On Hamiltonian formulations of the C₁(m,a,b) equations

In this letter we re-address a class of genuinely nonlinear third order dispersive equations; C₁(m,a,b): u_t+(u^m)_x+1/b[u^a(u^b)_xx]_x=0, which among other solitary structures admit compactons, and demonstrate that certain subclasses of these equations may be cast into Hamiltonian and Lagrangian formulations resulting in new conservation laws, some of which are nonlocal. In particular, the new nonlocal conservation law of the K(n,n) equations enables us to prove that the response to a certain class of excitations cannot contain only compactons.