We give a heat kernel proof of the equivariant holomorphic Morse inequalities. We use some techniques developed by Bismut and Lebeau. These inequalities, first obtained by Witten using a different argument, produce bounds on the multiplicities of weights occurring in the twisted Dolbeault cohomologies in terms of the data of the fixed points. The spatial discretization is based on a collocation technique. Periodic boundary conditions are considered.
Numerical results show that oscillations are confined in the vicinity of the shock.
If symmetry is known the oscillations are easily eliminated by decreasing the scale. Moreover when the method is associated with a coordinate transform, accurate results are obtained with relatively small number of points. Implications of such a scale for the unification of gauge couplings with small Majorana neutrino masses are discussed. New physical effects, arising due to the non-normality of linear dynamics, are described. The influence of the medium dispersion, rising due to the violation of quasineutrality for perturbations, is examined.