  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.
  A deterministic model of turbulence-like evolution is considered. We con-
struct 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 velocity field. We also notice that such
multiscale waves may be constructed for multidimensional Burgers' equation.
  This solution has scaling exponent h=-(1+n)/2 and its evolution in time is
self-similar with logarithmic periodicity and with the same average law L(t) as
for Burgers turbulence. Shocklines form self-similar regular tree-like struc-
tures. This model also describes important properties of the Burgers turbulence
such as the self-preservation of the evolution of large scale structures in the
presence of small scales perturbations.
