Universal breaking point asymptotic for energy spectrum of Riemann waves in weakly nonlinear non-dispersive media

TL;DR

This paper investigates the energy spectrum evolution of Riemann waves in weakly nonlinear, non-dispersive media, revealing a universal asymptotic behavior characterized by an exponential spectrum transitioning to a -8/3 power law at wave breaking.

Contribution

It demonstrates the universal asymptotic form of the energy spectrum of Riemann waves at breaking in nonlinear non-dispersive media, applicable across various physical systems.

Findings

Exponential energy spectrum deforms into a -8/3 power law at breaking.

Universal asymptotic behavior is observed for quadratic and cubic nonlinearities.

Results are relevant for magneto-hydrodynamics, oceanography, and nonlinear acoustics.

Abstract

In this Letter we study the form of the energy spectrum of Riemann waves in weakly nonlinear non-dispersive media. For quadratic and cubic nonlinearity we demonstrate that the deformation of an Riemann wave over time yields an exponential energy spectrum which turns into power law asymptotic with the slope being approximately -8/3 at the last stage of evolution before breaking. We argue, that this is the universal asymptotic behaviour of Riemann waves in any nonlinear non-dispersive medium at the point of breaking. The results reported in this Letter can be used in various non-dispersive media, e.g. magneto-hydro dynamics, physical oceanography, nonlinear acoustics.