We consider a system of nonlinear Klein-Gordon equations in one space dimension with quadratic nonlinearities
(∂t2+∂x2+ mj2)uj = Nj(∂u),
j = 1, . . . , l. We show the existence of solutions in an analytic function space. When the nonlinearity satisfies a strong null condition introduced by Georgiev we prove the global existence and obtain the large time asymptotic behavior of small solutions.
"Nonlinear scattering for a system of one dimensional nonlinear Klein-Gordon equations." Hokkaido Math. J. 37 (4) 647 - 667, November 2008. https://doi.org/10.14492/hokmj/1249046362