Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2022 / Section B / 5 / c / 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 givesHence the most general equation implied by this substitution isThe 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 $e^{-{\int }^{t}C(s)ds}\Psi$ sets $C=0$. With this conventional normalization,the heat equation.

For each $j=1,2$, the function ${\Psi }_j=e^{\nu {\kappa }_j^{2}t+{\kappa }_{j}x}$ 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,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.