• DOI:10.1137/040619090
• Corpus ID: 2387503

A Simple Illustration of a Weak Spectral Cascade

@article{Muraki2007ASI,
  title={A Simple Illustration of a Weak Spectral Cascade},
  author={David J. Muraki},
  journal={SIAM J. Appl. Math.},
  year={2007},
  volume={67},
  pages={1504-1521}
}

• D. Muraki
• Published 24 August 2007
• Physics, Mathematics
• SIAM J. Appl. Math.

The textbook first encounter with nonlinearity in a partial differential equation (PDE) is the first-order wave equation: $u_t + u u_x = 0$. Often referred to as the inviscid Burgers equation, this equation is familiar to many in the theoretical contexts of characteristics, wavebreaking, or shock propagation. Another canonical behavior contained within this simplest of PDEs is the spectral cascade. Surprisingly, buried in a little-known 1964 article by G.W. Platzman is an elegant example of an exact Fourier series solution associated with a purely sinusoidal initial condition. This Fourier representation, valid prior to wavebreaking, is generalized to arbitrary continuous initial conditions on both the periodic and infinite domains. For the specific example of Platzman's original problem, the Fourier coefficients decay exponentially with increasing wavenumber, and the decay rate flattens to zero precisely at the time of wavebreaking. It is demonstrated that two simplified descriptions, a downscale truncat...