  Maxwell equations in the absence of free charges require initial data with a
divergence free displacement field $\mathcal D$. In materials in which the
dependence $\mathcal D=\mathcal D(\mathcal E)$ is nonlinear the quasilinear
problem $\nabla\cdot\mathcal D(\mathcal E)=0$ is hence to be solved. In many
applications, e.g. in the modelling of wave-packets, an approximative
asymptotic ansatz of the electric field $\mathcal E$ is used, which satisfies
this divergence condition at $t=0$ only up to a small residual. We search then
for a small correction of the ansatz to enforce $\nabla\cdot\mathcal D(\mathcal
E)=0$ at $t=0$ and choose this correction in the form of a gradient field. In
the usual case of a power type nonlinearity in $\mathcal D(\mathcal E)$ this
leads to the sum of the Laplace and $p$-Laplace operators. We also allow for
the medium to consist of two different materials so that a transmission problem
across an interface is produced. We prove the existence of the correction term
for a general class of nonlinearities and provide regularity estimates for its
derivatives, independent of the $L^2$-norm of the original ansatz. In this way,
when applied to the wave-packet setting, the correction term is indeed
asymptotically smaller than the original ansatz. We also provide numerical
experiments to support our analysis.
