Time normalization in the Cole-Hopf transformation (${\Psi }_t-\nu {\Psi }_{xx}=C(t)\Psi$)

Substituting $u=-2\nu {\partial }_x\log \Psi$ into the viscous Burgers equation gives ${\partial }_x[({\Psi }_t-\nu {\Psi }_{xx})/\Psi ]=0$. Integration permits the arbitrary function $C(t)$ in the title. Multiplication of $\Psi$ by $e^{-\int C(t)dt}$ preserves $u$ and reduces its equation to the heat equation. Thus the heat-equation representation uses a permissible time normalization rather than a condition uniquely imposed on every representative $\Psi$.