Recently some essentially nonlinear proper structural models were developed in [1,2] for continua with internal structure.An important feature of these systems is the existence of exact traveling localized strain wave solutions of the governing equations for macro-and micro-strains and the dependence of their existence on the intervals of the velocity of the wave [3] The solutions obtained require specific initial conditions, and a natural question arises whether rather arbitrary initial disturbance evolves into a sequence of localized waves.