Climb of a bore on a beach Part 3. Run-up

When a bore travels shoreward into water at rest on a beach, then according to the first-order non-linear long-wave theory, the bore accelerates and decreases in height, until it collapses at the shore. The investigation here reported concerns the question, what happens next? It is formulated as a singular characteristic boundary-value problem with somewhat unusual mathematical properties. Its asymptotic solution predicts a rather thin sheet of run-up and back-wash with some unexpected features.