Two-dimensional intraband solitons in lattice potentials with local defects and self-focusing nonlinearity

It is commonly known that stable bright solitons in periodic potentials,\nwhich represent gratings in photonics/plasmonics, or optical lattices in\nquantum gases, exist either in the spectral semi-infinite gap (SIG) or in\nfinite bandgaps. Using numerical methods, we demonstrate that, under the action\nof the cubic self-focusing nonlinearity, defects in the form of "holes" in\ntwo-dimensional (2D) lattices support continuous families of 2D solitons\n\\textit{embedded} into the first two Bloch bands of the respective linear\nspectrum, where solitons normally do not exist. The two families of the\n\\textit{embedded defect solitons} (EDSs) are found to be continuously linked by\nthe branch of \\textit{gap defect solitons} (GDSs) populating the first finite\nbandgap. Further, the EDS branch traversing the first band links the GDS family\nwith the branch of regular defect-supported solitons populating the SIG. Thus,\nwe construct a continuous chain of regular, embedded, and gap-mode solitons\n("superfamily") threading the entire bandgap structure considered here. The\nEDSs are stable in the first Bloch band, and partly stable in the second. They\nexist with the norm exceeding a minimum value, hence they do not originate from\nlinear defect modes. Further, we demonstrate that double, triple and quadruple\nlattice defects support stable dipole-mode solitons and vortices.\n