Analytical modeling of essentially nonlinear strain waves in solids is developed based on a proper assumption about their complex internal structure. The exact travelling solitary wave solutions are used as a tool to study nonlinear and dispersive features of the solid. Various scenarios are found of the propagation of the macro-strain localized waves and corresponding waves of structural deviations. It is found by explicit analytical relationships that they are realized depending upon the velocity of the wave. This may help to study the material properties by dynamical methods.