We study the moving compacton-like kink in the system of a purely anharmonic lattice with symmetric onsite potential by a direct algebraic method. It is found that the localization of the compactons is related to the nonlinear coupling parameter Cnl and the potential barrier height V0 of the double well potential, and the velocity of the propagation of the compacton is determined by the localization parameter q and the potential barrier height V0. Numerical calculation results demonstrate that th...