# Maxwell's equations and nonlinear media

Are there analytical methods to analyze electromagnetic fields or magnetic diffusion in materials which are not linear using (or starting from) Maxwell’s equations?

Nonlinear material could be described as $B=\mu(H)H$, where $B$ is the flux density, $\mu$ the permeability, and $H$ the magnetic field strength. But how to consider that in the Maxwell’s equations?

Furthermore, even for simple geometry (cube, cylinder, ...) and even for one dimensional problems, e.g, one dimensional diffusion equation, i would not know how to consider nonlinear behavior analytically.

To be more specific, consider the magnetic field governing PDE for a long rod with radius $\rho$ for the axial component of $H$. In polar coordinates it reads:

$\partial_{\rho}^2 H(\rho,t)+\frac{1}{\rho}\partial_{\rho}H(\rho,t)-\mu \kappa \partial_t H(\rho, t)=0$, with conductivity $\kappa$. Now insert $\mu(H)$:

$\partial_{\rho}^2 H(\rho,t)+\frac{1}{\rho}\partial_{\rho}H(\rho,t)-\mu(H)\kappa \partial_t H(\rho, t)=0$.

Now, $\mu(H)$ might be approximated by a polynomial in $H$. Even in this simple case, an analytical solution might be difficult to obtain. I am wondering if there is maybe a finite dimensional analytic approximation of ODEs to describe this problem. For linear PDEs a mode decomposition is often made, but clearly for nonlinear systems this doesn’t works. Are there other methods to analyze equations like that? Does Maxwell’s equations indeed allow to just replace the $\mu$ by a field dependent permeability?

Any hints, experience with that or references are appreciated.

• A very general approach is probably expanding around the current value of H using e.g. Taylor expansion or maybe in some subspace of the target space you choose, I think Krylov subspace method is something like that but I have never used it myself. Then iteratively try to find a solution B using a numerical method. Sorry missed the analytical part... But if you do it symbolically and take some limit of small mesh size in the end it is maybe analytical again. – Emil Mar 3 '18 at 8:33
• @Emil :thanks, can you please provide more details with the symbolic approach? Is this found in some reference? – Carlos Mar 3 '18 at 8:45
• Carlos: I don't understand your question. I just meant simulate the numerical outline I wrote symbolically. – Emil Mar 3 '18 at 8:50
• @Emil : sorry, i thought symbolically like symbolic calculations as in tools as Mathematica :) – Carlos Mar 3 '18 at 8:57
• Treating electromagnetism in nonlinear media is equivalent to solving nonlinear PDEs. Even over simple volumes, no analytic method exists. In fact, it's not clear that most nonlinear PDEs even have solutions to begin with. If you want analytic methods, you're going to have to specify the particular nonlinear function you're interested in. – probably_someone Mar 3 '18 at 21:41