Conserved norms and related conservation laws for multi-peakon equations

Abstract

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_xh(u, u_x)m + u(h(u, u_x)m)_x = 0 for which the H^1 norm of u is conserved; m_t − (1/2)*u_xh'(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.