Analytic methods in the stability of periodic wavetrains for nonlinear wave equations of Klein-Gordon type

In this talk I present a detailed analysis of stability properties of
periodic traveling wave solutions of nonlinear Klein-Gordon equations with
periodic potential. Stability is considered from the point of view of
spectral analysis of the linearized problem (spectral stability), from the
point of view of wave modulation theory (the strongly nonlinear theory due
to Whitham as well as the weakly nonlinear theory of wave packets), and in
the orbital (nonlinear) framework. I will review a recent trend in the
stability analysis of periodic waves based on Evans function techniques,
which provide rigorous results on cases previously studied by the (formal)
physical modulation theory of Whitham. The connection between these two
different approaches is made through a modulational instability index. In
the nonlinear Klein-Gordon case, we analyse waves of both subluminal and
superluminal propagation velocities, as well as waves of both librational
and rotational types. We prove that only subluminal rotational waves are
spectrally stable and establish exponential instability in the other three
cases. The proof corrects a frequently cited result given by Scott (1965).
In addition, I will show that the spectral information is crucial in the
establishment of the orbital, nonlinear stability for subluminal rotational
wavetrains. This is joint work with C.K.R.T. Jones (North Carolina), P.D.
Miller (Michigan), J. Angulo (Sao Paulo) and R. Marangell (Sydney).