Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere

Abstract

We propose a method for construction of Darboux transformations, which is a new development of the dressing method for Lax operators invariant under a reduction group. We apply the method to the vector sine-Gordon equation and derive its Bäcklund transformations. We show that there is a new Lax operator canonically associated with our Darboux transformation resulting an evolutionary differential-difference system on a sphere. The latter is a generalised symmetry for the chain of Bäcklund transformations. Using the re-factorisation approach and the Bianchi permutability of the Darboux transformations we derive new vector Yang-Baxter map and integrable discrete vector sine-Gordon equation on a sphere.

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• Matrix factorizations and pentagon maps
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Birational solutions to the set-theoretical 4-simplex equation
• Drinfel'd-Sokolov construction and exact solutions of vector modified KdV hierarchy
• Discrete Lax pairs and hierarchies of integrable difference systems
• Multisoliton solutions of the vector nonlinear Schrödinger equation (Kulish-Sklyanin model) and the vector mKdV equation
• Integrable multi-component difference systems of equations
• Discrete integrable systems, Darboux transformations, and Yang-Baxter maps
• Integrable extensions of Adler's map via Grassmann algebras
• On refactorization problems and rational Lax matrices of quadrirational Yang-Baxter maps