Analysis and comparative study of non-holonomic and quasi-integrable deformations of the nonlinear Schrödinger equation

  • Kumar AbhinavEmail author
  • Partha Guha
  • Indranil Mukherjee
Original paper


The non-holonomic deformation of the nonlinear Schrödinger equation, uniquely obtained from both the Lax pair and Kupershmidt’s bi-Hamiltonian (Kupershmidt in Phys Lett A 372:2634, 2008) approaches, is compared with the quasi-integrable deformation of the same system (Ferreira et al. in JHEP 2012:103, 2012). It is found that these two deformations can locally coincide only when the phase of the corresponding solution is discontinuous in space, following a definite phase-modulus coupling of the non-holonomic inhomogeneity function. These two deformations are further found to be not gauge equivalent in general, following the Lax formalism of the nonlinear Schrödinger equation. However, the localized solutions corresponding to both these cases converge asymptotically as expected. Similar conditional correspondence of non-holonomic deformation with a non-integrable deformation, namely due to locally scaled amplitude of the solution to the nonlinear Schrödinger equation, is further obtained.


Nonlinear Schrödinger equation Quasi-integrable deformation Non-holonomic deformation Solitons 



The authors are grateful to Professors Luiz. A. Ferreira and Wojtek J. Zakrzewski for their encouragement, various useful discussions and critical reading of the draft.


Kumar Abhinav’s research is done at and supported by the IF, Naresuan University, Thailand. Partha Guha’s research is partially supported by FAPESP through Grant numbered 2016/06560-6. A pre-print of this manuscript is available online at arXiv:1611.00961 [nlin.SI].

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.


