Non-linearity arises when one takes the limit of quantum dynamics in some sense. Two standard examples are:

1) the semiclassical approximation (i.e. Born's correspondence principle) where in the limit of large quantum numbers ("$\hslash\to 0$") quantum linear dynamics becomes the classical (usually non-linear) one;

2) Mean field approximation (i.e. the limit of a very large number of particles), where the dynamics of each component of the system is modelled by an effective non-linear dynamics (i.e. Hartree or Gross-Pitaevskii equations as the mean field limit of many-bosons systems in condensed matter).

The subject of analyzing rigorously this classical or mean field limit is a very active subject in the domain of mathematical physics/analysis of PDEs.