  In this paper, we study the nonlinear Klein-Gordon systems arising from
relativistic physics and quantum field theories
  $$\left\{\begin{array}{lll} u_{tt}- u_{xx} +bu + \varepsilon v + f(t,x,u)
=0,\; v_{tt}- v_{xx} +bv + \varepsilon u + g(t,x,v) =0
  \end{array}\right.
  $$ where $u,v$ satisfy the Dirichlet boundary conditions on spatial interval
$[0, \pi]$, $b>0$ and $f$, $g$ are $2\pi$-periodic in $t$. We are concerned
with the existence, regularity and asymptotic behavior of time-periodic
solutions to the linearly coupled problem as $\varepsilon$ goes to 0. Firstly,
under some superlinear growth and monotonicity assumptions on $f$ and $g$, we
obtain the solutions $(u_\varepsilon, v_\varepsilon)$ with time-period $2\pi$
for the problem as the linear coupling constant $\varepsilon$ is sufficiently
small, by constructing critical points of an indefinite functional via
variational methods. Secondly, we give precise characterization for the
asymptotic behavior of these solutions, and show that as
$\varepsilon\rightarrow 0$, $(u_\varepsilon, v_\varepsilon)$ converge to the
solutions of the wave equations without the coupling terms. Finally, by careful
analysis which are quite different from the elliptic regularity theory, we
obtain some interesting results concerning the higher regularity of the
periodic solutions.
