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 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.
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Chvartatskyi, O., Dimakis, A. & Müller-Hoissen, F. Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations. Lett Math Phys 106, 1139–1179 (2016). https://doi.org/10.1007/s11005-016-0859-1
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DOI: https://doi.org/10.1007/s11005-016-0859-1