= 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 contribution
$$
\int\bigl(s'(t)[u]-[f]\bigr)\phi(s(t),t)\,dt.
$$
For example the normal to the interface in $(x,t)$ coordinates is proportional to $(1,-s')$, so the jump in the conserved spacetime flux $(u^2/2,u)$ has normal component $[f]-s'[u]$. Since the traces of the <test function> are arbitrary, the interior weak equation holds if and only if $s'[u]=[f]$. At a genuine jump $[u]\ne0$, this is
$$
\boxed{s'(t)=\frac{u_l(t)+u_r(t)}2.}
$$
This 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.
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