We have Burgers equation
$$ u_t - Du_{xx} - u u_x = 0$$
We want to prove that we can transform this PDE into the heat equation $v_t = v_{xx}$ if we use $u = \phi_x$ where $\phi = - 2 D \ln v $.
thoughts
Notice
$$ u_t = \frac{ \partial }{\partial x} \phi_t = \left( - 2 D \frac{ v_t }{v} \right)_x $$
and our equation becomes
$$ \left( - 2 D \frac{ v_t }{v} \right)_x = \left( D u_x + \frac{u^2}{2} \right)_x \implies - 2 D \frac{ v_t }{v} = D u_x + \frac{u^2}{2} +C(t)$$
where $C$ is any arbitrary function. Any ideas in how can we further simplify the above?