The equations of nonlinear elastodynamics have attracted the attention of pure and applied mathematicians since the second half of the last century. Nevertheless, many aspects of this theory are still obscure or need a more systematic analysis. In this paper we study the wave propagation in an isotropic elastic solid by taking into account the effects due to dispersion. Specifically, we determine a class of global (in both time and space) solutions, find exact solutions for longitudinal travelling waves within the fourth-order theory of elasticity, and derive two asymptotic reductions of the governing equations for waves of small amplitudes. •We study the wave propagation in a nonlinear isotropic elastic solid by taking into account the effects due to dispersion. We determine a class of global (in both time and space) solutions.•We find exact solutions for longitudinal travelling waves within the fourth-order theory of elasticity.•We derive two asymptotic reductions of the governing equations for waves of small but finite amplitudes.