  The features of propagation of intense waves are of great interest for theory
and experiment in electrodynamics and acoustics. The behavior of nonlinear
waves in a bounded volume is of especial importance and, at the same time, is
an extremely complicated problem. It seems almost impossible to find a rigorous
solution to such a problem even for any model of nonlinearity. We obtain the
first exact solution of this type. We present a new method for deriving exact
solutions of the Maxwell equations in a nonlinear medium without dispersion and
give examples of the obtained solutions that describe propagation of
cylindrical electromagnetic waves in a nonlinear nondispersive medium and free
electromagnetic oscillations in a cylindrical cavity resonator filled with such
a medium.
