We consider the Schrödinger system with point vortex-type interactions that was derived by R. Klein, A. Majda and K. Damodaran and by V. Zakharov to modelize the dynamics of N nearly parallel vortex filaments in a 3-dimensional homogeneous incompressible fluid. The known large time existence results are due to C. Kenig, G. Ponce and L. Vega and concern the case of same circulations for two filaments and for a class of configurations of three filaments. We prove large time existence results for particular configurations of four nearly parallel filaments and for a class of configurations of N filaments for any N larger or equal to 2. We also show the existence of travelling wave type dynamics, and we describe configurations leading to collision for N larger or equal to 3. Finally we consider the problem of collisions for perturbations of antiparallel translating pairs of filaments. These results are joint works with E. Faou and E. Miot.