= Solution
Write the shock as $x=s(t)$ and let the plus trace be on $x>s(t)$. Locally its discontinuous part can be represented with the Heaviside factor $H(x-s(t))$. Distributional differentiation gives $\partial_tH=-s'(t)\delta(x-s(t))$ and $\partial_xH=\delta(x-s(t))$. The singular part of the <conservation law> is therefore
$$
\bigl(-s'(t)[\mathbf P]_-^++[\mathbf Q]_-^+\bigr)\delta(x-s(t)).
$$
It must vanish, so the <Rankine-Hugoniot conditions> are
$$
\boxed{V[\mathbf P]_-^+=[\mathbf Q]_-^+,\qquad V=s'(t).}
$$
The smooth parts of the fields still satisfy the classical <conservation law> on each side. The jump condition ensures that the moving discontinuity creates no extra source of the conserved quantities.
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