In this work we study the orbital stability/instability in the energy space of a specific family of periodic wave solutions of the general $\phi^{ 4n}$-model for all $n\in\mathbb{ N}$. This family of periodic solutions are orbiting around the origin in the corresponding phase portrait and, in the standing case, are related (in a proper sense) with the aperiodic Kink solution that connect the states $-\tfrac{ 1}{ 2}$ with $\tfrac{ 1}{ 2}$. In the traveling case, we prove the orbital instability in the whole energy space for all $n\in\mathbb{ N}$, while in the standing case we prove that, under some additional parity assumptions, these solutions are orbitally stable for all $n\in\mathbb{ N}$.
Categories: math.AP