  1. 1.
    Das, A.: Integrable Models. World Scientific, Singapore (1989)CrossRefzbMATHGoogle Scholar
  2. 2.
    Cazenave, T.: An Introduction to Nonlinear Schrödinger Equations. Instituto de Matemética, UFRJ, Rio de Janeiro (1993)Google Scholar
  3. 3.
    Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467 (1968)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Zh. Exp. Teor. Fiz. 61, 118 (1971) Google Scholar
  5. 5.
    Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62 (1972)MathSciNetGoogle Scholar
  6. 6.
    Yadav, O.P., Jiwari, R.: Some soliton-type analytical solutions and numerical simulation of nonlinear Schrödinger equation. Nonlinear Dyn. 95, 2825 (2019)CrossRefGoogle Scholar
  7. 7.
    Mukam, S.P.T., Souleymanou, A., Kuetche, V.K., Bouetou, T.B.: Generalized Darboux transformation and parameter-dependent rogue wave solutions to a nonlinear Schrödinger system. Nonlinear Dyn. 93, 373 (2018)CrossRefzbMATHGoogle Scholar
  8. 8.
    Ferrara, S., Girardello, L., Sciuto, S.: An infinite set of conservation laws of the supersymmetric sine-Gordon theory. Phys. Lett. B 76, 303 (1978)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Liu, W., Zhang, Y., Luan, Z., Zhou, Q., Mirzazadeh, M., Ekici, M., Biswas, A.: Dromion-like soliton interactions for nonlinear Schrödinger equation with variable coefficients in inhomogeneous optical fibers. Nonlinear Dyn. 96, 729 (2019)CrossRefGoogle Scholar
  10. 10.
    Zhang, Y., Yang, C., Yu, W., Mirzazadeh, M., Zhou, Q., Liu, W.: Interactions of vector anti-dark solitons for the coupled nonlinear Schrödinger equation in inhomogeneous fibers. Nonlinear Dyn. 94, 1351 (2018)CrossRefGoogle Scholar
  11. 11.
    Ferreira, L.A., Zakrzewski, W.J.: The concept of quasi-integrability: a concrete example. JHEP 2011, 130 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Blas, H., Zambrano, M.: Quasi-integrability in the modified defocusing non-linear Schrödinger model and dark solitons. JHEP 2016, 005 (2016)CrossRefzbMATHGoogle Scholar
  13. 13.
    Blas, H., Callisaya, H.F.: Quasi-integrability in deformed sine-Gordon models and infinite towers of conserved charges. Commun. Nonlinear Sci. Numer. Simul. 55, 105 (2018)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Blas, H., do Bonfim, A.C.R., Vilela, A.M.: Quasi-integrable non-linear Schrödinger models, infinite towers of exactly conserved charges and bright solitons. JHEP 2017, 106 (2017)CrossRefzbMATHGoogle Scholar
  15. 15.
    Ferreira, L.A., Luchini, G., Zakrzewski, W.J.: The concept of quasi-integrability for modified non-linear Schrödinger models. JHEP 2012, 103 (2012)CrossRefzbMATHGoogle Scholar
  16. 16.
    Karasu-Kalkani, A., Karasu, A., Sakovich, A., Sakovich, S., Turhan, R.: A new integrable generalization of the Korteweg–de Vries equation. J. Math. Phys. 49, 073516 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Kupershmidt, B.A.: KdV6: an integrable system. Phys. Lett. A 372, 2634 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Kundu, A.: Two-fold integrable hierarchy of nonholonomic deformation of the derivative nonlinear Schrödinger and the Lenells–Fokas equation. J. Math. Phys. 51, 022901 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Guha, P.: Nonholonomic deformation of generalized KdV-type equations. J. Phys. A Math. Theor. 42, 345201 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Guha, P.: Nonholonomic deformation of coupled and supersymmetric KdV equations and Euler–Poincaré–Suslov method. Rev. Math. Phys. 27, 1550011 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Fetter, A.L., Walecka, J.D.: Quantum Theory of Many-Particle System. Dover Publications, New York (2012)Google Scholar
  22. 22.
    Lakshmanan, M.: Continuum spin system as an exactly solvable dynamical system. Phys. Lett. A 61, 53 (1977)CrossRefGoogle Scholar
  23. 23.
    Hasimoto, R.: A soliton on a vortex filament. J. Fluid Mech. 51, 477 (1972)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Balakrishnan, R.: On the inhomogeneous Heisenberg chain. J. Phys. C Solid State Phys. 15, L1305 (1982)CrossRefGoogle Scholar
  25. 25.
    Balakrishnan, R.: Dynamics of a generalised classical Heisenberg chain. Phys. Lett. 92, 243 (1982)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Balakrishnan, R.: Inverse spectral transform analysis of a nonlinear Schrödinger equation with x-dependent coefficients. Physica D 16, 405 (1985)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Abhinav, K., Guha, P.: Inhomogeneous Heisenberg spin chain and quantum vortex filament as non-holonomically deformed NLS systems. Eur. Phys. J. B 91, 52 (2018)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Shivamoggi, B.K.: Vortex motion in superfluid \(^4\)He: effects of normal fluid flow. Eur. Phys. J. B 86, 275 (2013)CrossRefGoogle Scholar
