Drift cancellation in a parabolic similarity reduction ($u_t=u_{xx}-\frac{x}{2t}{u}_x-u^3,u=t^{-1/2}f(x/\sqrt{t})$)

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 givesConsequently the nonautonomous partial differential equation reduces to the autonomous ordinary differential equation $f^{\prime \prime }=f^3-f/2$. 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.