It mainly studies the Cauchy problem of the cubic quasilinear shallow water wave equation. This equation is derived from the asymptotic expansion of the Euler equation in the full range of water waves. By using the transport equation theory and the classical Friedrichs regularization method, the local well-posedness of the solutions of this shallow water wave model in the critical Besov space is established. That is, the existence and uniqueness of the solutions of this model and the continuous dependence of the solutions on the initial values are obtained.