Abstract
This paper investigates soliton families in the two-dimensional Kerr
nonlinear Schrödinger equation with a complex linear lattice. Two
kinds of complex lattices are considered, both consisting of square
unit cells and each compatible with more than one spatial symmetry.
The first complex lattice exhibits one parity-time (PT) symmetry and
one partial PT (PPT) symmetry, whereas the second one possesses double
PPT symmetry. The band structures and linearized spectra of these two
lattices display significant differences. In the presence of
self-focusing nonlinearity, multiple types of solitons are found
within the semi-infinite band gaps. The stability of these localized
gap modes is examined using two numerical approaches: linear stability
analysis and direct perturbed simulations. The results indicate that
all solitons possess stable regions within the first complex square
lattice, whereas they are all unstable in the second complex lattice.
This instability is attributed to the more intricate imaginary part,
leading to the appearance of nonzero real parts in their stability
eigenvalue spectra.