Multidimensional inverse scattering for the nonlinear Klein-Gordon equation with a potential

In this paper, e solve the multidimensional inverse scattering problem for the nonlinear Klein Gordon equation on R-n, n greater than or equal to 2: partial derivative(2)/partial derivativet(2) u(x,t) - Deltau(x,t) + u(x,t) + V-0(x)u(x,t) + Sigma(j=1)(n) V-0(x)\u\(2Vo(x)) u(x,t) = 0. We prove that the small-amplitude limit of the scattering operator determines uniquely all the V-j, j = 0, 1,... . Our proof gives, as well, a method for the reconstruction of the V-j, j = 0, 1,... . (C) 2002 Elsevier Science (USA).