Nonlocal Symmetries, Compacton Equations, and Integrability

We review the theory of nonlocal symmetries of nonlinear partial differential equations and, as examples, we present infinite-dimensional Lie algebras of nonlocal symmetries of the Fokas-Qiao and Kaup-Kupershmidt equations. Then, we consider nonlocal symmetries of a family which contains the Korteweg-de Vries (KdV) and (a subclass of) the Rosenau-Hyman compacton-bearing K(m, n) equations. We find that the only member of the family which possesses nonlocal symmetries (of a kind specified in Sec. 3 below) is precisely the KdV equation. We take this fact as an indication that the K(m, n) equations are not integrable in general, and we use the formal symmetry approach of Shabat to check this claim: we prove that the only integrable cases of the full K(m, n) family are the KdV and modified KdV equations. © 2013 World Scientific Publishing Company.