= Time normalization in the Cole-Hopf transformation
{title2=$\Psi_t-\nu\Psi_{xx}=C(t)\Psi$}
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 $\Psi$.
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