= Solution
Set $u=-2\nu\partial_x\log\phi$. Direct differentiation gives
$$
u_t+uu_x-\nu u_{xx}=-2\nu\partial_x\left(\frac{\phi_t-\nu\phi_{xx}}{\phi}\right).
$$
Thus the <Cole-Hopf transformation> maps every nonvanishing solution of the <heat equation> to a solution of the <viscous Burgers equation>. Conversely the Burgers equation makes the ratio inside the derivative a function of time alone; a time-dependent rescaling of $\phi$, which leaves $u$ unchanged, removes that function and gives the heat equation exactly.
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