Periodic potentials, such as photonic crystals and optical lattices, have shown great ability to manipulate the dynamics of photonic and atomic waves. The interplay of the periodic potentials and material nonlinearity (self-focusing or defocusing) can create and stabilize several types of solitons, including ordinary and gap solitons, which populate, respectively, in the semi-infinite gap and finite bandgaps of the corresponding linear spectrum. Besides, lattice defects have also been used to construct solitons. This review reports the generation of two-dimensional (2D) solitons in lattice potentials with local defects, under the self-focusing nonlinearity. The numerical analysis demonstrates a novel kind of embedded solitons (or intraband solitons), which are continuous families of 2D localized modes (different from isolated solutions reported before in usual embedded-soliton models) embedded into the first and second Bloch bands of the underlying linear spectrum, and pinned to the defect, which determines the spatial position of the modes. We call these modes embedded defect solitons. Further, double, triple, and quadruple lattice defects can also support stable dipole-mode solitons and vortices. In addition to that, combined linear and nonlinear lattice potentials are also used to construct stable fundamental solitons, gap ones, and solitary vortices.
Part of the book: Vortex Structures in Fluid Dynamic Problems