  29. 29.
    Van Gorder, R.A.: Quantum Hasimoto transformation and nonlinear waves on a superfluid vortex filament under the quantum local induction approximation. Phys. Rev. E 91, 053201 (2015)CrossRefGoogle Scholar
  30. 30.
    Nian, J.: Note on Nonlinear Schrödinger Equation, KdV Equation and 2D Topological Yang–Mills–Higgs Theory. arXiv:1611.04562 [hep-th] (2016)
  31. 31.
    Guo, R., Zhao, H.-H., Wang, Y.: A higher-order coupled nonlinear Schrödinger system: solitons, breathers, and rogue wave solutions. Nonlinear Dyn. 83, 2475 (2016)CrossRefzbMATHGoogle Scholar
  32. 32.
    Kumar, S.S., Balakrishnan, S., Sahadevan, R.: Integrability and Lie symmetry analysis of deformed N-coupled nonlinear Schrödinger equations. Nonlinear Dyn. 90, 2783 (2017)CrossRefzbMATHGoogle Scholar
  33. 33.
    Abhinav, K., Guha, P., Mukherjee, I.: Study of quasi-integrable and non-holonomic deformation of equations in the NLS and DNLS hierarchy. J. Math. Phys. 59, 101507 (2018)CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)zbMATHGoogle Scholar
  35. 35.
    Kamiński, M.: The Stochastic Perturbation Method for Computational Mechanics. Wiley, New York (2013)CrossRefzbMATHGoogle Scholar
  36. 36.
    Krupková, O.: Mechanical systems with nonholonomic constraints. J. Math. Phys. 38, 5098 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  37. 37.
    Gervais, Jean-Loup, Saveliev, Mikhail V.: Higher grading generalizations of the Toda systems. Nucl. Phys. B 453(1–2), 449 (1995)CrossRefGoogle Scholar
  38. 38.
    Ferreira, Luiz A., Gervais, Jean-Loup, Guillen, Joaquin Sanchez, Saveliev, Mikhail V.: Affine Toda systems coupled to matter fields. Nucl. Phys. B 470(1–2), 236 (1995)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Kundu, A.: Nonlinearizing linear equations to integrable systems including new hierarchies with nonholonomic deformations. J. Math. Phys. 50, 102702 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  40. 40.
    Fordy, A.P., Holm, D.D.: A tri-Hamiltonian formulation of the self-induced transparency equations. Phys. Lett. A 160, 143 (1991)CrossRefMathSciNetGoogle Scholar
  41. 41.
    Kundu, A.: Non-holonomic deformation of the DNLS equation for controlling optical soliton in doped fibre media. IMA J. Appl. Math. 77, 382 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  42. 42.
    Nakazawa, M., Kimura, Y., Kurokawa, K., Suzuki, K.: Self-induced-transparency solitons in an erbium-doped fiber waveguide. Phys. Rev. A 45(1), 23 (1992)CrossRefGoogle Scholar
  43. 43.
    Nakazawa, M., Yamada, E., Kubota, H.: Coexistence of self-induced transparency soliton and nonlinear Schrödinger soliton. Phys. Rev. Lett. 66, 2625 (1991)CrossRefGoogle Scholar
  44. 44.
    Ferreira, L.A., Luchini, G., Zakrzewski, W.J.: The concept of quasi-integrability, nonlinear and modern mathematical physics. AIP Conf. Proc. 1562, 43 (2013)CrossRefGoogle Scholar
  45. 45.
    Abhinav, K., Guha, P.: Quasi-integrability in supersymmetric sine-Gordon models. EPL 116, 10004 (2016)CrossRefGoogle Scholar
  46. 46.
    Abhinav, K., Guha, P.: Quasi-Integrability of the KdV System. arXiv:1612.07499 [math-ph]
  47. 47.
    Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer, Berlin (1999)zbMATHGoogle Scholar
  48. 48.
    Gerdjikov, V.S., Uzunov, I.M., Evstatiev, E.G., Diankov, G.L.: Nonlinear Schrödinger equation and N-soliton interactions: generalized Karpman–Solov’ev approach and the complex Toda chain. Phys. Rev. E 55, 6039 (1997)CrossRefMathSciNetGoogle Scholar
  49. 49.
    Wu, J.: Integrability aspects and multi-soliton solutions of a new coupled Gerdjikov–Ivanov derivative nonlinear Schrödinger equation. Nonlinear Dyn. 96, 789 (2019)CrossRefGoogle Scholar
  50. 50.
    Yang, B., Chen, Y.: Dynamics of high-order solitons in the nonlocal nonlinear Schrödinger equations. Nonlinear Dyn. 94, 489 (2018)CrossRefzbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.The Institute for Fundamental Study (IF)Naresuan UniversityPhitsanulokThailand
  2. 2.S. N. Bose National Centre for Basic SciencesSalt Lake, KolkataIndia
  3. 3.Department of Natural ScienceM A K Azad University of TechnologyHaringhata, NadiaIndia

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