Dynamical Hamiltonian–Hopf instabilities of periodic traveling waves in Klein–Gordon equations

• We study the stability of periodic traveling waves of Klein–Gordon equations.
• We analyze the unstable spectrum near the imaginary axis in the complex plane.
• Dynamical Hamiltonian–Hopf instabilities are imaginary limits of unstable spectrum.
• They are located via the discriminant of Hill’s equation.
• Their existence is related to certain known instability indices.

We study the unstable spectrum close to the imaginary axis for the linearization of the nonlinear Klein–Gordon equation about a periodic traveling wave in a co-moving frame. We define dynamical Hamiltonian–Hopf instabilities as points in the stable spectrum that are accumulation points for unstable spectrum, and show how they can be determined from the knowledge of the discriminant of Hill’s equation for an associated periodic potential. This result allows us to give simple criteria for the existence of dynamical Hamiltonian–Hopf instabilities in terms of instability indices previously shown to be useful in stability analysis of periodic traveling waves. We also discuss how these methods can be applied to more general nonlinear wave equations.