`Relativistic' propagation of instability fronts in nonlinear Klein-Gordon equation dynamics

We consider propagation of instability fronts in conservative nonlinear wave systems by the Whitham method. Whitham modulation equations for periodic solutions of the generalized Klein-Gordon equation are solved in the limit of asymptotically large times, when the size of the instability wave region is much greater than the size of the initial localized disturbance, so the solution reaches the self-similar regime. The general self-similar solution is illustrated by two typical examples of the nonlinearity function. It is shown that in these models the instability fronts propagate with maximal group velocity.