Spectral properties of inviscid solutions of the burgers equation with random initial conditions

Abstract

The Burgers equation with random self-similar initial conditions is investigated numerically in the inviscid limit by a parallel fast Legendre transform algorithm, using Connection Machine CM-200. The use of this equation for solving the problem of nervous impulse propagation through axons is discussed. An attempt is made to simulate recent experiments where the form of the density of propagated nerve impulses, which initially had a power spectrum close to a white noise distribution, appeared similar to the triangular pulses that arise in the inviscid Burgers equation and where the 1/f power law was observed on scales larger than the typical time interval between pulses. It is shown that in the inviscid Burgers equation model the power spectra for different types of initial conditions in the developed “Burgers turbulence” regime (i.e., at a sufficiently large time) consists of two parts with a rather sharp transition between them: The spectrum virtually coincides with the initial spectra for low wavenumbers, and the 1/f² law holds for high wavenumbers. There is no interval with an intermediate power law dependence such as 1/f. It is inferred that the true 1/f spectrum of nerve impulses propagating through axons cannot be explained in terms of the Burgers equation model and that other mechanisms must be taken into account.