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

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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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Authors and Affiliations

  1. School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, UK

    Georgios Papamikos & Jing Ping Wang

  2. School of Mathematics, University of Leeds, Leeds, UK

    Alexander V. Mikhailov

Authors
  1. Alexander V. Mikhailov
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  2. Georgios Papamikos
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  3. Jing Ping Wang
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Correspondence to Jing Ping Wang.

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Mikhailov, A.V., Papamikos, G. & Wang, J.P. Darboux Transformation for the Vector Sine-Gordon Equation and Integrable Equations on a Sphere. Lett Math Phys 106, 973–996 (2016). https://doi.org/10.1007/s11005-016-0855-5

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  • DOI: https://doi.org/10.1007/s11005-016-0855-5

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