  We present N-body simulations of unstable spiral modes in a dynamically cool
collisionless disc. We show that spiral modes grow in a thin collisionless disk
in accordance with the analytical perturbation theory. We use the particle-mesh
code SUPERBOX with nested grids to follow the evolution of unstable spirals
that emerge from an unstable equilibrium state. We use a large number of
particles (up to 40 million particles) and high-resolution spatial grids in our
simulations (128^3 cells). These allow us to trace the dynamics of the unstable
spiral modes until their wave amplitudes are saturated due to nonlinear
effects. In general, the results of our simulations are in agreement with the
analytical predictions. The growth rate and the pattern speed of the most
unstable bar-mode measured in N-body simulations agree with the linear
analysis. However the parameters of secondary unstable modes are in lesser
agreement because of the still limited resolution of our simulations.
