= Solution
In the moving stretched coordinates, $\partial_t=\partial_\tau-s'(\tau)\varepsilon^{-1}\partial_\xi$ and $\partial_x=\varepsilon^{-1}\partial_\xi$. Multiplying the <damped Burgers equation> by $\varepsilon$ gives
$$
\varepsilon(u_\tau+u)+(u-s')u_\xi=u_{\xi\xi}.
$$
Discarding the order-$\varepsilon$ terms yields the leading <Viscous Burgers travelling shock> equation $\boxed{(u-s')u_\xi=u_{\xi\xi}}$. Integrate across a nonconstant transition whose derivative vanishes at either end:
$$
\boxed{\left[\frac{u^2}{2}-s'u\right]_-^+=0.}
$$
Multiplication by $u$ and a further <integration by parts> give
$$
\boxed{\left[\frac{u^3}{3}-\frac{s'u^2}{2}\right]_-^+=-\int_{-\infty}^{\infty}u_\xi^2\,d\xi<0.}
$$
The first identity is the <Rankine-Hugoniot condition>, giving $s'=(u_-+u_+)/2$. 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 $s'=e^{-\tau}$. Matching to $s(\log2)=0$ yields
$$
\boxed{s(\tau)=\frac12-e^{-\tau},\qquad\tau>\log2.}
$$
These 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.
Back to article page
