 Solution

Write $[u]=u_r-u_l$ and $[f]=u_r^2/2-u_l^2/2$. Splitting the test-function integral along $x=s(t)$ and integrating by parts gives a remaining interface contributionFor example the normal to the interface in $(x,t)$ coordinates is proportional to $(1,-s^{\prime })$, so the jump in the conserved spacetime flux $(u^2/2,u)$ has normal component $[f]-s^{\prime }[u]$. Since the traces of the test function are arbitrary, the interior weak equation holds if and only if $s^{\prime }[u]=[f]$. At a genuine jump $[u]\ne 0$, this isThis is the Rankine-Hugoniot condition. An initial-value weak formulation also requires the prescribed initial trace; the jump condition alone concerns the interior equation. The source's reference to “(1)” is to the displayed Burgers equation, not a different PDE.