  We consider the full set of Maxwell equations in a slab or cylindrical
waveguide with a cubically nonlinear material law for the polarization of the
electric field. The nonlinear polarization may be instantaneous or retarded,
and we assume it to be confined inside the core of the waveguide. We prove
existence of infinitely many spatially localized, real-valued and time-periodic
solutions (breathers) propagating inside the waveguide by applying a
variational minimization method to the resulting scalar quasilinear
elliptic-hyperbolic equation for the profile of the breathers. The temporal
period of the breathers has to be carefully chosen depending on the linear
properties of the waveguide. As an example, our results apply if a two-layered
linear axisymmetric waveguide is enhanced by a third core region with low
refractive index where also the nonlinearity is located. In this case we can
also connect our existence result with a bifurcation result. We illustrate our
results with numerical simulations. Our solutions are polychromatic functions
in general, but for some special models of retarded nonlinear material laws,
also monochromatic solutions can exist. In this case the numerical simulations
raise an interesting open question: are the breather solutions with minimal
energy monochromatic or polychromatic?
