Nonlocal Trends in the Geometry of Differential Equations: Symmetries, Conservation Laws, and Bäcklund Transformations

Geometric Integrability of the Camassa–Holm Equation. II

“…Nonlocal symmetries of (systems of) partial differential equations were first studied rigorously from a geometric point of view by Vinogradov and Krasü'shchik in their seminal paper [53], and full accounts of their research appear in [30,31]. 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.…”

“…If we define the matrix S + as the standard formal linearization of the system E a = 0 "lifted" to the covering n (see [30,31] and references therein), that is,…”

“…Note that Equation (18) depends only on the vector G and the system S a = 0, and that it is the obvious "nonlocal generalization" of the condition for local symmetries [1,2,9,31,40]. Following Krasü'shchik and Vinogradov [30,31,53], we cali G the TT-shadow of the nonlocal 7r-symmetry X. We also remark that nonlocal 7r-symmetries form a Lie algebra, and that this structure exists because we work on a fixed covering of the equation at hand.…”

“…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].…”

“…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.…”