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D. Pelinovsky, E. Pelinovsky, E. Kartashova, T. Talipova, and A. Giniyatulin,

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Universal power law for the energy spectrum of breaking Riemann waves,

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JETP Letters 98, 237-241 (2013)

**Abstract:**

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.

**Keywords**:

Burgers equation, wave breaking, Korteweg-de Vries equation, power law.