Ricardo Weder
Inverse Scattering for the Nonlinear Schroedinger Equation. Reconstruction of the Potential and the Nonlinearity.
(32K, Latex)
ABSTRACT. In this paper we consider the inverse scattering problem for the nonlinear Schroedinger equation on the line:
$$ i \frac{\partial}{\partial t} u(t,x)=- \frac{d^2}{dx^2} u(t,x)
+V_0(x) u(t,x) +\sum_{j=1}^{\infty} V_j(x) |u|^{2(j_0+j)} u(t,x).$$
We prove, under appropriate conditions, that the small-amplitude scattering operator determines uniquely, $V_j, j=0,1, \cdots $. Our proof gives also a method for the reconstruction of the $V_j, j=0,1,\cdots $.