  Bifurcations of solitary waves are classified for the generalized nonlinear
Schr\"odinger equations with arbitrary nonlinearities and external potentials
in arbitrary spatial dimensions. Analytical conditions are derived for three
major types of solitary wave bifurcations, namely saddle-node bifurcations,
pitchfork bifurcations and transcritical bifurcations. Shapes of power diagrams
near these bifurcations are also obtained. It is shown that for pitchfork and
transcritical bifurcations, their power diagrams look differently from their
familiar solution-bifurcation diagrams. Numerical examples for these three
types of bifurcations are given as well. Of these numerical examples, one shows
a transcritical bifurcation, which is the first report of transcritical
bifurcations in the generalized nonlinear Schr\"odinger equations. Another
shows a power loop phenomenon which contains several saddle-node bifurcations,
and a third example shows double pitchfork bifurcations. These numerical
examples are in good agreement with the analytical results.
