explain in the sequel. These differences lead to opposite velocity-amplitude relation for the vector solitary waves in each system.

Here, we carry out a comprehensive study of the vector solitary waves that were observed by @ziv2019b, by analyzing a large set of numerical experiments that were generated using the method they developed in that work, with the following findings. First, we find that these vector solitary waves can be divided into two types which differ in their velocity and strain profile. We term the two types *quasi*-pressure and *quasi*-shear solitary waves, since in the limiting case when the heterogeneity vanishes and the strains are small, they reduce to the standard pressure and shear waves, respectively. We show that the profile of quasi-pressure vector solitary waves resembles the sech$^{2}$ function, which is the solitary wave solution of the well-known KdV equation [@korteweg1895xli]. We find that while quasi-pressure solitary waves maintain an identical wave profile as they pass between adjacent unit cells, the profile and velocity of quasi-shear solitary waves change between different cells in a periodic and permanent manner. This oscillation occurs through a periodic transfer of energy between the two polarizations of the vector solitary wave, which we consider as 