  Non-standard parabolic regularization of gradient catastrophes for the
Burgers-Hopf equation is proposed. It is based on the analysis of all (generic
and higher order) gradient catastrophes and their step by step regularization
by embedding the Burgers-Hopf equation into multi-component parabolic systems
of quasilinear PDEs with the most degenerate Jordan block. Probabilistic
realization of such procedure is presented. The complete regularization of the
Burgers-Hopf equation is achieved by embedding it into the infinite parabolic
Jordan chain. It is shown that the Burgers equation is a particular reduction
of the Jordan chain. Gradient catastrophes for the parabolic Jordan systems are
also studied.
