Bridging trails in reflectionless potential deformation: Two paths and one horizon

Abstract

In contrast to the conventional one-parameter class of isospectral deformation using translation, we calculate the two- and three-parameter classes of isospectral deformation of the well-known reflectionless potential by utilizing a far different approach of scaling methodology. Subsequently, using these results, we find that this more general class of deformations is not unique but instead subsume in the same class of conventional one-parameter translational deformation. We also provide a theoretical foundation for how these two incredibly different approaches converge at the same destination. Finally, we show that the most generic class of potentials, obtained by scaling deformation, are solutions of the nonlinear KdV equation.