Evolution of the primary pulse in one-dimensional granular crystals subject to on-site perturbations: analytical study.

The propagation of the primary pulse through an uncompressed granular chain subject to external on-site perturbation is studied. An 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. An additional interesting finding made in the present paper corresponds to the chains subject to a special type of perturbation, including the terms leading to dissipation and those acting as an energy source. In the paper, it is shown that an application of such a perturbation may lead to the formation of stable stationary primary pulses propagating with constant amplitudes and acting as attractors for the initially unperturbed Nesterenko solitary waves. Interestingly enough, the developed analytical procedure provides an extremely close estimation for the amplitudes of these stationary primary pulses as well as predicts zones of their stability. In conclusion, we would like to stress that the developed analytical model has demonstrated spectacular agreement with the results of the direct numerical simulations, and this is for various configurations considered in the current paper.