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Solitary waves bifurcated from edges of Bloch bands in two-dimensionalperiodic media are determined both analytically and numerically in the contextof a two-dimensional nonlinear Schr\"odinger equation with a periodicpotential. Using multi-scale perturbation methods, envelope equations ofsolitary waves near Bloch bands are analytically derived. These envelopeequations reveal that solitary waves can bifurcate from edges of Bloch bandsunder either focusing or defocusing nonlinearity, depending on the signs ofsecond-order dispersion coefficients at the edge points. Interestingly, at edgepoints with two linearly independent Bloch modes, the envelope equations leadto a host of solitary wave structures including reduced-symmetry solitons,dipole-array solitons, vortex-cell solitons, and so on -- many of which havenever been reported before. It is also shown analytically that the centers ofenvelope solutions can only be positioned at four possible locations at orbetween potential peaks. Numerically, families of these solitary waves aredirectly computed both near and far away from band edges. Near the band edges,the numerical solutions spread over many lattice sites, and they fully agreewith the analytical solutions obtained from envelope equations. Far away fromthe band edges, solitary waves are strongly localized with intensity and phaseprofiles characteristic of individual families.Comment: 23 pages, 15 figures. To appear in Phys. Rev.

Solitary waves bifurcated from Bloch-band edges in two-dimensional periodic media