Orbital stability and instability of periodic wave solutions for ϕ4n -models

Abstract In this work we study the orbital stability/instability in the energy space of a specific family of periodic wave solutions of the general ϕ 4 n -model for all n ∈ 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 − v 2 with v 2 . In the traveling case, we prove the orbital instability in the whole energy space for all n ∈ N , while in the standing case we prove that, under some additional parity assumptions, these solutions are orbitally stable for all n ∈ N . Furthermore, as a by-product of our analysis, we are able to extend the main result in (de Loreno and Natali 2020 arXiv: 2006.01305 ) (given for a different family of equations) to traveling wave solutions in the whole space, for all n ∈ N .