Stability of solitary waves in nonlinear composite media

The system of equations for planar waves in elastic composite media in the presence of anisotropy is considered. In anisotropic case two two-parametric families of solitary waves are found in an explicit form. In case of the absence of anisotropy these two families coalesce into the unique three parametric family. The solitary wave solutions are found to be orbitally stable in a certain range of their phase speeds (range of stability) both in an anisotropic as well as in an isotropic materials. It is also shown that the initial value problem for the governing equations is locally well posed which is needed to prove the stability result. The local well-posedness of the initial value problem along with stability of solitary waves implies global existence result provided the initial data lie in a neighbourhood of a stable solitary wave. This complements the previous results of blow-up for this type of equations.