Nonlocal Symmetries and Ghosts

The local theory of symmetries of differential equations has been well-established since the days of Sophus Lie. Generalized, or higher order symmetries can be traced back to the original paper of Noether, [32], but were not exploited until the discovery that they play a critical role in integrable (soliton) partial differential equations, cf. [30, 33, 35]. While the local theory is very well developed, the theory of nonlocal symmetries of nonlocal differential equations remains incomplete. Particular results on certain classes of nonlocal symmetries and nonlocal differential equations have been developed by several groups, including Abraham–Shrauner et. al., [1, 2, 3, 13], Bluman et. al., [5, 6, 7], Chen et. al., [8, 9, 10], Fushchich et. al., [17], Guthrie and Hickman, [20, 21, 22], Ibragimov et. al., [4], [23; Chapter 7], and many others, [11, 12, 16, 18, 19, 24, 28, 29, 31, 37]. Perhaps the most promising proposed foundation for a general theory of nonlocal symmetries is the Krasilshchik-Vinogradov theory of coverings, [25, 26, 27, 38, 39]. However, their construction relies on the a priori specification of the underlying differential equation, and so, unlike local jet space, does not form a universally valid foundation for the theory.