Ramp and Cliff Solutions to the Viscous Burgers Equation: Explicit Formula?

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.

• without viscosity the cliffs have the simple form $u(z)=\text{sign}\,(z)(1-e^{-|z|})$, with $z=x-ct$, see arXiv:nlin/0202059 ; I don't know of an explicit solution with viscosity. Commented May 31, 2021 at 14:55
• @Carlo Beenakker Thank you for your comment. This is the formula for the cliffs in the case of the $b$-family when $b=0$, which is not the Burgers equation. However, if more general formulas exist for the $b$-family's ramp-cliff solutions, I would be as interested, if not more, as I am interested in the Burgers case. Commented May 31, 2021 at 17:36

I think this refers to the solution of the Burgers equation $$u_t + uu_x = \gamma u_{xx}$$ with a Dirac delta initial condition $$u(x,0) = \delta(x)$$. This is derived in many textbooks, for example Whitham's Linear and Nonlinear Waves (Sect. 4.4) or Olver's Introduction to Partial Differential Equations (Sect. 8.4). Equation (8.89) in the latter source gives the solution as $$u(x,t) = 2 \sqrt{\frac{\gamma}{\pi t}} \, \frac{e^{-x^2/(4\gamma t)}}{\coth\left(\dfrac{1}{4\gamma}\right) - \operatorname{erf}\left(\dfrac{x}{2\sqrt{\gamma t}}\right)} .$$ For small $$t>0$$ the wave looks similar to a Gaussian curve, but as $$t$$ grows it spreads out and tips over to the right, taking a more triangular shape where it rises like a “ramp” and decays back to zero like a “cliff”:
(Here $$\gamma=\frac{1}{10}$$ and $$t \in \{1,2,3,\dots,10\}$$.)
• I don't think that his comment quite captured what Holm & Staley are trying to say with their equation (33) (in the arXiv version that he referred to). Their “cliff” solution in Def. 3.2 is $u(z) = c e^z$ for $z \le 0$ and $u(z) = c(2-e^{-z})$ for $z \ge 0$, or the negative of that function. The “peakon” solution in Def. 3.1 is $u(z) = c e^z$ for $z \le 0$ (same as the cliff) but $u(z) = c e^{-z}$ for $z \ge 0$ (cliff solution reflected across the line $u=c$), or the negative of that function. I wouldn't call either a shock solution, since they are continuous. Commented Jun 3, 2021 at 17:08