Soliton families in the complex square lattice with double spatial symmetry

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.