Nonlinear scattering for a system of nonlinear Klein-Gordon equations

We consider the initial value problem for systems of nonlinear Klein-Gordon equations with quadratic nonlinearities. We prove the existence of scattering states, namely, the asymptotic stability of small solutions in the neighborhood of the free solutions for small initial data in the weighted Sobolev space H^4,3(R³) × H^3,3(R³). If nonlinearities satisfy the strong null condition, then the same result is true in two space dimensions for small initial data in H^5,4(R²) × H^4,4(R²). A system of massive Dirac-massless Klein-Gordon equations in three space dimensions is also considered by our method.