They're linear coupled PDE s are they not? which allow for a significantly wider range of behaviours that linear ODEs? As far as I understand, there is no 'clo

They're linear coupled PDE s are they not? which allow for a significantly wider range of behaviours that linear ODEs? As far as I understand, there is no 'closed form solution' for the initial value problem of Maxwell's equations, despite them being linear? So I'm not sure it's correct to say that "linearity ~ simplicity" in PDE theory. My understanding is the correct analog of "linear" in PDEs is "elliptic" [which is more akin to 'operator with full spectrum'], which Maxwell's equations are not?