A self-consistent system of the Boltzmann kinetic equation and the Poisson equation is used to study the dynamical evolution of Saturn's main A, B and C rings composed of discrete mutually gravitating particles. The simplified case of relatively rare collisions between identical particles, when the collision frequency is smaller than (compared with) the orbital frequency, is examined. Equations describing the quasi-linear stage of Jeans instability of small-amplitude gravity perturbations in Saturn's rings are derived and solved analytically. Conditions under which the quasi-linearization of the Boltzmann equation can be used to describe a wave-particle interaction are calculated with reference to the excitation of Jeans-type perturbations. The theory, as applied to Saturn's rings, predicts for several features, such as numerous irregular Jeans-unstable wakes, with size and spacing between them of the order of 2pih less than or equal to 100 m, where h is the typical thickness of the system. The interaction of particles with these almost aperiodically growing gravity perturbations increases both the radial spread of the disc and random velocities of particles in a very short time-scale of only two to three disc orbital revolutions. The latter leads to an eventual stabilization of the system, unless some effective cooling mechanism exists, reducing the magnitude of the relative velocity of particles. It is suggested that inelastic (dissipative) interparticle impacts provide such a cooling mechanism, leading to the recurrent density waves activity. We predict that the high-resolution images from the forthcoming (2004) Cassini spacecraft will reveal this fine-scale recurrent similar to100 m or even less spiral structure in low and moderately high optical depth regions (tau less than or equal to 1, where tau is the normal optical depth) of Saturn's main rings.
