  We study phase-imprinted solitons of ultracold bosons in an optical lattice
with a harmonic trap, which shows the superfluid (SF) and Mott-insulator (MI)
shell structures. The earlier study [Konstantin V. Krutitsky, J. Larson, and M.
Lewenstein, Phys. Rev. A 82, 033618 (2010).] reported three types of
phase-imprinted solitons in the Bose-Hubbard model: in-phase soliton,
out-of-phase soliton, and wavelet. In this paper, we uncover the dynamical
phase diagram of these phase-imprinted solitons, and find another type of the
phase-imprinted soliton namely, the hybrid soliton. In the harmonically trapped
system, the solitonic excitations created at the SF core cannot penetrate into
the outer SF shell. This repulsion at the surface of the outer SF shell can be
cured by inposing a repulsive potential at the center of the trap. These
results can be interpreted as a kind of the impedance matching of excitations
in BECs in terms of the effective chemical potentials or the local particle
numbers in the shell, and the analogous results can be observed also in the
sound wave created by the local single-shot pulse potential.
