  We report results of systematic numerical studies of 2D matter-wave soliton
families supported by an external potential, in a vicinity of the junction
between stable and unstable branches of the families, where the norm of the
solution attains a minimum, facilitating the creation of the soliton. The model
is based on the Gross-Pitaevskii equation for the self-attractive condensate
loaded into a quasiperiodic (QP) optical lattice (OL). The same model applies
to spatial optical solitons in QP photonic crystals. Dynamical properties and
stability of the solitons are analyzed with respect to variations of the depth
and wavenumber of the OL. In particular, it is found that the single-peak
solitons are stable or not in exact accordance with the Vakhitov-Kolokolov (VK)
criterion, while double-peak solitons, which are found if the OL wavenumber is
small enough, are always unstable against splitting.
