Previous work has shown that in a two-dimensional periodic medium under focusing or defocusing cubic
nonlinearities, gap solitons in the form of low-amplitude and slowly modulated single-Bloch-wave packets can
bifurcate out from the edges of Bloch bands. In this paper, linear stability properties of these gap solitons near
band edges are determined both analytically and numerically. Through asymptotic analysis, it is shown that
these gap solitons are linearly unstable if the slope of their power curve at the band edge has the opposite sign
of nonlinearity here focusing nonlinearity is said to have a positive sign, and defocusing nonlinearity to have
a negative sign. An equivalent condition for linear instability is that the power of the gap solitons near the
band edge is lower than the limit power value on the band edge. Through numerical computations of the power
curves, it is found that this condition is always satisfied, thus two-dimensional gap solitons near band edges are
linearly unstable. The analytical formula for the unstable eigenvalue of gap solitons near band edges is also
asymptotically derived. It is shown that this unstable eigenvalue is proportional to the cubic power of the
soliton’s amplitude, and it induces width instabilities of gap solitons. A comparison between this analytical
eigenvalue formula and numerically computed eigenvalues shows excellent agreement.