Bäcklund and Reciprocal Transformations: Gauge Connections

Abstract

This chapter discusses Backlund and reciprocal transformations. The role of Backlund transformations (BTs) is well established in the analysis of nonlinear evolution equations amenable to the inverse scattering transform (IST). Thus, in particular, BTs are routinely used to construct nonlinear superposition principles, whereby multisoliton solutions may be generated. BTs can be constructed by a variety of means. In the context of integrable systems, perhaps the most direct approach is via the dressing method (DM). This is linked to the classical Darboux transformation (CDT) that generates solutions to related Schrodinger equations. The relation between the BT and DM methods for 1 + 1 dimensional nonlinear integrable systems has been detailed by Levi, Ragnisco, and Sym. Gauge transformations, in conjunction with reciprocal transformations, also play an important part in links among scattering schemes associated with the classes of nonlinear integrable systems. In the chapter, reciprocal transformations in 1 + 1- and 2 + 1- dimensions are introduced. The link between the 1 + 1-dimensional Zakharov–Shabat (ZS)-AKNS and WKI spectral schemes via a combination of gauge and reciprocal transformations is established in the chapter. In 2 + 1-dimensions, a reciprocal link between the Kadomtsev–Petviashvili and a Dym-type equation is revealed. This allows a novel invariance of the latter to be constructed.