A renormalization method or modulational stability of quasi-steady patterns in dispersive systems

We employ global quasi-steady manifolds to rigorously reduce forced, linearly damped dispersive partial differential equations to finite dimensional flows. The manifolds we consider are not invariant, but through a renormalization group method we capture the long-time volution of the full system as a flow on the manifold. For the parametric nonlinear Schrodinger equation we consider a manifold describing N well-separated pulses and derive an explicit system of ordinary differential equations for the flow on the manifold which captures the leading order pulse motion through the tail-tail interactions. W also outline a rigorous connection between the slow volution in the hyperbolic PNLS and the fourth-order parabolic phase sensitive amplification equation for fiber optic systems.