  We present a phenomenological approach to dispersion in nonlinear elasticity.
A simple, thermomechanically sound, constitutive model is proposed to describe
the (non-dissipative) properties of a hyperelastic dispersive solid, without
recourse to a microstructure or a special geometry. As a result, nonlinear and
dispersive waves can travel in the bulk of such solids, and special waves
emerge, some classic (periodic waves or pulse solitary waves of infinite
extend), some exotic (kink or pulse waves of compact support). We show that for
incompressible dispersive power-law solids and forth-order elasticity solids,
solitary waves can however only exist in the case of linear transverse
polarization. We also study the influence of pre-stretch and hardening. We
provide links with other (quasi-continuum, asymptotic) theories; in particular,
an appropriate asymptotic multiscale expansion specializes our exact equations
of motion to the vectorial MKdV equation, for any hyperelastic material.
