  Klein-Gordon equations describe the dynamics of waves/particles in sub-atomic
scales. For nonlinear Klein-Gordon equations, their breather solutions are
usually known as time periodic solutions with the vanishing spatial-boundary
condition. The existence of breather solution is known for the Sine-Gordon
equations, while the Sine-Gordon equations are also known as the soliton
equation. The breather solutions is a certain kind of time periodic solutions
that are not only play an essential role in the bridging path to the chaotic
dynamics, but provide multi-dimensional closed loops inside phase space. In
this paper, based on the high-precision numerical scheme, the appearance of
breather mode is studied for nonlinear Klein-Gordon equations with periodic
boundary condition. The spatial periodic boundary condition is imposed, so that
the breathing-type solution in our scope is periodic with respect both to time
and space. In conclusion, the existence condition of space-time periodic
solution is presented, and the compact manifolds inside the
infinite-dimensional dynamical system is shown. The space-time breather
solutions of Klein-Gordon equations can be a fundamental building block for the
sub-atomic nonlinear dynamics.
