  Soliton dynamics in a large variety of longitudinally modulated lattices are
studied in terms of phase space analysis for an effective particle approach and
direct numerical simulations. Complex soliton dynamics are shown to depend
strongly on both their power/width and their initial momentum as well as on
lattice parameters. A rish set of qualitatively distinct dynamical features of
soliton propagation that have no counterpart in longitudinally uniform lattices
is illustrated. This set includes cases of enhanced soliton mobility, dynamical
switching, extended trapping in several transverse lattice periods, and
quasiperiodic trapping, which are promising for soliton control applications.
