Paper detail

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

One of the most interesting phenomena occuring in nonlinear media models is the existence of wave patterns, such as kinks, solitons, compactons, peakons and 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 of completely integrable ones, do not enable to analyze the properties of solitary waves interaction using only qualitative methods. Therefore it is instructive, when dealing with the non-integrable PDEs, to combine the qualitative treatment with numerical simulations. In this report we are going to present the results of studying compacton solutions in the continual models for granular pre-stressed chains. The model is shown to possess a pair of compacton TW solutions which are the bright and dark compactons. First we consider the stability properties of the compacton solutions and show that both the bright and the dark compactons pass the stability test. Next we analyze the dynamics of the compactons, simulating numerically the temporal evolution of a single compacton, a well as the interection of pairs of compactons, including bright-bright, dark-dark and bright-dark pairs. To be able to simulate the evolutin of interacting compactons, we have modified the numerical scheme built by J. de Frutos, M. A. Lopez-Marcos, and J. M. Sanz-Serna. Results of simulations are compared with that of evolution of corresponding impulse in the granular pre-stressed chain.