Skip to main content
SpringerLink
Log in

Six-component semi-discrete integrable nonlinear Schrödinger system

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

We suggest the six-component integrable nonlinear system on a quasi-one-dimensional lattice. Due to its symmetrical form, the general system permits a number of reductions; one of which treated as the semi-discrete integrable nonlinear Schrödinger system on a lattice with three structural elements in the unit cell is considered in considerable details. Besides six truly independent basic field variables, the system is characterized by four concomitant fields whose background values produce three additional types of inter-site resonant interactions between the basic fields. As a result, the system dynamics becomes associated with the highly nonstandard form of Poisson structure. The elementary Poisson brackets between all field variables are calculated and presented explicitly. The richness of system dynamics is demonstrated on the multi-component soliton solution written in terms of properly parameterized soliton characteristics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ablowitz, M.J., Ladik, J.F.: Nonlinear differential–difference equations. J. Math. Phys. 16, 598–603 (1975)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Ablowitz, M.J., Ladik, J.F.: Nonlinear differential–difference equations and Fourier analysis. J. Math. Phys. 17, 1011–1018 (1976)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Ablowitz, M.J., Musslimani, Z.H.: Discrete spatial solitons in a diffraction-managed nonlinear waveguide array: a unified approach. Physica D 184, 276–303 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Bonilla, L.L., Grahn, H.T.: Non-linear dynamics of semiconductor superlattices. Rep. Prog. Phys. 68, 577–683 (2005)

    Article  ADS  Google Scholar 

  5. Brizhik, L.S., Piette, B.M.A.G., Zakrzewski, W.J.: Donor-acceptor electron transport mediated by solitons. Phys. Rev. E 90, 052915 (2014)

    Article  ADS  Google Scholar 

  6. Caudrey, P.J.: The inverse problem for a general \(N \times N\) spectral equation. Physica D 6, 51–66 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Caudrey, P.J.: Differential and discrete spectral problems and their inverses. In: Rogers, C., Moodie, T.B. (eds.) Wave Phenomena: Modern Theory and Applications. North-Holland Mathematics Studies, vol. 97, pp. 221–232. Elsevier, Amsterdam (1984)

    Chapter  Google Scholar 

  8. Christodoulides, D.N., Joseph, R.I.: Discrete self-focusing in nonlinear arrays of coupled waveguides. Opt. Lett. 13, 794–796 (1988)

    Article  ADS  Google Scholar 

  9. Davydov, A.S.: Teoriya Molekulyarnykh Eksitonov. Nauka, Moskva (1968)

    Google Scholar 

  10. Davydov, A.S.: Theory of Molecular Excitons. Plenum Press, New York (1971)

    Book  Google Scholar 

  11. Davydov, A.S., Eremko, A.A., Sergienko, A.I.: Solitony v \(\alpha \)-spiral’nykh belkovykh molekulakh (Solitons in \(\alpha \)-helix protein molecules). Ukr. J. Phys. 23, 983–993 (1978)

    Google Scholar 

  12. Davydov, A.S.: Solitony v Molekulyarnykh Sistemakh. Naukova Dumka, Kyïv (1984)

    Google Scholar 

  13. Davydov, A.S.: Solitons in Molecular Systems. Kluwer Academic, Dordrecht (1991)

    Book  MATH  Google Scholar 

  14. Dubrovin, B.A., Novikov, S.P., Fomenko, A.F.: Sovremennaya Geometriya. Metody i Prilozheniya. Nauka, Moskva (1986)

    MATH  Google Scholar 

  15. Dubrovin, B.A., Fomenko, A.F., Novikov, S.P.: Modern Geometry–Methods and Applications. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  16. Garanovich, I.L., Longhi, S., Sukhorukov, A.A., Kivshar, YuS: Light propagation and localization in modulated photonic lattices and waveguides. Phys. Rep. 518, 1–79 (2012)

    Article  ADS  Google Scholar 

  17. Heeger, A.J., Kivelson, S., Schrieffer, J.R., Su, W.-P.: Solitons in conducting polymers. Rev. Mod. Phys. 60, 781–850 (1988)

    Article  ADS  Google Scholar 

  18. Kako, F., Mugibayashi, N.: Complete integrability of general nonlinear differential–difference equations solvable by the inverse method. II. Prog. Theor. Phys. 61, 776–790 (1979)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Kulish, P.P.: Quantum difference nonlinear Schrödinger equation. Lett. Math. Phys. 5, 191–197 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Leo, K.: Interband optical investigation of Bloch oscillations in semiconductor superlattices. Semicond. Sci. Technol. 13, 249–263 (1998)

    Article  ADS  Google Scholar 

  21. Martini, R., Klose, G., Roskos, H.G., Kurz, H., Grahn, H.T., Hey, R.: Superradiant emission from Bloch oscillations in semiconductor superlattices. Phys. Rev. B 54, 14325–14328 (1996)

    Article  ADS  Google Scholar 

  22. Marquié, P., Bilbault, J.M., Remoissenet, M.: Nonlinear Schrödinger models and modulational instability in real electrical lattices. Physica D 87, 371–374 (1995)

    Article  ADS  Google Scholar 

  23. Marquié, P., Bilbault, J.M., Remoissenet, M.: Observation of nonlinear localized modes in an electrical lattice. Phys. Rev. E 51, 6127–6133 (1995)

