An hyperbolic system of P.D.E. relevant in general relativity

Assuming as starting point the validity of the Einstein-Rosen metric, we study the hyperbolic system of P.D.E. to which the Einstein’s field’s equations can be reduced. We prove using the implicit function theorem in Banach spaces, the existence and uniqueness of gravitational waves of small amplitude. A class of solutions, not necessarily small is also constructed. In the last Section a theorem of existence and uniqueness is given for the corresponding stationary problem.