Compactons in Higher-Order Nesterenko's-Type Equations

Abstract

A model describing propagation of waves in a prestressed granular media is considered. The model, having the form of evolutionary partial differential equation (PDE), is obtained from the system of ordinary differential equations (ODEs) describing dynamics of a chain of prestressed granules by means of formal asymptotic expansion. It is shown in our previous papers that in the lowest asymptotic approximation, in which both nonlinear effects and the presence of media structure are taken into account, the model equation possesses traveling wave (TW) solutions with compact support (compactons) manifesting soliton properties. In this paper, we study a higher-order evolutionary PDE obtained by taking into account previously discarded terms of the asymptotic expansion, as well as another PDE (called analogue), differing from the original one in the values of parameters and having compacton solutions expressed in analytical form. Numerical and analytical studies of both the higher-order model and its analogue allow to conclude that both models have compacton solutions exhibiting some properties of “true” solitons. This, in turn, testifies the stability of the previously used model with respect to the inclusion of the discarded terms of the asymptotic expansion.