We introduce a system with one or two amplified nonlinear sites ("hot spots",
HSs) embedded into a two-dimensional linear lossy lattice. The system describes
an array of evanescently coupled optical or plasmonic waveguides, with gain
applied at selected HS cores. The subject of the analysis is discrete solitons
pinned to the HSs. The shape of the localized modes is found in
quasi-analytical and numerical forms, using a truncated lattice for the
analytical consideration. Stability eigenvalues are computed numerically, and
the results are supplemented by direct numerical simulations. In the case of
self-focusing nonlinearity, the modes pinned to a single HS are stable or
unstable when the nonlinearity includes the cubic loss or gain, respectively.
If the nonlinearity is self-defocusing, the unsaturated cubic gain acting at
the HS supports stable modes in a small parametric area, while weak cubic loss
gives rise to a bistability of the discrete solitons. Symmetric and
antisymmetric modes pinned to a symmetric set of two HSs are considered too.Comment: Philosophical Transactions of the Royal Society A, in press (a
special issue on "Localized structures in dissipative media"
