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.