An integrable equation with nonsmooth solitons

We present the bi-Hamiltonian structure and Lax pair of the equation $\rho_t= bu_x+(1/2)[(u^2-u_x^2)\rho]_x$, where $\rho=u-u_{xx}$ and $b=\mathrm{const}$, which guarantees its integrability in the Lax pair sense. We study nonsmooth soliton solutions of this equation and show that under the vanishing boundary condition $u\to0$ at the space and time infinities, the equation has both “W/M-shape” peaked soliton (peakon) and cusped soliton (cuspon) solutions.