  I utilize the Petrov-Galerkin formulation and develop a new method for
solving the unsteady collisionless Boltzmann equation in both the linear and
nonlinear regimes. In the first order approximation, the method reduces to a
linear eigenvalue problem which is solved using standard numerical methods. I
apply the method to the dynamics of a model stellar disk which is embedded in
the field of a soft-centered logarithmic potential. The outcome is the full
spectrum of eigenfrequencies and their conjugate normal modes for prescribed
azimuthal wavenumbers. The results show that the fundamental bar mode is
isolated in the frequency space while spiral modes belong to discrete families
that bifurcate from the continuous family of van Kampen modes. The population
of spiral modes in the bifurcating family increases by cooling the disk and
declines by increasing the fraction of dark to luminous matter. It is shown
that the variety of unstable modes is controlled by the shape of the dark
matter density profile.
