A new integrable equation with no smooth solitons

In this paper, we propose a new completely integrable equation:mt=121m2xxx-121m2x,which has no smooth solitons. This equation is shown to have bi-Hamiltonian structure and Lax pair, which imply integrability of the equation. Studying this new equation, we develop two new kinds of soliton solutions under the inhomogeneous boundary condition lim|x|→∞m=B where B is nonzero constant. One is continuous and piecewise smooth “W/M”-shape-peaks solitary solution and the other one-single-peak soliton. The two new kinds of peaked solitons can not be written as the regular type peakon: ce-|x-ct|, where c is a constant. We will provide graphs to show those new kinds of peaked solitons.