Ghost Symmetries

We introduce the notion of a ghost characteristic for nonlocal differential equations. Ghosts are essential for maintaining the validity of the Jacobi identity for the characteristics of nonlocal vector fields. 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, [24], and received added importance after the discovery that they play a critical role in integrable (soliton) partial differential equations, cf. [25]. While the local theory is very well developed, the theory of nonlocal symmetries of nonlocal differential equations remains incomplete. Several groups, including Chen et. al., [5, 6, 7], Ibragimov et. al., [1], [16, Chapter 7], Fushchich et. al., [11], Guthrie and Hickman, [13, 14, 15], Kaptsov, [17], Bluman et. al., [2, 3, 4], and others, [8, 10, 12, 21, 27], have proposed a foundation for such a theory. Perhaps the most promising is the KrasilshchikVinogradov theory of coverings, [18, 19, 20, 28, 29], but this has the disadvantage that 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. Recently, the second and third author made a surprising discovery that the Jacobi identity for nonlocal vector fields appears to fail for the usual characteristic computations! This observation arose during an attempt to systematically investigate the symmetry properties of the Kadomtsev–Petviashvili (KP) equation, previously studied in [6, 7, 9, 22, 23]. The observed violation of the näıve version of the Jacobi identity applies to all of the preceding nonlocal symmetry calculi, and, consequently, many statements about the “Lie algebra” of nonlocal symmetries of differential equations are, by in large, not valid as stated. This indicates the need for a comprehensive re-evaluation of all earlier results on nonlocal symmetry algebras. In this announcement, we show how to resolve the Jacobi paradox through the introduction of what we name “ghost characteristics”. Ghost characteristics are genuinely Copyright c © 2001 by P.J. Olver, J.A. Sanders and J.P. Wang 2 P.J. Olver, J.A. Sanders and J.P. Wang nonlocal objects that have no counterpart in the local theory, but serve to provide missing terms that resolve apparent contradictions that have, albeit unnoticed, plagued the nonlocal theory. Details of the construction and proofs will appear in a forthcoming paper. We shall assume that the reader is familiar with the basic theory of generalized symmetries in the local jet bundle framework. We adopt the notation and terminology of [25] without further comment. We specify p independent variables x = (x1, . . . , xp) and q dependent variables u = (u1, . . . , uq), with uJ = D J(uα) denoting the induced jet space coordinates. Here DJ = D1 1 · · ·Dp p denotes the corresponding total derivative operator. In the local version, multi-indices J = (j1, . . . , jp) are assumed to be non-negative, J ≥ 0, meaning jν ≥ 0 for ν = 1, . . . , p. We begin with the usual local generalized vector fields in evolutionary form