Nonlocal symmetries. Heuristic approach

Geometric Integrability of the Camassa–Holm Equation. II

“…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].…”

“…As pointed out by Krasü'shchik, Vinogradov, and their coworkers (see [30,31] and references therein), standard Bácklund transformations, zero curvature representations, conservation laws, Wahlquist-Eastbrook pseudo-potentials, and pseudo-differential recursion operators give rise to coverings and so, we may expect, to nonlocal symmetries; see, for instance, [2,6,9,21,23,24,[29][30][31]. The symmetries presented below depend on pseudo-potentials of the equations at hand, which we introduce as follows:…”

“…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.…”

Similarity: generalizations, applications and open problems

“…[37][38][39][40][41][42]); -extend the spaces of admitted symmetries and hence construct further similarity solutions (cf. [10,11,33,[42][43][44][45][46][47]); -extend the applications to construct solutions from admitted symmetries to include admitted "symmetries" arising from generalizations (similarity solutions arising from the nonclassical and other related methods) (cf. [10,42,[48][49][50][51][52][53][54][55]); -efficiently solve the (overdetermined) linear systems of determining equations for symmetries or conservation law multipliers and solve the nonlinear systems of determining equations for the nonclassical and related methods through the development of symbolic manipulation software (cf.…”

A symmetry-based method for constructing nonlocally related partial differential equation systems

“…A related heuristic approach to find nonlocal symmetries, called quasilocal symmetries, was presented in [6,7], where rich sets of examples were exhibited, especially those involving the gas dynamics equations.…”