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

Abstract

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.

Figures (3)

• Figure 1: Envelopes of amplitudes $a$ for the sine-Gordon instability wave at different moments of time: (a) $t=0.1$; (b) $t=0.5$; (c) $t=1$; (d) $t=2$; (e) $t=3$. The dashed line corresponds to the maximal amplitude $\varphi_m=\pi$.
• Figure 2: Dependence of the amplitude $a(0,t)$ at the center of the expanding wave structure on time $t$. Stars correspond to the asymptotic formula (\ref{['eq20']}), so it provides a good enough approximation for $t\gtrsim0.5$.
• Figure 3: Profile of the instability waves at $t=20$ in the two-well potential model. The modulated wave (\ref{['eq26']}) is depicted by a blue line and the enveloped $a_{\pm}$ following from Eq. (\ref{['eq27']}) are shown by dashed lines.