  We consider the $(1+1)$-dimensional quasilinear wave equation
$g(x)w_{tt}-w_{xx}+h(x) (w_t^3)_t=0$ on $\mathbb{R}\times\mathbb{R}$ which
arises in the study of localized electromagnetic waves modeled by
Kerr-nonlinear Maxwell equations. We are interested in time-periodic, spatially
localized solutions. Here $g\in L^{\infty}(\mathbb{R})$ is even with
$g\not\equiv 0$ and $h(x)=\gamma\,\delta_0(x)$ with
$\gamma\in{\mathbb{R}}\backslash\{0\}$ and $\delta_0$ the delta-distribution
supported in $0$. We assume that $0$ lies in a spectral gap of the operators
$L_k=-\frac{d^2}{dx^2}-k^2\omega^2g$ on $L^2(\mathbb{R})$ for all $k\in
2\mathbb{Z}+1$ together with additional properties of the fundamental set of
solutions of $L_k$. By expanding $w$ into a Fourier series in time we transfer
the problem of finding a suitably defined weak solution to finding a minimizer
of a functional on a sequence space. The solutions that we have found are
exponentially localized in space. Moreover, we show that they can be well
approximated by truncating the Fourier series in time. The guiding examples,
where all assumptions are fulfilled, are explicitly given step potentials and
periodic step potentials $g$. In these examples we even find infinitely many
distinct breathers.
