  To clarify certain nonlinear properties of strong gravitational field, we
investigate cylindrically symmetric gravitational waves that are localized as
regular wave packets in the space of radial and time coordinates. The waves are
constructed by applying a certain kind of harmonic mapping method to the seed
solutions with linear polarization, which are generalizations of the solution
representing a cylindrical gravitational pulse wave discussed by Weber-Wheeler
and Bonnor. The solutions obtained here, though their form is rather simple,
show occurrence of strong mutual conversion between a linear mode and a cross
mode apparently. The single localized wave shows the conversion in the vicinity
of the symmetric axis where the self-interaction is strengthened, and the
collision between multiple waves also causes the conversion. These phenomena
can be thought to be the emergence of genuine nonlinearity that the Einstein
gravity holds. Finally we discuss a simple, but interesting application of the
solutions to the case of the Einstein-Maxwell system.
