the sign of axial strain in the laminate may be positive. Similarly, we term the former *tensile* solitary waves, notwithstanding the fact that the sign of axial strain may be negative. Interestingly, such a separation is absent from the map for quasi-shear solitary waves, since these waves can only decrease the axial strain, as explained later.

The rest of the paper is organized as follows. Sec.$\ $\[sec:Governing-equations\] contains a description of the relevant elastodynamics problem, together with its governing vectorial equations. Sec.$\ $\[sec:Method-of-solution\] first revisits solutions of the differential relations between the strain components in the benchmark problem of a homogeneous medium [@ZIV2019mom]. In the second part of this Sec., the computational study of the heterogeneous medium is carried out, where we also show that solutions for the homogeneous medium serve as an estimators for the differential relations in the main problem. A recap of our main results together with comments on future work close the paper in Sec.$\ $\[sec:Summary\].

Problem statement and governing equations\[sec:Governing-equations\]
====================================================================

We consider an infinite periodic repetition in the $X_{1}$ direction of two hyperelastic phases, namely, $a$ and $b$, governed by the same strain energy density function $\hyper$, and different initial mass density $\refden$. We 