The decay of multiscale signals – deterministic model of the Burgers turbulence

This work is devoted to the study of the decay of multiscale deterministic solutions of the unforced Burgers’ equation in the limit of vanishing viscosity. It is well known that Burgers turbulence with a power law energy spectrum E0(k) ∼ |k| has a self-similar regime of evolution. For n < 1 this regime is characterised by an integral scale L(t) ∼ t2/(3+n), which increases with the time due to the multiple mergings of the shocks, and therefore, the energy of a random wave decays more slowly than the energy of a periodic signal. In this paper a deterministic model of turbulence-like evolution is considered. We construct the initial perturbation as a piecewise linear analog of the Weierstrass function. The wavenumbers of this function form a ”Weierstrass spectrum”, which accumulates at the origin in geometric progression. ”Reverse” sawtooth functions with negative initial slope are used in this series as basic functions, while their amplitudes are chosen by the condition that the distribution of energy over exponential intervals of wavenumbers is the same as for the continuous spectrum in Burgers turbulence. Combining these two ideas allows us to obtain an exact analytical solution for the