= Finite-exponential Cole-Hopf solution
{title2=$q=\frac{\sum_jq_je^{A_j}}{\sum_je^{A_j}},\quad A_j=\frac{q_j(\theta-\theta_j)+q_j^2Z/2}{2\epsilon}$}
For the negative-flux <viscous Burgers equation> $q_Z-qq_\theta=\epsilon q_{\theta\theta}$, set $q=2\epsilon\partial_\theta\log\psi$. A sum of exponentials in the initial heat data evolves into $\psi=\sum_je^{A_j}$, 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.
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