On Reciprocal Bäcklund Transformations of Inverse Scattering Schemes

The notion of reciprocally related inverse scattering schemes is introduced and is shown to be a key component in the link between the AKNS and WKI schemes. Reciprocal auto-Bäcklund transformations are presented both for a generalised Harry-Dym equation and an equation descriptive of nonlinear oscillation of elastic beams. Further, the N-loop soliton solution of the KIW equation is generated in a convenient parametric form via reciprocal Bäcklund transformations. Finally, an important reduction to canonical spectral form is shown to be a reciprocal transformation.