A Renormalization Method for Modulational Stability of Quasi-Steady Patterns in Dispersive Systems

We employ global quasi-steady manifolds to rigorously reduce innnite dimensional dynamical systems to nite dimensional ows. The manifolds we construct are not invariant, but through a renormalization group method we capture the long-time evolution of the full system as a ow on the manifold up to a small residual. For the parametric nonlinear Schrr odinger equation (PNLS) we consider a manifold describing N well separated pulses and derive an explicit system of ordinary diierential equations for the ow on the manifold which captures the leading order pulse motion through the tail-tail interactions. As a consequence of our analysis we obtain a rigorous connection between the slow evolution in the hyperbolic PNLS and the fourth order parabolic Phase Sensitive Ampliication equation for ber optic systems. The renormalization group method is presented in an abstract setting which includes many linearly damped, forced dispersive systems.