Self-Consistent Sources for Integrable Equations via Deformations of Binary Darboux Transformations

TL;DR

This paper introduces a method to derive self-consistent source extensions of various integrable equations using deformations of binary Darboux transformations, unifying their structure through bidifferential calculus.

Contribution

It presents a novel deformation approach to generate self-consistent sources for integrable systems via binary Darboux transformations, encompassing multiple classical equations.

Findings

Derived matrix versions of self-consistent source extensions for several integrable equations.

Revealed a (2+1)-dimensional Yajima-Oikawa system from pKP hierarchy deformation.

Established a universal framework using bidifferential calculus for these systems.

Abstract

We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As examples, we obtain in particular matrix versions of self-consistent source extensions of the sine-Gordon, nonlinear Schrodinger, KdV, Boussinesq, KP, Davey-Stewartson, two-dimensional Toda lattice and discrete KP systems. We also recover a (2+1)-dimensional version of the Yajima-Oikawa system from a deformation of the pKP hierarchy. By construction, these systems are accompanied by a hetero binary Darboux transformation, which generates solutions of such a system from a solution of the source-free system and additionally solutions of an associated linear system and its adjoint. The essence of all this is encoded in universal equations in the framework of bidifferential calculus.