The onset of the bipolar flavor conversion of supernova neutrinos
  The study of supernova neutrinos result an interesting non-linear phenomenon, consisting of three phases: synchronized oscillation phase, bipolar flavor conversion phase and the phase of spectral split. In the collective oscillation of supernova neutrino the self energy is not a constant but varies adiabatically, which is responsible to have such different phases. In this article the transition point from synchronized oscillation to bipolar phase is studied numerically as well as analytically. The numerical results yielding different graphs depending on different values of possible small but non-vanishing mixing angles show the onset of the bipolar phase from the synchronized phase varies as the mixing angle. But the analytical study in terms of a spinning top model results a unique onset condition, which is independent of the choice of mixing angle. Such discrepancy between numerical results and analytical results is explained properly. 
Indranath Bhattacharyya
physics.gen-ph