  Interactions of dispersive shock (DSWs) and rarefaction waves (RWs)
associated with the Korteweg-de Vries equation are shown to exhibit multiphase
dynamics and isolated solitons. There are six canonical cases: one is the
interaction of two DSWs which exhibit a transient two-phase solution, but
evolve to a single phase DSW for large time; two tend to a DSW with either a
small amplitude wave train or a finite number of solitons, which can be
determined analytically; two tend to a RW with either a small wave train or a
finite number of solitons; finally, one tends to a pure RW.
