Self-Consistent Sources for Integrable Equations via Deformations of Binary Darboux Transformations
arXiv:1510.05166 · doi:10.1007/s11005-016-0859-1
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.
35 pages, 1 figure, second version: some amendments, additional references, section 5 added
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- Tropical limit of matrix solitons and entwining Yang-Baxter maps
- Matrix Boussinesq solitons and their tropical limit
- The nonlinear Schrödinger equation with forcing involving products of eigenfunctions