Ricardo Weder
Multidimensional Inverse Scattering for the Nonlinear Klein-Gordon 
Equation with a Potential
(37K, Latex)

ABSTRACT.  In this paper we solve the multidimensional inverse scattering problem for the nonlinear Klein-Gordon equation on $\ER^n, n \geq 2$: 
$$ \frac{\partial^2}{\partial t^2} u(x,t) -\Delta u(x,t)+ u(x,t) + V_0(x) 
u(x,t) +\sum_{j=1}^{\infty} V_j(x) |u|^{2(j_0+j)} u(x,t)=0. 
$$ 
We prove that the small-amplitude limit of the scattering operator determines 
uniquely all the $V_j, j=0,1, \cdots $. Our proof gives, as well, a method for the 
reconstruction of the $V_j, j=0,1, \cdots$.