Two-dimensional Solitons in Irregular Lattice Systems

Solitons are localized nonlinear waves and occur in many branches of physics. Their properties have provided a deep and fundamental understanding of complex nonlinear systems. In recent years, there has been considerable interest in studying solitons in systems with periodic potentials or lattices, in particular, those that can be generated in nonlinear optical materials [1]–[5]. In periodic lattices, solitons can form when their propagation constant, or eigenvalue, is within certain regions, often called gaps, a concept that is borrowed from the Floquet–Bloch theory for linear propagation. But the external potential of complex systems can be much more general and physically richer than a periodic lattice. For example, atomic crystals can have various irregularities, such as defects and edge dislocations, and also quasicrystal structures, which have long-range orientational order but no translational symmetry [6]–[8]. In general, when the lattice periodicity is slightly perturbed, the band–gap structure and soliton properties also become slightly perturbed, but solitons are otherwise expected to exist in much the same way as in the perfectly periodic case [9], [10]. But the existence and properties of multidimensional solitons when the external potential has large variations from periodicity has remained largely unexplored. In this study, we find two-dimensional (2D) solitons in lattices that have vacancy defects, edge dislocations, and quasicrystal structures. We use a fixed-point spectral method to compute the ground states of the underlying nonlinear Schrödinger (NLS) equation. We find solitons on irregular lattices similarly to solitons on periodic lattices. A comparative study of the eigenvalue dependence of the soliton power leads to important observations regarding soliton power, gap edge, and stability properties. We investigate the evolution by direct numerical simulation, showing that slightly perturbed solitary waves in irregular