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

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when toggle format	what		by	license	comment
Jun 3, 2021 at 17:08	comment	added	Hans Lundmark		The “ramp and cliff” solutions in their Fig. 1 are computed numerically; as far as I'm aware, no explicit expressions are known.
Jun 3, 2021 at 17:08	comment	added	Hans Lundmark		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.
Jun 3, 2021 at 14:54	comment	added	Gateau au fromage		Thank you! I will take a look at that. That's really useful. I was expecting it to be a shock. Such as the solution Carlo Beenakker suggested above for the b-family. If you have any insight abut that, I would appreciate. But, if not, please know that I am really appreciative for your response.
Jun 3, 2021 at 14:49	vote	accept	Gateau au fromage
Jun 3, 2021 at 12:05	history	answered	Hans Lundmark	CC BY-SA 4.0