  In our study we consider nonlinear, power-law field-dependent electrical
permitivity and magnetic permeability and investigate the time-dependent
Maxwell equations with the self-similar Ansatz. This is a first-order
hyperbolic PDE system which can conserve non-continuous initial conditions
describing electromagnetic shock-waves. Besides shock-waves other interesting
solutions (e.g. with localized compact support) can be found with delicate
physical properties. Such phenomena may happen in complex materials induced by
the planned powerful Extreme Light Infrastructure(ELI) laser pulses.
