Nonlocal symmetries and the Kaup–Kupershmidt equation

Abstract

Computational and geometric aspects of nonlocal (infinitesimal) symmetries of nonlinear partial differential equations are considered. In particular, the relation of nonlocal symmetries with classical, generalized and internal symmetries is briefly discussed. A nonlocal symmetry for the Kaup–Kupershmidt equation is introduced and studied in some detail. Some explicit particular solutions are found with its help, and a Darboux-like transformation is also obtained.

Geometric Integrability of the Camassa–Holm Equation. II

“…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] …”

“…(1) The ACH equation admits a pseudo-potential y determined by the compatible equations (47)- (52) of the associated CH equation is that of Figure 2 whenever u, p, y, 5, /9, X solve the augmented ACH system (41)- (45). D…”

“…We essentially follow Krasü'shchik and Vinogradov's approach [30,31], although we formúlate their theory as explicitly as we can. We partially base this section on the expositions appearing in the papers [45,46] by the second-named author. We then present Lie algebras of nonlocal symmetries for the CH and ACH equations in Section 3.…”

“…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].…”

New Nonlocal Symmetries of Diffusion-Convection Equations and Their Connection with Generalized Hodograph Transformation

“…For this purpose we use the additional Lie symmetries (that are additional operators of invariance Lie algebras of higher dimension) of the intermediate equations arising at steps of this transformation, which consists of potential substitution and usual hodograph transformation. One can find many references and the extensive bibliography in researches devoted to studying the potential symmetries of nonlinear partial differential equations and systems [2][3][4][5][6][7][8]. The notion of potential symmetries of differential equations was introduced by Bluman et al [2,3].…”