On inverse scattering

In a previous paper the inverse problem associated with a hyperbolic dispersive partial differential equation with smooth coefficients was considered. The inverse problem (the determination of the coefficients) was formulated in terms of a dual set of integral equations involving measurable quantities, the kernels of the transmission, and reflection operators. These equations contained an unknown parameter which occurs in a linear manner. A better approach to determine this parameter is presented here. It involves an auxiliary equation, which is used to eliminate the unknown parameter from the integral equations. It is shown that the resulting system has a unique solution for a certain class of scattering problems. These uniqueness results are then strengthened when an additional equation is employed to reduce the dual set of integral equations to a single integral equation.