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.
Highlights
The study of soliton equations with self-consistent sources has been pursued in particular in the work of Mel’nikov [1,2,3,4,5,6]
We obtain in particular matrix versions of self-consistent source extensions of the KdV, Boussinesq, sine-Gordon, nonlinear Schrodinger, KP, Davey–Stewartson, two-dimensional Toda lattice and discrete KP equation
We show that self-consistent source extensions arise via a simple deformation of the “potential” that appears in the binary Darboux transformation method
Summary
The study of soliton equations with self-consistent sources has been pursued in particular in the work of Mel’nikov [1,2,3,4,5,6]. 2. Via Binary Darboux Transformation to Self-Consistent Source Extensions of the pKP Equation. The above procedure provides us with a hetero binary Darboux transformation from the pKP equation and its associated linear system to any of the pKP systems with self-consistent sources (modified by ω).. In the framework of bidifferential calculus, we can abstract the underlying structure from the specific example (here pKP) and obtain corresponding self-consistent source extensions of quite a number of other integrable equations. We can choose different bidifferential calculi and obtain self-consistent source extensions of other integrable equations. The resulting equations, together with (3.18), constitute the (Miura-) dual of (3.14) and (3.15) It is another generating system for further self-consistent source extensions of integrable equations. A corresponding extension of the whole pKP hierarchy is presented in Appendix A
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