Orbital stability of harmonic waves in the Klein–Gordon equation against localized perturbations

We investigate the stability and long-term behavior of spatially periodic harmonic waves in the complex nonlinear Klein–Gordon equation under localized perturbations. Such perturbations render the wave neither localized nor periodic, placing its stability analysis outside the scope of the classical orbital stability theory for Hamiltonian systems developed by Grillakis, Shatah, and Strauss. Inspired by Zhidkov’s work on the stability of time-periodic, spatially homogeneous states in the nonlinear Schrödinger equation, we develop an alternative method that relies on an amplitude-phase decomposition and leverages conserved quantities tailored to the perturbation equation. We establish an orbital stability result of harmonic waves that is locally uniform in space, accommodating L2-localized perturbations as well as unbounded phase modulations. Our result is sharp in the sense that it holds up to the spectral stability boundary.