Splitting and focusing of neutrino collective states

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.