Conserved norms and related conservation laws for multi-peakon equations

All nonlinear dispersive wave equations in the general class $m_t+f(u,u_x)m+(g(u,u_x)m{)}_x=0$ are known to possess multi-peakon weak solutions. A classification is presented for families of multi-peakon equations in this class that possess conserved momentum; conserved $H^1$ norm of $u$; conserved $H^2$ norm of $u$; conserved $L^2$ norm of $m$; related conservation laws. The results yield, among others, two interesting wide families of equations: $m_t+2u_{x}h(u,u_x)m+u(h(u,u_x)m{)}_x=0$ for which the $H^1$ norm of $u$ is conserved; $m_t-\frac{1}{2}{u}_x{h}^{\mathrm{\prime }}(u)m+(h(u)m{)}_x=0$ for which the $L^2$ norm of $m$ is conserved. The overlap of these two families yields a singular equation which is nevertheless found to possess both smooth solitary wave solutions and peakon travelling wave solutions.