Abstract. The numerical simulation of the nonlinear evolution of the parallel propagating Alfvén waves in a radially expanding plasma is performed by using a kinetic-fluid model (the Vlasov–MHD model). In our study, both the nonlinear evolution of the Alfvén waves and the radial evolution of the velocity distribution function (VDF) are treated simultaneously. On the other hand, important ion kinetic effects such as ion cyclotron damping and instabilities driven by the non-equilibrium ion velocity distributions are not included in the present model. The results indicate that the steepened Alfvén wave packets outwardly accelerate ions, which can be observed as the beam components in the interplanetary space. The energy of imposed Alfvén waves is converted into the longitudinal fluctuations by the nonlinear steepening and the nonlinear Landau damping. The wave shoaling due to the inhomogeneity of the phase velocity is also observed.