Stimulated Raman scattering in the transient limit is an integrable system. In contrast, however, to the usual behavior in integrable systems, solitons are transient and the behavior of the system at long distances is dominated by self-similar solutions which may be found by symmetry reduction. It is shown for fairly general initial conditions precisely which self-similar solution the system tends toward at long distances, and the system evolution is studied numerically. It is argued that this behavior in which self-similar solutions dominate the long-distance evolution should often appear in nonlinear optical systems with memory. A possible experiment is proposed.