Solitary waves bifurcated from edges of Bloch bands in two-dimensional periodic media are determined
both analytically and numerically in the context of a two-dimensional nonlinear Schrödinger equation with a
periodic potential. Using multiscale perturbation methods, the envelope equations of solitary waves near Bloch
bands are analytically derived. These envelope equations reveal that solitary waves can bifurcate from edges of
Bloch bands under either focusing or defocusing nonlinearity, depending on the signs of the second-order
dispersion coefficients at the edge points. Interestingly, at edge points with two linearly independent Bloch
modes, the envelope equations lead to a host of solitary wave structures, including reduced-symmetry solitons,
dipole-array solitons, vortex-cell solitons, and so on—many of which have not been reported before to our
knowledge. It is also shown analytically that the centers of envelope solutions can be positioned at only four
possible locations at or between potential peaks. Numerically, families of these solitary waves are directly
computed both near and far away from the band edges. Near the band edges, the numerical solutions spread
over many lattice sites, and they fully agree with the analytical solutions obtained from the envelope equations.
Far away from the band edges, solitary waves are strongly localized, with intensity and phase profiles characteristic
of individual families.