Nonlinear stability of oscillatory pulses in the parametric nonlinear Schrodinger equation

We extend the renormalization group method, developed for the study of pulse interaction in damped wave equations, to the study of oscillatory motion of supercritical pulses in the parametrically forced nonlinear Schrodinger equation (PNLS). We construct a global manifold which asymptotically attracts the flow into an O( r(4)) neighbourhood in the H-1 norm, where r is the amplitude of the internal oscillations. The oscillatory and translational dynamics of the pulses are rigorously recovered as a finite-dimensional flow on the manifold. The normal form for the projected dynamics of the oscillatory pulse shows that it is created in a supercritical Poincare-Hopf bifurcation.