Ghost Symmetries

Abstract

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] [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 naïve 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

Geometric Integrability of the Camassa–Holm Equation. II

“…For instance, a subtle result proved in [3] states that if a system of partial differential equations E a = 0 satisfies a technical condition (dubbed the "descent property"), then every infernal symmetry of E a = 0 is obtained by restriction to S (l> of a first order generalized symmetry of E a = 0. Now infernal symmetries form a Lie algebra on S (l> but if they are not restrictions of external symmetries, there is no reason why they should form a Lie algebra on the jet space J (l> E. As (35) shows, the same phenomenon appears in the realm of nonlocal symmetries, reflecting the fact that they are defined on coverings of the equation manifold of a (system of) differential equations, and not on some "universal" jet space as it happens with local symmetries [40,42]. Thus, it appears to us that (see also [45] …”

“…Other approaches to nonlocal symmetries (of formal, algebraic, or heuristic nature) have also been advanced in the literature (see, e.g., [1,2,9,41,42]), but we believe that the Krasü'shchikVinogradov geometric viewpoint remains the most satisfactory approach to nonlocalities at our disposal. We mention three interesting earlier references on applications of the Krasü'shchik-Vinogradov theory: van Bemmelen [6], Dodd [18], and Kirnasov [29].…”

“…We do not expect the "space of all nonlocal symmetries" of a given equation to possess a Lie algebra structure. This important point is discussed in [30] and also in [41,42], in which a proposal for transforming a "space of nonlocal evolutionary vector fields" into a Lie algebra is made.…”

Time-Dependent Recursion Operators and Symmetries

“…18) where p x = xu, n x = u 2 , r x = xv, w x = u − v, g x = uh and h x = uw. More generally the multicomponent nonautonomous JKdV system [24] is 19) where s i jk are constants, symmetric in the lower indices and satisfy the Jordan identities 20) with F i plj = s i jk s k lp − s i lk s k jp .…”

Nonlocal Symmetries and Generation of Solutions for Partial Differential Equations

“…Nonlocal symmetries of the pseudopotential type in sense of prolongation structures of Estabrook and Wahlquist were considered in [21]. It was discovered in the paper [22] that the Jacobi identity for characteristics of nonlocal vector fields "appears to fail for the usual characteristic computations" that led to the notion called "ghost symmetries". That also showed impossibility of discussion of Lie algebraic properties for general nonlocal symmetries of DEs.…”