  Propagation of primary pulse through an un-compressed granular chain subject
to external on-site perturbation is studied. Analytical procedure predicting
the evolution of the primary pulse is devised for the general form of the
on-site perturbation applied on the chain. The validity of the analytical model
is confirmed with several specific granular setups such as, chains mounted on
the nonlinear elastic foundation, chains perturbed by the dissipative forces as
well as randomly perturbed chains. Additional interesting finding made in the
present study corresponds to the chains subject to a special type of
perturbations including the terms leading to dissipation and those acting as an
energy source. It is shown in the study that application of such perturbation
may lead to formation of stable stationary shocks acting as attractors for the
initially unperturbed, propagating Nesterenko solitary waves. Interestingly
enough the developed analytical procedure provides an extremely close
estimations for the amplitudes of these stationary shocks as well as predicts
zones of their stability. In conclusion we would like to stress that the
developed analytical model have demonstrated spectacular correspondence to the
results of direct numerical simulations for all the setups considered in the
study.
