  We numerically study solitary waves in the coupled nonlinear Schr\"odinger
equations. We detect pitchfork bifurcations of the fundamental solitary wave
and compute eigenvalues and eigenfunctions of the corresponding eigenvalue
problems to determine the spectral stability of solitary waves born at the
pitchfork bifurcations. Our numerical results demonstrate the theoretical ones
which the authors obtained recently. We also compute generalized eigenfunctions
associated with the zero eigenvalue for the bifurcated solitary wave exhibiting
a saddle-node bifurcation, and show that it does not change its stability type
at the saddle-node bifurcation point.
