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Letters in Mathematical Physics

, Volume 106, Issue 7, pp 973–996 | Cite as

Darboux Transformation for the Vector Sine-Gordon Equation and Integrable Equations on a Sphere

  • Alexander V. Mikhailov
  • Georgios Papamikos
  • Jing Ping WangEmail author
Article

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.

Keywords

the vector sine-Gordon equation Lax representations Darboux transformations Bäcklund transformations Yang–Baxter maps integrable equations on a sphere 

Mathematics Subject Classification

35Q51 37K10 37K35 39A14 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Alexander V. Mikhailov
    • 2
  • Georgios Papamikos
    • 1
  • Jing Ping Wang
    • 1
    Email author
  1. 1.School of Mathematics, Statistics and Actuarial ScienceUniversity of KentCanterburyUK
  2. 2.School of MathematicsUniversity of LeedsLeedsUK

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