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


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.


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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  1. 1.
    Matveev J.P., Salle M.A.: Darboux transformations and solitons. Springer series in nonlinear dynamics, vol. 4. Springer-Verlag, Berlin (1991)CrossRefzbMATHGoogle Scholar
  2. 2.
    Rogers, C.; Schief, W.K.: Bäcklund and Darboux transformations. Cambridge Texts in Applied Mathematics, Geometry and modern applications in soliton theory. Cambridge University Press, Cambridge (2002)Google Scholar
  3. 3.
    Bobenko, A.I., Suris, Yu.B.: Integrable systems on quad-graphs. Int. Math. Res. Not. 11:573–611Google Scholar
  4. 4.
    Khanizadeh F., Mikhailov A.V., Wang J.P.: Darboux transformations and recursion operators for differential-difference equations. Theor. Math. Phys. 177(3), 1606–1654 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Mikhailov A.V.: Integrability of a two-dimensional generalization of the Toda chain. JETP Lett. 30(7), 414–418 (1979)ADSGoogle Scholar
  6. 6.
    Mikhailov A.V.: Reduction in integrable systems. The reduction group. JETP Lett. 32(2), 187–192 (1980)Google Scholar
  7. 7.
    Mikhailov A.V.: The reduction problem and the inverse scattering method. Phys. D. 3(1&2), 73–117 (1981)CrossRefzbMATHGoogle Scholar
  8. 8.
    Konstantinou-Rizos S., Mikhailov A.V., Xenitidis P.: Reduction groups and related integrable difference systems of nonlinear Schrödinger type. J. Math. Phys. 56(8), 082701 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Konstantinou-Rizos S., Mikhailov A.V.: Darboux transformations, finite reduction groups and related Yang-Baxter maps. J. Phys. A Math. Theor. 46(42), 425201 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mikhailov, A.V., Papamikos, G., Wang, J.P.: Darboux transformation with dihedral reduction group. J. Math. Phys. 55(11), 113507 (2014). arXiv:1402.5660
  11. 11.
    Pohlmeyer K., Rehren K.H.: Reduction of the two-dimensional O(n) nonlinear \({\sigma }\)-model. J. Math. Phys. 20(12), 2628–2632 (1979)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Zakharov V.E., Mikhailov A.V.: Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method. Zh. Èksper. Teoret. Fiz. 74(6), 1953–1973 (1978)MathSciNetGoogle Scholar
  13. 13.
    Eichenherr H., Pohlmeyer K.: Lax pairs for certain generalizations of the sine-Gordon equation. Phys. Lett. B. 89(1), 76–78 (1979)ADSCrossRefGoogle Scholar
  14. 14.
    Bakas I., Park Q.-H., Shin H.-J.: Lagrangian formulation of symmetric space sine-Gordon models. Phys. Lett. B. 372(12), 45–52 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wang, J.P.: Generalized Hasimoto transformation and vector Sine-Gordon equation. In: Abenda, S., Gaeta, G., Walcher, S. (eds.) SPT 2002: symmetry and perturbation theory (Cala Gonone). World Sci. Publ, River Edge (2003)Google Scholar
  16. 16.
    Mikhailov, A.V., Papamikos, G., Wang, J.P.: Dressing method for the vector sine-Gordon equation and its soliton interactions. Phys. D Nonlinear Phenom. 325:53–62 (2016). arXiv:1506.01878
  17. 17.
    Budagov A.S., Takhtajan L.A.: A nonlinear one-dimensional model of classical field theory with internal degrees of freedom. Dokl. Akad. Nauk SSSR. 235(4), 805–808 (1977)MathSciNetGoogle Scholar
  18. 18.
    Budagov A.S.: Completely integrable model of classical field theory with nontrivial particle interaction in two-dimensional space-time. In questions of quantum field theory and statistical physics. Zap. Nauchn. Sem. LOMI. 77, 24–56 (1978)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Park Q.-H., Shin H.J.: Darboux transformation and Crums formula for multi-component integrable equations. Phys. D Nonlinear Phenom. 157, 1–15 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Adler V.E.: Classification of integrable Volterra-type lattices on the sphere: isotropic case. J. Phys. A Math. Theor. 41(14), 145201 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tsuchida, T.: Integrable discretizations of the vector/matrix nonlinear Schrödinger equations and the associated Yang-Baxter map. (2015). arXiv:1505.07924
  22. 22.
    Ragnisco O., Santini P.M.: A unified algebraic approach to integral and discrete evolution equations. Inverse Probl. 6(3), 441 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Bobenko A.I., Suris Yu.B.: Discrete time lagrangian mechanics on lie groups, with an application to the lagrange top. Commun. Math. Phys. 204, 147–188 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Drinfel’d, V.G., Sokolov, V.V.: Lie algebras and equations of Korteweg–de Vries type. In: Current problems in mathematics, vol. 24, Itogi Nauki i Tekhniki, pp. 81–180. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (1984)Google Scholar
  25. 25.
    Mikhailov, A.V.: Formal diagonalisation of the Lax-Darboux scheme and conservation laws of integrable partial differential, differential-difference and partial difference.
  26. 26.
    Adler V.E., Postnikov V.V.: On vector analogs of the modified volterra lattice. J. Phys. A Math. Theor. 41(45), 455203 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Adler V.E., Svinolupov S.I., Yamilov R.I.: Multi-component Volterra and Toda type integrable equations. Phys. Lett. A. 254(1–2), 24–36 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wang J.P.: Representations of \({{\mathfrak{sl}}(2,{\mathbb{C}})}\) in category \({{\mathcal{O}}}\) and master symmetries. Theor. Math. Phys. 184(2), 1078–1105 (2015)CrossRefzbMATHGoogle Scholar
  29. 29.
    Nijhoff F., Capel H.: The discrete Korteweg-de Vries equation. Acta Appl. Math. 39(1–3), 133–158 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Suris Yu.B., Veselov A.P.: Lax matrices for Yang-Baxter maps. J. Nonlinear Math. Phys. 10, 223–230 (2003)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Kouloukas T., Papageorgiou V.: Poisson Yang-Baxter maps with binomial Lax matrices. J. Math. Phys. 52(12), 404012 (2011)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Goncharenko, V.M., Veselov, A.P.: Yang-baxter maps and matrix solitons. In: Shabat, A.B., González-López, A., Mañas, M., Martnez Alonso, L., Rodriguez, M.A. (eds.) New trends in integrability and partial solvability, NATO Science Series, vol. 132, pp. 191–197. Springer Netherlands (2004)Google Scholar
  33. 33.
    Mikhailov, A.V., Wang, J.P., Xenitidis, P.: Recursion operators, conservation laws and integrability conditions for difference equations. Theor. Math. Phys. 167, 421–443 (2011). arXiv:1004.5346

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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