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One of the most interesting phenomena occuring in nonlinear media models isthe existence of wave patterns, such as kinks, solitons, compactons, peakonsand many others. There are known numerous nonlinear evolutionary PDEs,supporting soliton (multi-soliton) and compacton traveling wave (TW) solutions.Unfortunately, the vast majority of the models, with the exception ofcompletely integrable ones, do not enable to analyze the properties of solitarywaves interaction using only qualitative methods. Therefore it is instructive,when dealing with the non-integrable PDEs, to combine the qualitative treatmentwith numerical simulations. In this report we are going to present the resultsof studying compacton solutions in the continual models for granularpre-stressed chains. The model is shown to possess a pair of compacton TWsolutions which are the bright and dark compactons. First we consider thestability properties of the compacton solutions and show that both the brightand the dark compactons pass the stability test. Next we analyze the dynamicsof the compactons, simulating numerically the temporal evolution of a singlecompacton, a well as the interection of pairs of compactons, includingbright-bright, dark-dark and bright-dark pairs. To be able to simulate theevolutin of interacting compactons, we have modified the numerical scheme builtby J. de Frutos, M. A. Lopez-Marcos, and J. M. Sanz-Serna. Results ofsimulations are compared with that of evolution of corresponding impulse in thegranular pre-stressed chain.

Solitary waves in one-dimensional pre-stressed lattice and its continual analog