  Dynamical and self-trapping properties of two-dimensional (2D) binary
mixtures of Bose-Einstein condensates (BECs) in cross-combined lattices
consisting of a one-dimensional (1D) linear optical lattice (LOL) in the $x-$
direction for the first component and a 1D non linear optical lattice (NOL) in
the $y$-direction for the second component, are analytically and numerically
investigated. The existence and stability of 2D binary matter wave solitons in
these settings is demonstrated both by variational analysis and by direct
numerical integration of the coupled Gross-Pitaevskii equations (GPE). We find
that in absence of the NOL binary solitons, stabilised by the action of the 1D
LOL and by the attractive inter-component interaction can freely move in the
$y-$direction. In the presence of the NOL we find, quite remarkably, the
existence of threshold curves in the parameter space separating regions where
solitons can move, from regions where the solitons become dynamically
self-trapped. The mechanism underlying the dynamical self-trapping phenomenon
(DSTP) is qualitatively understood in terms of a dynamical barrier induced by
the the NOL similar to the Peirls-Nabarro barrier of solitons in discrete
lattices. DSTP is numerically demonstrated for binary solitons that are put in
motion both by phase imprinting and by the action of external potentials
applied in the $y-$direction. In the latter case we show that the trapping
action of the NOL allows maintaining a 2D binary soliton at rest in a
non-equilibrium position of a parabolic trap, or to prevent it from falling
under the action of gravity. Possible applications of the results are also
briefly discussed.
