= Drift cancellation in a parabolic similarity reduction
{title2=$u_t=u_{xx}-\frac{x}{2t}u_x-u^3,\quad u=t^{-1/2}f(x/\sqrt t)$}
The drift $-x u_x/(2t)$ exactly cancels the coordinate-motion term of the <similarity solution> $u=t^{-1/2}f(\eta)$, $\eta=x/\sqrt t$. Direct differentiation gives
$$
u_t=t^{-3/2}(-f/2-\eta f'/2),\qquad u_{xx}=t^{-3/2}f'',\qquad -\frac{x}{2t}u_x=-t^{-3/2}\eta f'/2.
$$
Consequently the nonautonomous <partial differential equation> reduces to the autonomous <ordinary differential equation> $f''=f^3-f/2$. The balance fixes the same length scale $x\sim\sqrt t$ as <diffusion>, but removes the first-derivative term that usually appears in the profile equation. A first-integral argument can then determine a whole profile family without solving a nonautonomous second-order equation.
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