In this paper, we investigate the dynamical behaviour of the following one-dimensional dispersive viscoelasticity equation: The nonlinear function is smooth, non-convex, unbounded and satisfies some general conditions of growth. By introducing equivalent norms, based on the Hamiltonian structure of the limit dispersive problem with , we can prove the existence of global absorbing balls, and a global attractor. Depending on the parameters , we make different transformations depending on whether is positive or negative; the latter case is the most interesting one as the dissipation is not strong enough to dominate dispersion. The semigroup S(t) generated by satisfies a spectral barrier property, and the existence of inertial manifolds is proved in both cases.