Panayotis Panayotaros
Quartic normal forms for the periodic
nonlinear Schr\"odinger equation with dispersion management
(298K, pdf)
ABSTRACT. We investigate Birkhoff normal forms
for the periodic
nonlinear Schr\"odinger 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.