Darboux Transformations for Higher-Rank Kadomtsev—Petviashvili and Krichever—Novikov Equations

It is shown that the action of a special ‘rank 2’ or ‘rank 3’ Darboux transformation, called transference, upon a pair of commuting ordinary differential operators of orders 4 and 6 implements the Abelian sum on their elliptic joint spectrum. A consequence of this is that, under the deformation of these commuting operators by the KP flow, every ‘rank 2’ KP solution corresponds to a solution of the Krichever—Novikov (KN) equation, and vice versa, with the transference process providing the correspondence between (2 + 1) and (1 + 1) dimensions. For a singular joint spectrum, it is further shown that transference at the singular point produces a correspondence between solutions of the singular KN equation and those of the KdV equation. These correspondences are illustrated by considering examples of a nondecaying rational KdV or Boussinesq solutions and the corresponding rational, singular-KN and rational KP solutions which the transference process generates.