  This work is devoted to the decay ofrandom solutions of the unforced Burgers
equation in one dimension in the limit of vanishing viscosity. The initial
velocity is homogeneous and Gaussian with a spectrum proportional to $k^n$ at
small wavenumbers $k$ and falling off quickly at large wavenumbers. In physical
space, at sufficiently large distances, there is an ``outer region'', where the
velocity correlation function preserves exactly its initial form (a power law)
when $n$ is not an even integer. When $1<n<2$ the spectrum, at long times, has
three scaling regions : first, a $|k|^n$ region at very small $k$\ms1 with a
time-independent constant, stemming from this outer region, in which the
initial conditions are essentially frozen; second, a $k^2$ region at
intermediate wavenumbers, related to a self-similarly evolving ``inner region''
in physical space and, finally, the usual $k^{-2}$ region, associated to the
shocks. The switching from the $|k|^n$ to the $k^2$ region occurs around a wave
number $k_s(t) \propto t^{-1/[2(2-n)]}$, while the switching from $k^2$ to
$k^{-2}$ occurs around $k_L(t)\propto t^{-1/2}$ (ignoring logarithmic
corrections in both instances). The key element in the derivation of the
results is an extension of the Kida (1979) log-corrected $1/t$ law for the
energy decay when $n=2$ to the case of arbitrary integer or non-integer $n>1$.
A systematic derivation is given in which both the leading term and estimates
of higher order corrections can be obtained. High-resolution numerical
simulations are presented which support our findings.
