Compacton equations and integrability: The Rosenau-Hyman and Cooper-Shepard-Sodano equations

Abstract

We study integrability –in the sense of admitting recursion operators– of two nonlinear equations which are known to possess compacton solutions: the K(m, n) equation introduced by Rosenau and Hyman Dt(u) + Dx(um) + Dx3 (un) = 0 , and the CSS equation introduced by Coooper, Shepard, and Sodano, Dt(u) + ul−2Dx(u) + αpDx(up−1u2x) + 2αDx2 (upux) = 0 . We obtain a full classification of integrable K(m, n) and CSS equations; we present their recursion operators, and we prove that all of them are related (via nonlocal transformations) to the Korteweg-de Vries equation. As an application, we construct isochronous hierarchies of equations associated to the integrable cases of CSS.