I read an article in which the authors describe an observed phenomenon as being related to the ``classical ramp and cliff Burgers solutions''. Those are described as Burgers solutions that behave asymptotically like a combination of a ‘‘ramp’’-like solution proportional to $x/t$, together with an exponentially-decaying tail (‘‘cliff’’). I am now interested in those Burgers solutions.

I looked at several articles where those Burgers solutions are shown (with a mixed of numerics and analysis) to appear several at a time and such occurrence is related to turbulence. But that's not what I am interested in.

I am interested in an expression describing only one of those solutions. I imagine there would be a formula for the ramp ($x/t$ in the case of classical Viscous Burgers equation) and another one for the cliff. Then, there would be a Rankine–Hugoniot condition to connect the two of them. If anyone knows of such an explicit description, I would be grateful for a reference.