  It is shown that the collective nonlinear interactions between intense
neutrino or anti-neutrino fluxes and a dense neutrino plasma are governed by a
multi-dimensional coupled cubic Schr\"odinger equation in which the interaction
potential is positive or negative depending on the neutrino type. The cubic
Schr\"odinger equation describes the splitting and focusing of intense neutrino
beams due to the nonlinear excitations associated with the modifications of the
individual neutrino energies in a dense neutrino background.
