© 2021 Elsevier B.V.In this paper, we first consider the Rosenau equation with the quadratic nonlinearity and identify its Lie symmetry algebra. We obtain reductions of the equation to ODEs, and find periodic analytical solutions in terms of elliptic functions. Then, considering a general power-type nonlinearity, we prove the non-existence of solitary waves for some parameters using Pohozaev type identities. The Fourier pseudo-spectral method is proposed for the Rosenau equation with this single power type nonlinearity. In order to investigate the solitary wave dynamics, we generate the initial solitary wave profile by using the Petviashvili's method. Then the evolution of the single solitary wave and overtaking collision of solitary waves are investigated by various numerical experiments.