A REMARK ON LONG RANGE SCATTERING FOR THE NONLINEAR KLEIN–GORDON EQUATION

We consider the scattering problem for the nonlinear Klein–Gordon Equation with long range nonlinearity in one dimension. We prove that for all prescribed asymptotic solutions there is a solution of the equation with such behavior, for some choice of initial data. In the case the nonlinearity has the good sign (repulsive) the result hold for arbitrary size asymptotic data. The method of proof is based on reducing the long range phase effects to an ODE; this is done via an appropriate ansatz. We also find the complete asymptotic expansion of the solutions.