Trans-Alfv¿nic shocks waves (TASWs), at which the flow velocity passes over the Alfv¿n velocity, are considered in the two-fluid approximation. In the ideal medium, such shocks are nonevolutionary. This means that a small perturbation of the shock causes finite variation of the initial flow. Under the action of a small perturbation, the shock disintegrates or transforms to some other nonsteady configuration. In the present paper, it is shown that the dissipative and dispersive processes that form the shock structure do not alter the conclusions based on the principle of evolutionarity. At the same time, an Alfv¿n discontinuity becomes nonevolutionary in the presence of arbitrarily small but nonzero dissipation. The disintegration configuration of a TASW includes an Alfv¿n discontinuity or another TASW, which takes the magnetic field reversal given at the initial shock. Because of this, the time evolution of a nonevolutionary shock has the form of oscillatory disintegration, i.e., reversible transformation of one nonevolutionary discontinuity to the other. Therefore, most of the time of its evolution, the flow is in the intermediate state between these two and it cannot be identified as one of the known types of discontinuities. The implication of this result for interplanetary collisionless shocks is discussed. The transition of the flow velocity over the Alfv¿n velocity remains crucial, when the fluid approximation is violated, and makes collisionless TASWs nonevolutionary. It is suggested that the nonsteady behavior is the reason why the TASWs are observed in the real plasma much more rarely than fast, evolutionary, shocks. ¿ 1999 American Geophysical Union