Quartic normal forms for the periodic nonlinear Schrِdinger equation with dispersion management

We investigate Birkhoff normal forms for the periodic nonlinear Schrِdinger equation with dispersion management. The normalization we describe is related to averaging arguments considered in the literature, and has the advantage of producing fewer resonant couplings between high spatial frequency modes. One consequence is that the normal form equations have invariant subspaces of large but finite dimension, where we can find several classes of periodic orbits. The formal arguments apply to other related dispersive systems, and to normal forms of high order. We also present a rigorous version of the normal form calculation and show that solutions of the quartic normal form equations remain close to solutions of the full system over a time that is inversely proportional to a small nonlinearity parameter.