Exact multisolitons of noncommutative and commutative Lakshmanan–Porsezian–Daniel equation


We propose and study a noncommutative generalization of Lakshmanan–Porsezian–Daniel equation. For this generalization, we demonstrate its integrability by constructing its zero curvature representation, Darboux transformation and multisoliton solutions. The soliton solutions up to fifth order are presented explicitly within framework of quasideterminants.

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  1. 1.

    M.J. Ablowitz, B. Prinari, A.D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems (Cambridge University Press, Cambridge, 2004)

    Google Scholar 

  2. 2.

    Y.S. Kivshar, G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, Cambridge, 2003)

    Google Scholar 

  3. 3.

    G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, Cambridge, 2001)

    Google Scholar 

  4. 4.

    A. Hasegawa, Optical Solitons in Fibers (Springer, Berlin, 1990)

    Google Scholar 

  5. 5.

    M. Daniel, K. Deepamala, Davydov soliton in alpha helical proteins: higher order and discreteness effects. Phys. A 221, 241 (1995)

    Article  Google Scholar 

  6. 6.

    M. Lakshmanan, K. Porsezian, M. Daniel, Effect of discreteness on the continuum limit of the Heisenberg spin chain. Phys. Lett. A 133, 483 (1988)

    ADS  Article  Google Scholar 

  7. 7.

    K. Porsezian, M. Daniel, M. Lakshmanan, On the integrability aspects of the one-dimensional classical continuum isotropic biquadratic Heisenberg spin chain. J. Math. Phys. 33, 1807 (1992)

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    K. Porsezian, Completely integrable nonlinear Schrödinger type equations on moving space curves. Phys. Rev. E 55, 3785 (1997)

    ADS  MathSciNet  Article  Google Scholar 

  9. 9.

    P. Wang, B. Tian, Y. Jiang, Y.F. Wang, Integrability and soliton solutions for an inhomogeneous generalized fourth-order nonlinear Schrödinger equation describing the inhomogeneous alpha helical proteins and Heisenberg ferromagnetic spin chains. Phys. B 411, 166 (2013)

    ADS  Article  Google Scholar 

  10. 10.

    J.W. Yang, Y.T. Gao, Q.M. Wang, C.Q. Su, Y.J. Feng, X. Yu, Bilinear forms and soliton solutions for a fourth-order variable-coefficient nonlinear Schrödinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain or an alpha helical protein. Phys B 481, 148 (2016)

    ADS  Article  Google Scholar 

  11. 11.

    W. Liu, D.Q. Qiu, Z.W. Wu, J.S. He, Dynamical behavior of solution in integrable nonlocal Lakshmanan–Porsezian–Daniel equation. Commun. Theor. Phys. 65, 671 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    P.G. Estévez, E. Díaz, F.D. Adame, J.M. Cerveró, E. Diez, Lump solitons in a higher-order nonlinear equation in 2+1 dimensions. Phys. Rev E 93, 062219 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    M. Lakshmanan, Continuum spin system as an exactly solvable dynamical system. Phys. Lett. A 61, 53 (1977)

    ADS  Article  Google Scholar 

  14. 14.

    S. Minwalla, M.V. Raamsdonk, N. Seiberg, Noncommutative perturbative dynamics. J. High Energy Phys. 02, 020 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    K. Furuta, T. Inami, Ultraviolet property of noncommutative Wess–Zumino—Witten model. Mod. Phys. Lett. A 15, 997 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  16. 16.

    N. Seiberg, E. Witten, String theory and noncommutative geometry. J. High Energy Phys. 09, 032 (1999)

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    M. Hamanaka, Noncommutative Ward’s conjecture and integrable systems. Nucl. Phys. B 741, 368 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  18. 18.

    Yu. Fa-Jun, Noncommutative AKNS equation hierarchy and its integrable couplings with Kronecker product. Chin. Phys. Lett. 25, 359 (2008)

    ADS  Article  Google Scholar 

  19. 19.

    A. Dimakis, F. Müller-Hoissen, Bicomplexes, integrable models, and noncommutative geometry. Int. J. Mod. Phys. B 14, 2455 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    O. Lechtenfeld, A.D. Popov, Noncommutative multi-solitons in 2+1 dimensions. JHEP 11, 040 (2001)

    ADS  Article  Google Scholar 

  21. 21.

    O. Lechtenfeld, L. Mazzanti, S. Penati, A.D. Popov, L. Tamassia, Integrable noncommutative sine-Gordon model. Nucl. Phys. B 705, 477 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  22. 22.

    C.R. Gilson, J.J.C. Nimmo, On a direct approach to quasideterminant solutions of a noncommutative KP equation. J. Phys. A 40, 3839 (2007)

    ADS  MathSciNet  Article  Google Scholar 

  23. 23.

    C.R. Gilson, S.R. Macfarlane, Dromion solutions of noncommutative Davey–Stewartson equations. J. Phys. A 42, 235202 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  24. 24.

    N. Mushahid, M. Hassan, A noncommutative coupled dispersionless system, Darboux transformation and explicit solutions. Mod. Phys. Lett. A 29, 1450206 (2014)

    ADS  MathSciNet  Article  Google Scholar 

  25. 25.

    A. Dimakis, F. Müller-Hoissen, Noncommutative NLS equation. Czech. J. Phys. 51, 1285 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  26. 26.

    I. Gelfand, V. Retakh, Determinants of matrices over noncommutative rings. Funct. Anal. Appl. 25(2), 91–102 (1991)

    MathSciNet  Article  Google Scholar 

  27. 27.

    I.M. Gelfand, S. Gelfand, V.M. Retakh, R.L. Wilson, Quasideterminants. Adv. Math. 193, 56 (2005)

    MathSciNet  Article  Google Scholar 

  28. 28.

    H.W.A. Riaz, M. Hassan, Multisoliton solutions of integrable discrete and semi-discrete principal chiral equations. Commun. Nonlinear Sci. Numer. Simulat. 54, 416 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  29. 29.

    H.W.A. Riaz, M. Hassan, Generalized lattice Heisenberg model and its quasideterminant soliton solutions. Theor. Math. Phys. 195, 665 (2018)

    MathSciNet  Article  Google Scholar 

  30. 30.

    H.W.A. Riaz, M. Hassan, Multi-component noncommutative coupled dispersionless system and its quasideterminant solutions. Mod. Phys. Lett. A 33, 1850086 (2018)

    ADS  MathSciNet  Article  Google Scholar 

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The author would like to thank Professor QP Liu for reading the paper and giving many suggestions. The author would also like to thank Department of Physics, University of the Punjab, Lahore, Pakistan for providing the research facilities where the most part of this work was carried out. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11871471, 11331008 and 11931017), Foreign Experts Scientific Cooperation Fund.

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Riaz, H.W.A. Exact multisolitons of noncommutative and commutative Lakshmanan–Porsezian–Daniel equation. Eur. Phys. J. Plus 135, 508 (2020). https://doi.org/10.1140/epjp/s13360-020-00505-6

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