wide variety of nonlocal systems, and suggests the concept of a ghost symmetry that resolves the Jacobi paradox of the failure of the Jacobi identity for nonlocal vector fields. The following topics of investigation are now of importance: (a) A complete reevaluation of earlier works on nonlocal symmetries of local and nonlocal partial differential equations, (b) The establishment of a complete nonlocal variational calculus on the nonlocal differential algebra \(\mathfrak{L}^{(\infty)}\), including nonlocal conservation laws and a nonlocal form of Noether’s theorem, (c) Implementation of the ghost calculus in standard computer algebra packages.
For the entire collection see [Zbl 1050.35003].

Reviewer: Boris V. Loginov (Ul’yanovsk)

MSC:

35R10	Functional partial differential equations
58J70	Invariance and symmetry properties for PDEs on manifolds
35A30	Geometric theory, characteristics, transformations in context of PDEs
76M60	Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics

Keywords:

nonlocal vector fields; nonlocal symmetries; nonlocal differential algebra; ghost symmetries; nonlocal differential equations; nonlocal conservation laws