Orbital stability and instability of periodic wave solutions for the $\phi^4$-model

In this work we find explicit periodic wave solutions for the classical $\phi^4$-model, and study their corresponding orbital stability/instability in the energy space. In particular, for this model we find at least four different branches of spatially-periodic wave solutions, which can be written in terms of Jacobi elliptic functions. Two of these branches correspond to superluminal waves, a third-one corresponding to a sub-luminal wave and the remaining one corresponding to a stationary complex-valued wave. In this work we prove the orbital instability of both, real-valued sub-luminal traveling waves and stationary complex-valued waves. Furthermore, we prove that under some additional hypothesis the zero-speed sub-luminal case is stable. This latter case is related (in some sense) to the classical Kink solution.