A complete list of conservation laws for non-integrable compacton equations of K(m, m) type

In 1993, P Rosenau and J M Hyman introduced and studied Korteweg-de-Vries-like equations with nonlinear dispersion admitting compacton solutions, u(t) + D-x(3)(u(n))+D-x(u(m)) = 0, m, n > 1, which are knownas theK(m, n) equations. In this paper we consider a slightly generalized version of theK(m, n) equations for m = n, namely, u(t) = aD(x)(3) (u(m)) + bD(x)(u(m)), where m, a, b are arbitrary real numbers. We describe all generalized symmetries and conservation laws thereof for m not equal -2,-1/2, 0, 1; for these four exceptional values of m the equation in question is either completely integrable (m = -2,-1/2) or linear (m = 0, 1). It turns out that for m not equal -2,-1/2, 0, 1 there are only three symmetries corresponding to x- and t-translations and scaling of t and u, and four non-trivial conservation laws, one of which expresses the conservation of energy, and the other three are associated with the Casimir functionals of the Hamiltonian operator D = aD(x)(3) + bD(x) admitted by our equation. Our result provides inter alia a rigorous proof of the fact that the K(2, 2) equation has just four conservation laws from the paper of P Rosenau and J M Hyman.