On the spectral and modulational stability of periodic wavetrains for nonlinear Klein-Gordon equations

In this contribution, we summarize recent results [8, 9] on the stability analysis of periodicwavetrains for the sine-Gordon and general nonlinearKlein-Gordon equations. Stability is considered both from the point of view of spectral analysis of the linearized problem and from the point of view of the formal modulation theory of Whitham [12]. The connection between these two approaches is made through a modulational instability index [9], which arises from a detailed analysis of the Floquet spectrum of the linearized perturbation equation around the wave near the origin. We analyze waves of both subluminal and superluminal propagation velocities, as well as waves of both librational and rotational types. Our general results imply in particular that for the sine-Gordon case only subluminal rotationalwaves are spectrally stable. Our proof of this fact corrects a frequently cited one given by Scott [11].