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Stability of solitary waves in nonlinear composite media. (English) Zbl 1013.74044
Summary: We consider the system of equations for planar waves in elastic composite media in the presence of anisotropy. In anisotropic case two two-parametric families of solitary waves are found in an explicit form. In the absence of anisotropy these two families coalesce into a unique three-parameter family. The solitary wave solutions are found to be orbitally stable in a certain range of their phase speeds (range of stability) both in anisotropic as well as in isotropic materials. It is also shown that the initial value problem for 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 completes previous results on blow-up for this type of equations.

74J35 Solitary waves in solid mechanics
74E30 Composite and mixture properties
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