Letters in Mathematical Physics

, Volume 106, Issue 8, pp 1139–1179 | Cite as

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

  • Oleksandr Chvartatskyi
  • Aristophanes Dimakis
  • Folkert Müller-HoissenEmail author
Open Access


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 KdV, Boussinesq, sine-Gordon, nonlinear Schrödinger, KP, Davey–Stewartson, two-dimensional Toda lattice and discrete KP equation. 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.


bidifferential calculus Darboux transformation integrable systems self-consistent sources KP discrete KP nonlinear Schrödinger Davey–Stewartson Toda lattice Yajima–Oikawa 

Mathematics Subject Classification

35C08 37K10 70H06 


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© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Oleksandr Chvartatskyi
    • 1
    • 2
  • Aristophanes Dimakis
    • 3
  • Folkert Müller-Hoissen
    • 2
    Email author
  1. 1.Mathematisches InstitutGeorg-August Universität GöttingenGöttingenGermany
  2. 2.Max Planck Institute for Dynamics and Self-OrganizationGöttingenGermany
  3. 3.Department of Financial and Management EngineeringUniversity of the AegeanChiosGreece

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