  We perform novel energy and norm density resolved wave packet spreading
studies in the disordered Gross-Pitaevskii (GP) lattice to confine energy
density fluctuations. We map the locations of GP regimes of weak and strong
chaos subdiffusive spreading in the 2D density control parameter space and
observe strong chaos spreading over several decades. We obtain a
renormalization of the ground state due to disorder, which allows for a new
disorder-induced phase of disconnected insulating puddles of matter due to
Lifshits tails. Inside this Lifshits phase, the wave packet spreading is
substantially slowed down.
