  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 solitary wave profile as an initial
condition 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.
