A Normal Form for Hamiltonian--Hopf Bifurcations in Nonlinear Schrödinger Equations with General External Potentials

A normal form is derived for Hamiltonian--Hopf bifurcations of solitary waves in nonlinear Schrödinger equations with general external potentials. This normal form is a simple second-order nonlinear ordinary differential equation (ODE) 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 be 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 partial differential equation (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.