The solution of the Cauchy problem for the equation of longitudinal displacement wave propagation in an infinitely long elastic rod is considered taking the physical and geometric nonlinearities of the material, the wave dispersion, and inhomogeneities of the second and third order elastic moduli into account. A slow change in the elastic moduli in the wave propagation direction results in a perturbation of the equation of the problem solvable by the method of multiscale decomposition. It is shown that for certain initial data the solution of the problem is a soliton in the longitudinal displacement velocity. The soliton parameters are determined by the elastic moduli of the material, and its propagation over the rod is accompanied by a low-amplitude long- wave (plateau). Relations are derived between the elastic moduli for which the soliton amplitude remains constant or the plateau is not formed behind the main impulse. Under other initial conditions the Cauchy problem is solved numerically, and shaping of the solitary waves is investigated. Soliton properties are detected in solutions of the solitary-wave type for the longitudinal velocity of displacement in the presence of slow and small changes in the elastic moduli of the material or the rod cross-sectional area.