Periodicity of the solution of the wave equation In a few references, I have found that if I have a wave equation
$$
\nabla^2 E=\left[\frac{n(x,t)}{c}\right]^2\frac{\partial^2E}{\partial t^2}
$$
where the refractive index is periodic in space and time
$$
n(x+\Lambda,t)=n(x,t) \\
n(x,t+T)=n(x,t)
$$
then, also the electric field $E(x,t)$ has the same periodicity. How can I prove it?
 A: The statement in question is the basis of Floquet theorem or Bloch theorem, depending on the application (the former is usually used in nonlinear theory, while the latter is the name used in condensed matter physics). This theorem not only states the periodicity of the solution, but also proposes its specific form.
The periodicity is easily verified by checking that $E(x + n\Lambda, t + mT)$, where $n,m$ are arbitrary integers, is the solution of the same equation.
Indeed , the change of variables $x \longrightarrow x + n\Lambda, t \longrightarrow t + mT$ leads us to equation
\begin{equation}
\frac{\partial^2}{\partial x^2} E(x + n\Lambda, t + mT)
=
\left[\frac{n(x + n\Lambda, t + mT)}{c}\right]^2
\frac{\partial^2}{\partial t^2} E(x + n\Lambda, t + mT).
\end{equation}
However the periodicity of $n(x,t)$ means that this is identical with 
\begin{equation}
\frac{\partial^2}{\partial x^2} E(x + n\Lambda, t + mT)
=
\left[\frac{n(x,t)}{c}\right]^2
\frac{\partial^2}{\partial t^2} E(x + n\Lambda, t + mT),
\end{equation}
which means that 
$E(x + n\Lambda, t + mT)$ is identical with $E(x, t)$.
A: It is because electric field is function of medium refractive index :
$$
\mathbf {E} (n(z,t)) =\operatorname {Re} \!\left[\mathbf {E} _{0}e^{i(2\pi (n(z,t)+i\kappa )z/\lambda _{0}-\omega t)}\right]
$$
