= Solution
For $\nu>0$ and positive $\Psi$, set $f=\log\Psi$, so the <Cole-Hopf transformation> is $u=-2\nu f_x$. Substitution into the <viscous Burgers equation> gives
$$
u_t+uu_x-\nu u_{xx}
=-2\nu\partial_x\left(f_t-\nu f_{xx}-\nu f_x^2\right)
=-2\nu\partial_x\left(\frac{\Psi_t-\nu\Psi_{xx}}{\Psi}\right)=0.
$$
Hence the most general equation implied by this substitution is
$$
\Psi_t-\nu\Psi_{xx}=C(t)\Psi.
$$
The <time normalization in the Cole-Hopf transformation> accounts for the arbitrary function $C(t)$: multiplying $\Psi$ by a nonzero function of time leaves $\partial_x\log\Psi$ and therefore $u$ unchanged. Replacing $\Psi$ by $e^{-\int^tC(s)ds}\Psi$ sets $C=0$. With this conventional normalization,
$$
\boxed{\partial_t\Psi=\nu\partial_x^2\Psi,}
$$
the <heat equation>.
For each $j=1,2$, the function $\Psi_j=e^{\nu\kappa_j^2t+\kappa_jx}$ obeys $\partial_t\Psi_j=\nu\kappa_j^2\Psi_j=\nu\partial_x^2\Psi_j$. By the <superposition principle> for this linear <partial differential equation>,
$$
\boxed{\Psi=e^{\nu\kappa_1^2t+\kappa_1x}+e^{\nu\kappa_2^2t+\kappa_2x}}
$$
is a solution, positive for all real $x,t,\kappa_1,\kappa_2$. Here $\kappa_j$ has units $\mathrm{m^{-1}}$ so both exponents are dimensionless. The positivity ensures that its logarithm and the resulting <velocity> are regular.
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