Double upscaling procedure for the Sine–Gordon equation with highly-oscillating coefficients: Homogenization and modulation equations

We study the sine-Gordon equation with $h$-periodic in space coefficients. Leading-order homogenization yields an effective sine-Gordon equation for which traveling wave periodic solutions of wavelength $\lambda \gg h$ can be determined. The periodic solutions are then modulated on a scale $\Lambda \gg \lambda$. As we know, the corresponding Whitham equations are elliptic, which ensures that the periodic solution is unstable. However, the instability scenarios are not universal. In this paper, such scenarios are described both in the low and high energy regimes and for supersonic compared to the averaged sound speed case. In the low energy case the space derivatives of the solutions “explode” in finite time (a caustic appears), while in the high energy case the solutions grow at most linearly in time.