part and a localized part about $X_{1}=0$, namely,

$$\epsilon_{i}\left(X_{1},t=0\right)=\begin{cases}
\initial{\epsilon_{i}}+\amp{\epsilon_{i}}\cos\frac{\pi X_{1}}{w}, & -\frac{w}{2}<X_{1}<\frac{w}{2},\\
\initial{\epsilon_{i}}, & \mathrm{otherwise},
\end{cases}\label{eq:initialState}$$

where $i=A,\,T$ denotes the axial and shear strains, respectively, and superscripts $\left(U\right)$ and $\left(L\right)$ correspond to the uniform and localized parts of the pre-strain, respectively.

\

As mentioned in Sec.$\ $\[sec:Governing-equations\], to solve the equations of motion we apply the finite-volume method of @ziv2019b, which was developed for analyzing motions comprising two coupled displacements in soft laminates.

We begin by showing in Fig.$\ $\[fig:pressureSoliton\](a) the distribution of the axial (left panel) and shear (right panel) strains as functions of space and time, when the laminate is subjected to a pre-strain with the parameters

$$\initial{\shearstrain}=2.2,\ \initial{\pressurestrain}=0,\ \amp{\shearstrain}=0.15,\ \amp{\pressurestrain}=0.3,\ w=0.08\,\mathrm{m}.\label{eq:quasi-pressure-strains}$$

We observe that a rightward propagating wave is generated. A movie of this propagation is also provided in the supplementary material online under the name S1. Remarkably, while the wave profile is different in phases $a$ and $b$, as it should, the respective profile in each phase is maintained in passing from one unit cell to another. Furthermore, by 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 