  A self-consistent quadratic theory is presented to account for nonlinear
contributions in quantum dynamics. Evolution equations are shown to depend on
higher-order gradients of the Hamiltonian, which are incorporated via their
equations of motion or through perturbative calculations. The dynamics is
proven trace-preserving, with the Hamiltonian acting as a constant of motion
for initial Gaussian states. Nonlinear response functions are calculated
perturbatively, and sufficient conditions are provided for the existence of
their classical limit.
