ABSTRACT
The results of inverse scattering problem associated with the initial-boundary value problem (IBVP) for the Korteweg-de Vries (KdV) equation with dominant surface tension are formulated. The solution of the IBVP is expressible through the found solution of the Gelfand–Levitan–Marchenko equation. This chapter aims to establish the necessary and sufficient conditions for given functions to be the left- and right-reflection coefficients. It also aims to reduce the problem of solving the considered IBVP to that of solving two linear inverse scattering problems (SPs). The first SP is associated with the KdV equation, the second SP is self-adjoint. The chapter seeks to study the properties of the scattering matrix s(k) and the left- and right-reflection coefficients for the first SP. It highlights the necessary and sufficient conditions for given functions to be the left- and right-reflection coefficients.
