  We study the pumping of matter-wave solitons formed in Bose-Einstein
condensates (BECs) with attractive atomic interactions that are loaded into
optical superlattices in which one of the lattices is moving with respect to
the other. We find that solitons exhibit the remarkably similar pumping
properties in both shallow and deep lattices, and that for exactly the same
soliton initially excited, switching between integer (fractional) pumping and
trapping can be achieved by simply adjusting the lattice parameters. In
addition, we find that the gap solitons, which bifurcate from the lowest energy
band in a semi-infinite band gap, also exhibit this lattice-parameter-dependent
pumping and trapping. The treatment of solitons as classical particles with
effective centre-of-mass equations of motion provides a good description of
this parameter-dependent integer (fractional) pumping and trapping of solitons.
