Abstract

On the Existence of Localized Invariant Solutions in Relaxing Hydrodynamics Models

V.Vladimirov (Faculty of Applied Mathematics, University of Mining and Metallurgy, Cracow, Poland); S.I.Skurativskyy (Division of Geodynamics of Explosion, S.I. Subbotin Institute of Geophysics, NAS of Ukraine, Kiev, Ukraine)

e-mail: vsan@rambler.ru;vladimir@mat.agh.edu.pl

We consider a set of invariant travelling wave (TW) solutions for two modelling systems, simulating long non-linear waves propagation in structured media. Using the methods of local non-linear analysis and numerical simulation, we show that in both cases the set of TW solutions contains localized soliton-like regimes, corresponding to homoclinic loops of three-dimensional dynamical systems obtained from the initial systems of PDE through the group theory reduction. We also show that a set of parameter vaues corresponding to the homoclinic loops appearance looks like a smooth curve in the case of the first, more simple modeling system, while the set corresponding to second system has a fractal structure

Section : 12