Since we consider nonlinear Maxwell and wave-type equations on unbounded space-time domains, localization of the solutions in some or all space directions is a natural requirement. Having started from standing or moving time-periodic, spatially localized solutions (breathers) we broaden our variational existence studies to breathers with a non-trivial profile at infinity as well as to solutions being localized both in space and time (rogue waves). For the nonlinear Maxwell problem we extend our results into the range of more realistic, bounded material coefficients. Here a key role is played by retarded material laws. Their regularizing effect facilitates the application of methods from bifurcation theory.
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