calculating the time of flight that takes the maximum of the strains to pass from a certain phase in one periodic cell to the same phase in adjacent cell, we find that the wave velocity oscillates about a constant value. These persistent oscillations of the amplitude and velocity occur through a periodic transfer of energy between the axial and transverse strains of the wave, which we consider as internal mode of the solitary wave. Similar long-lived oscillations were reported before in other fields [@Campbell1983ty; @Mantsyzov1995pra; @kivshar1998internal; @Szankowski2010prl], and recently were also reported in an acoustic system using the continuum limit of a discrete model based on mechanicalelectrical analogies [@zhang2017bright]. Our results are the first report of oscillating vector solitary waves that are generated from the equations of continuum elastodynamics.

Lastly, the axial and shear strains associated with the wave have an opposite sign, similarly to quasi-shear waves in a homogeneous compressible Gent medium (Fig.$\ $\[fig:integralCurves\]b). On account of these features, we term the wave *oscillating quasi-shear vector solitary wave*[^5]*.*

In order to characterize the dependency of the wave velocity and the relation between $\shearstrain$ and $\pressurestrain$ on the initial strain, we perform a set of numerical experiments with different initial 