  We analyze the stochastic scaling laws arising in the invicid limit of the
decaying solutions of the Burgers equation. The linear scaling of the velocity
structure functions is shown to reflect the domination by shocks of the
long-time asymptotics. We exhibit new self-similar statistics of solutions
describing phases with diluted shocks. Some speculations are included on the
nature of systems whose large time behavior is described by the new statistics.
