  We observe a crossover from strong to weak chaos in the spatiotemporal
evolution of multiple site excitations within disordered chains with cubic
nonlinearity. Recent studies have shown that Anderson localization is
destroyed, and the wave packet spreading is characterized by an asymptotic
divergence of the second moment $m_2$ in time (as $t^{1/3}$), due to weak
chaos. In the present paper, we observe the existence of a qualitatively new
dynamical regime of strong chaos, in which the second moment spreads even
faster (as $t^{1/2}$), with a crossover to the asymptotic law of weak chaos at
larger times. We analyze the pecularities of these spreading regimes and
perform extensive numerical simulations over large times with ensemble
averaging. A technique of local derivatives on logarithmic scales is developed
in order to quantitatively visualize the slow crossover processes.
