 Solution

Write $L=\log \psi$, with $\psi >0$. Under the Cole-Hopf transformation $q=2ϵL_{\theta }$, direct differentiation givesSince $L_{\theta \theta }+L_{\theta }^2={\psi }_{\theta \theta }/\psi$, Burgers' equation is equivalent toThe ratio is therefore a function $a(Z)$ alone. Multiplying $\psi$ by $\exp [-\int a(Z)dZ]$ leaves $q$ unchanged and removes this scalar freedom. Thus one may choose the normalization for whichThis is the heat equation. The positive sign in $q=2ϵ{\partial }_{\theta }\log \psi$ corresponds to the negative conservative flux $-q^2/2$ used here; the more usual positive-flux convention has the opposite sign.