  In this study, we investigate the existence and stability of in-phase and
out-of-phase dissipative solitons in a dual-waveguide lattice with linear
localized gain and nonlinear losses under both focusing and defocusing
nonlinearities. Numerical results reveal that both types of dissipative
solitons bifurcate from the linear amplified modes, and their nonlinear
propagation constant changes to a real value when nonlinearity, linear
localized gain, and nonlinear losses coexist. We find that increasing the
linear gain coefficient leads to an increase in the power and propagation
constant of both types of dissipative solitons. For defocusing nonlinearity,
in-phase solitons are stable across their entire existence region, while
focusing nonlinearity confines them to a small stable region near the lower
cutoff value in the propagation constant. In contrast, out-of-phase solitons
have a significantly larger stable region under focusing nonlinearity compared
to defocusing nonlinearity. The stability regions of both types of dissipative
solitons increase with increasing nonlinear losses coefficient. Additionally,
we validate the results of linear stability analysis for dissipative solitons
using propagation simulations, showing perfect agreement between the two
methods.
