  This paper is devoted to a generalisation of the quantum adiabatic theorem to
a nonlinear setting. We consider a Hamiltonian operator which depends on the
time variable and on a finite number of parameters and acts on a separable
Hilbert space of which we select a fixed basis. We study an evolution equation
in which this Hamiltonian acts on the unknown vector, while depending on
coordinates of the unknown vector in the selected basis, thus making the
equation nonlinear. We prove existence of solutions to this equation and
consider their asymptotics in the adiabatic regime, i.e. when the Hamiltonian
is slowly varying in time. Under natural spectral hypotheses, we prove the
existence of normalised time dependent vectors depending on the Hamiltonian
only, called instantaneous nonlinear eigenvectors, and show the existence of
solutions which remain close to these vectors, up to a rapidly oscillating
phase, in the adiabatic regime. We first investigate the case of bounded
operators and then exhibit a set of spectral assumptions under which the result
extends to unbounded Hamiltonians.
