 Solution

In the moving stretched coordinates, ${\partial }_t={\partial }_{\tau }-s^{\prime }(\tau ){\epsilon }^{-1}{\partial }_{\xi }$ and ${\partial }_x={\epsilon }^{-1}{\partial }_{\xi }$. Multiplying the damped Burgers equation by $\epsilon$ givesDiscarding the order-$\epsilon$ terms yields the leading Viscous Burgers travelling shock equation $\boxed{(u-s^{\prime })u_{\xi }=u_{\xi \xi }}$. Integrate across a nonconstant transition whose derivative vanishes at either end:Multiplication by $u$ and a further integration by parts giveThe first identity is the Rankine-Hugoniot condition, giving $s^{\prime }=(u_{-}+u_{+})/2$. Substitution into the second makes its left side $(u_{+}-u_{-}{)}^3/12$, so the viscosity-selected entropy shock has $u_{-}>u_{+}$.

After the first characteristic collapse, the shock is fed by the left constant initial state and the right zero state. Their damped values are $u_{-}=2e^{-\tau }$ and $u_{+}=0$. Therefore $s^{\prime }=e^{-\tau }$. Matching to $s(\log 2)=0$ yieldsThese matching states are consistent with the characteristic feet: the left foot is $s-2(1-e^{-\tau })=-3/2+e^{-\tau }<-1$, while the right foot is $s>0$. The formula describes the leading thin-layer position in the small-viscosity limit.