  We consider localized solutions of variants of the semilinear curl-curl wave
equation $s(x) \partial_t^2 U +\nabla\times\nabla\times U + q(x) U \pm V(x)
|U|^{p-1} U = 0$ for $(x,t)\in \mathbb{R}^3\times\mathbb{R}$ and arbitrary
$p>1$. Depending on the coefficients $s, q, V$ we can prove the existence of
three types of localized solutions: time-periodic solutions decaying to $0$ at
spatial infinity, time-periodic solutions tending to a nontrivial profile at
spatial infinity (both types are called breathers), and rogue waves which
converge to $0$ both at spatial and temporal infinity. Our solutions are weak
solutions and take the form of gradient fields. Thus they belong to the kernel
of the curl-operator so that due to the structural assumptions on the
coefficients the semilinear wave equation is reduced to an ODE. Since the space
dependence in the ODE is just a parametric dependence we can analyze the ODE by
phase plane techniques and thus establish the existence of the localized waves
described above. Noteworthy side effects of our analysis are the existence of
compact support breathers and the fact that one localized wave solution
$U(x,t)$ already generates a full continuum of phase-shifted solutions
$U(x,t+b(x))$ where the continuous function $b:\mathbb{R}^3\to\mathbb{R}$
belongs to a suitable admissible family.
