Universal power law for the energy spectrum of breaking Riemann waves

The universal power law for the spectrum of one-dimensional breaking Riemann waves is justified for the simple wave equation. The spectrum of spatial amplitudes at the breaking time $t = t_b$ has an asymptotic decay of $k^{-4/3}$, with corresponding energy spectrum decaying as $k^{-8/3}$. This spectrum is formed by the singularity of the form $(x-x_b)^{1/3}$ in the wave shape at the breaking time. This result remains valid for arbitrary nonlinear wave speed. In addition, we demonstrate numerically that the universal power law is observed for long time in the range of small wave numbers if small dissipation or dispersion is accounted in the viscous Burgers or Korteweg-de Vries equations.