On the Relation between Lie Symmetries and Prolongation Structures of Nonlinear Field Equations: Non-Local Symmetries

Abstract

An algebraic method is devised to look for non-local symmetries of the pseudopotential type of nonlinear field equations. The method is based on the use of an infinite-dimensional subalgebra of the prolongation algebra L associated with the equations under consideration. Our approach, which is applied by way of example to the Dym and the Korteweg-de Vries equation, allows us to obtain a general formula for the infinitesimal operator of non-local symmetries expressed in terms of elements of L. The method could be exploited to investigate the symmetry properties of other nonlinear field equations possessing nontrivial prolongations. * )

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] …”

“…It is well known that nonlocal symmetries are of interest: they carry information about the existence of linearizing transformations (Bluman and Kumei [9], Krasü'shchik and Vinogradov [30,31,53]) and Bácklund/Darboux transformations (Krasü'shchik and Vinogradov [30,31], Schiff [50], and Reyes [45]), and they also allow us to construct highly nontrivial families of solutions to the equations at hand (Galas [21], Schiff [50], Leo et al [34,35], and Reyes [45]). In this article, we continué the investigation of nonlocal symmetries of the Camassa-Holm equation: we construct an infinite-dimensional Lie algebra of nonlocal symmetries, we observe that it contains a semidirect sum of the loop algebra over si (2, R) and the centerless Virasoro algebra, and as applications we obtain explicit solutions anda Darboux transformation, and we re-derive the CH recursion operator appearing in [20].…”

Nonlocal Symmetries and Generation of Solutions for Partial Differential Equations

“…Further development of this approach was obtained in [19,20] via reversion of the recursion operator. 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".…”

Pseudo-potentials, nonlocal symmetries and integrability of some shallow water equations

“…However, it appears that it is only in [18,29,40] that they have been used to find explicit solutions. It is then of interest to further study the work carried out in these three papers, and to show novel applications of the theory.…”

“…A short introduction to the theory of coverings is also included here: it appears in Subsection 3.2. From the point of view of this paper, the pseudo-potentials of the scalar equations considered in [18,29,40] determine geodesics of the pseudo-spherical structures described by the equations at hand, and the nonlocal symmetries G for them are obtained by studying infinitesimal deformations u → u + τ u of the dependent variable u which preserve geodesics to first order in the deformation parameter τ . Section 4 contains the application of the work carried out in Sections 2 and 3 to the the Korteweg-de Vries [25], Camassa-Holm [10] and Hunter-Saxton [19,20] equations.…”