  We study the dynamics of a nonlinear one-dimensional disordered system from a
spectral point of view. The spectral entropy and the Lyapunov exponent are
extracted from the short time dynamics, and shown to give a pertinent
characterization of the different dynamical regimes. The chaotic and
self-trapped regimes are governed by log-normal laws whose origin is traced to
the exponential shape of the eigenstates of the linear problem. These
quantities satisfy scaling laws depending on the initial state and explain the
system behaviour at longer times.
