  We consider the Schr\"odinger system with Newton-type interactions that was
derived by R. Klein, A. Majda and K. Damodaran [18] to modelize the dynamics of
N nearly parallel vortex filaments in a 3-dimensional homogeneous
incompressible uid. The known large time existence results are due to C. Kenig,
G. Ponce and L. Vega [17] and concern the interaction of two filaments and
particular configurations of three filaments. In this article we prove large
time existence results for particular configurations of four nearly parallel
filaments and for a class of configurations of N nearly parallel filaments for
any N\geq 2. We also show the existence of travelling wave type dynamics.
Finally we describe configurations leading to collision.
