  We introduce a non-commutative generalization of the Gross-Pitaevskii
equation for one-dimensional quantum gasses and quantum liquids. This
generalization is obtained by applying the time-dependent variational principle
to the variational manifold of continuous matrix product states. This allows
for a full quantum description of many body system ---including entanglement
and correlations--- and thus extends significantly beyond the usual mean-field
description of the Gross-Pitaevskii equation, which is known to fail for
(quasi) one-dimensional systems. By linearizing around a stationary solution,
we furthermore derive an associated generalization of the Bogoliubov -- de
Gennes equations. This framework is applied to compute the steady state
response amplitude to a periodic perturbation of the potential.
