On the orbital stability of periodic snoidal waves for the $\phi^4-$equation

TL;DR

This paper investigates the global well-posedness and orbital stability of odd periodic traveling waves for the $\

Contribution

It introduces new results on global well-posedness and proves orbital stability using a Morse index theorem for the $\

Findings

Established global well-posedness of weak solutions.

Proved orbital stability of odd periodic waves.

Applied Morse index theorem to constrained linearized operator.

Abstract

The main purpose of this paper is to investigate the global well-posedness and orbital stability of odd periodic traveling waves for the $\phi^4$-equation in the Sobolev space of periodic functions with zero mean. We establish new results on the global well-posedness of weak solutions by combining a semigroup approach with energy estimates. As a consequence, we prove the orbital stability of odd periodic waves by applying a Morse index theorem to the constrained linearized operator defined in the Sobolev space with the zero mean property.