Orbital stability and instability of periodic wave solutions for $\phi^{4n}$-models

TL;DR

This paper investigates the stability and instability of specific periodic wave solutions in the general $\,phi^{4n}$-model, revealing conditions for their orbital stability or instability in the energy space.

Contribution

It provides a comprehensive analysis of orbital stability and instability for periodic solutions of the $\,phi^{4n}$-model across all natural numbers $n$, including new stability criteria.

Findings

Traveling solutions are orbitally unstable in the energy space for all $n$.

Standing solutions are orbitally stable under certain parity conditions.

The stability analysis connects periodic solutions with Kink solutions in the phase space.

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 $\phi^{4n}$-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 $n\in\mathbb{N}$.