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

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.

arXiv comment35 pages, 1 figure, second version: some amendments, additional references, section 5 added

References in corpus (11)

• Generalized Darboux transformations for the KP equation with self-consistent sources
• A new extended KP hierarchy
• Bi-differential calculi and integrable models
• Constrained KP Hierarchies: Additional Symmetries, Darboux-Bäcklund Solutions and Relations to Multi-Matrix Models
• Two-component Analogue of Two-dimensional Long Wave-Short Wave Resonance Interaction Equations: A Derivation and Solutions
• The Solutions of the NLS Equations with Self-Consistent Sources
• Exponentially Localized Solutions of Mel'nikov Equation
• Functional representations of integrable hierarchies
• Discrete KP equation with self-consistent sources
• A new multi-component two dimensional Toda lattice hierarchy and two dimensional Toda lattice with self-consistent sources
• A new extended matrix KP hierarchy and its solutions

Cited by in corpus (8)

• New integrable (3+1)-dimensional systems and contact geometry
• Matrix KP: tropical limit and Yang-Baxter maps
• A vectorial binary Darboux transformation of the first member of the negative part of the AKNS hierarchy
• NLS breathers, rogue waves, and solutions of the Lyapunov equation for Jordan blocks
• Generalized Volterra lattices: binary Darboux transformations and self-consistent sources
• 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