  Quantum mechanics is an inherently linear theory. However, collective effects
in many body quantum systems can give rise to effectively nonlinear dynamics.
In the present work, we analyze whether and to what extent such nonlinear
effects can be exploited to enhance the rate of quantum evolution. To this end,
we compute a suitable version of the quantum speed limit for numerical and
analytical examples. We find that the quantum speed limit grows with the
strength of the nonlinearity, yet it does not trivially scale with the
``degree'' of nonlinearity. This is numerically demonstrated for the parametric
harmonic oscillator obeying Gross-Piteavskii and Kolomeisky dynamics, and
analytically for expanding boxes under Gross-Pitaevskii dynamics.
