Inverse scattering problem for a cubic string having the shape of a step

Abstract: Problem for a cubic string having the shape of a step is studied. It is proved that here we have two scattering problems: the direct one and its dual. The direct problem describes scattering of the waves coming from $+\infty$'', and its dual, correspondingly, of the waves from$-\infty$''. The main systems of linear singular equations for both problems are obtained. It is shown how to unambiguously restore potential from the solutions to these equation systems.