    Article  ADS  Google Scholar 

  24. Maschke, B.M., Van Der Schaft, A.J., Breedveld, P.C.: An intrinsic Hamiltonian formulation of network dynamics: non-standard Poisson structures and gyrators. J. Franklin Inst. 329, 923–966 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Newell, A.C.: Solitons in Mathematics and Physics. SIAM Press, Philadelphia (1985)

    Book  MATH  Google Scholar 

  26. Salathiel, Ya., Amadou, Ya., Betchewe, G., Doka, S.Y., Crepin, K.T.: Soliton solutions and traveling wave solutions for a discrete electrical lattice with nonlinear dispersion through the generalized Riccati equation mapping method. Nonlinear Dyn. 87, 2435–2443 (2017)

    Article  MathSciNet  Google Scholar 

  27. Scharf, R., Bishop, A.R.: Properties of the nonlinear Schrödinger equation on a lattice. Phys. Rev. A 61, 6535–6544 (1991)

    Article  ADS  Google Scholar 

  28. Su, W.P., Schrieffer, J.R., Heeger, A.J.: Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698–1701 (1979)

    Article  ADS  Google Scholar 

  29. Su, W.P., Schrieffer, J.R., Heeger, A.J.: Soliton excitations in polyacetylene. Phys. Rev. B 22, 2099–2111 (1980)

    Article  ADS  Google Scholar 

  30. Takhtadzhyan, L.A., Faddeyev, L.D.: Gamil’tonov Podkhod v Teorii Solitonov. Nauka, Moskva (1986)

    MATH  Google Scholar 

  31. Faddeev, L.D., Takhtajan, L.A.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  32. Torres del Castillo, G.F., Velázquez Quesada, M.P.: Symplectic structures and dynamical symmetry groups. Rev. Mex. de Fís 50, 608–613 (2004)

    Google Scholar 

  33. Vakhnenko, O.O.: Solitons on a zigzag-runged ladder lattice. Phys. Rev. E 64, 067601 (2001)

    Article  ADS  Google Scholar 

  34. Vakhnenko, O.O.: A discrete nonlinear model of three coupled dynamical fields integrable by the Caudrey method. Ukr. J. Phys. 48, 653–666 (2003)

    MathSciNet  Google Scholar 

  35. Vakhnenko, O.O.: Inverse scattering transform for the nonlinear Schrödinger system on a zigzag-runged ladder lattice. J. Math. Phys. 51, 103518 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Vakhnenko, O.O.: Semidiscrete integrable nonlinear systems generated by the new fourth-order spectral operator. Local conservation laws. J. Nonlinear Math. Phys. 18, 401–414 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Vakhnenko, O.O.: Four-wave semidiscrete nonlinear integrable system with \(PT\)-symmetry. J. Nonlinear Math. Phys. 20, 606–622 (2013)

    Article  MathSciNet  Google Scholar 

  38. Vakhnenko, O.O.: Integrable nonlinear Schrödinger system on a triangular-lattice ribbon. J. Phys. Soc. Jpn. 84, 014003 (2015)

    Article  ADS  Google Scholar 

  39. Vakhnenko, O.O.: Nonlinear integrable model of Frenkel-like excitations on a ribbon of triangular lattice. J. Math. Phys. 56, 033505 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Vakhnenko, O.O.: Symmetry-broken canonizations of the semi-discrete integrable nonlinear Schrödinger system with background-controlled intersite coupling. J. Math. Phys. 57, 113504 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Vakhnenko, O.O.: Asymmetric canonicalization of the integrable nonlinear Schrödinger system on a triangular-lattice ribbon. Appl. Math. Lett. 64, 81–86 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  42. Vakhnenko, O.O.: Distinctive features of the integrable nonlinear Schrödinger system on a ribbon of triangular lattice. Ukr. J. Phys. 62, 271–282 (2017)

    Article  Google Scholar 

  43. Vakhnenko, O.O.: Semi-discrete integrable nonlinear Schrödinger system with background-controlled inter-site resonant coupling. J. Nonlinear Math. Phys. 24, 250–302 (2017)

    Article  MathSciNet  Google Scholar 

  44. Yulin, A.V., Konotop, V.V.: Conservative and PT-symmetric compactons in waveguide networks. Opt. Lett. 38, 4880–4883 (2013)

    Article  ADS  Google Scholar 

  45. Zakharov, V.E., Kuznetsov, E.A.: Gamil’tonovskiy formalizm dlya nelineynykh voln. Usp. Fiz. Nauk 167, 1137–1167 (1997)

    Article  Google Scholar 

  46. Zakharov, V.E., Kuznetsov, E.A.: Hamiltonian formalism for nonlinear waves. Phys. Uspekhi 40, 1087–1116 (1997)

    Article  ADS  Google Scholar 

  47. Zezyulin, D.A., Konotop, V.V., Abdullaev, F.K.: Discrete solitons in arrays of positive and negative index waveguides. Opt. Lett. 37, 3930–3932 (2012)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The work has been supported by the National Academy of Sciences of Ukraine within the Program No. 0115U005302.

Author information

Authors and Affiliations

  1. Department for Theory of Nonlinear Processes in Condensed Matter, Bogolyubov Institute for Theoretical Physics, 14-B Metrologichna Street, Kyïv, 03680, Ukraine

    Oleksiy O. Vakhnenko

Authors
  1. Oleksiy O. Vakhnenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oleksiy O. Vakhnenko.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vakhnenko, O.O. Six-component semi-discrete integrable nonlinear Schrödinger system. Lett Math Phys 108, 1807–1824 (2018). https://doi.org/10.1007/s11005-018-1049-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-018-1049-0

Keywords

Mathematics Subject Classification

Search

Navigation