  A normal form is derived for Hamiltonian-Hopf bifurcations of solitary waves
in generalized nonlinear Schr\"odinger equations. This normal form is a simple
second-order nonlinear ordinary differential equation that is asymptotically
accurate in describing solution dynamics near Hamiltonian-Hopf bifurcations.
When the nonlinear coefficient in this normal form is complex, which occurs if
the second harmonic of the Hopf bifurcation frequency falls inside the
continuous spectrum of the system, the solution of this normal form will blow
up to infinity in finite time, meaning that solution oscillations near
Hamiltonian-Hopf bifurcations will strongly amplify and eventually get
destroyed. When the nonlinear coefficient of the normal form is real, the
normal form can admit periodic solutions, which correspond to long-lasting
solution oscillations in the original PDE system. Quantitative comparison
between the normal form's predictions and true PDE solutions is also made in
several numerical examples, and good agreement is obtained.
