 Finite-exponential Cole-Hopf solution

For the negative-flux viscous Burgers equation $q_Z-q{q}_{\theta }=ϵq_{\theta \theta }$, set $q=2ϵ{\partial }_{\theta }\log \psi$. A sum of exponentials in the initial heat data evolves into $\psi ={\sum }_j{e}^{A_j}$, giving the displayed softmax function average. Two terms produce a traveling tanh shock from the lower to the higher state, with speed minus their mean.