Skip to main content
SpringerLink
Log in

Madelung fluid description on a generalized mixed nonlinear Schrödinger equation

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Within the framework of the Madelung fluid description, in the present paper, we will derive bright and dark (including gray- and black-soliton) envelope solutions for a generalized mixed nonlinear Schrödinger model \({\mathrm {i}}\,\dfrac{\partial \varPsi }{\partial t}=\dfrac{\partial ^2 \varPsi }{\partial x^2}+{\mathrm {i}}\,a\,|\varPsi |^{2}\,\dfrac{\partial \varPsi }{\partial x}+{\mathrm {i}}\,b\,\varPsi ^{2}\,\dfrac{\partial \varPsi ^*}{\partial x}+c\,|\varPsi |^{4}\varPsi +d\,|\varPsi |^{2}\varPsi \), by virtue of the corresponding solitary wave solutions for the generalized stationary Gardner equations. Via corresponding parametric constraints, our results are achieved under suitable assumptions for the current velocity associated with different boundary conditions of the fluid density \(\rho \), while we have only considered the motion with stationary-profile current velocity case and excluded the motion with constant current velocity case. Note that our model is a generalized one with the inclusion of multiple coefficients (\(a\), \(b\), \(c\) and \(d\)).

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. Madelung, E.: Quantum theory in hydrodynamical form. Z. Phys. 40, 332 (1926)

    Google Scholar 

  2. Korn, A.: Schrödingers Wllenmechanik und meine eigenen mechanischen Theorien. Berührungspunkte und Divergenzen Zeitschrift für Physik 44, 745 (1927)

    Article  MATH  Google Scholar 

  3. Auletta, G.: Foundation and Interpretation of Quantum Mechanics. World Scientific, Singapore (2000)

    Book  Google Scholar 

  4. Fedele, R., Anderson, D., Lisak, M.: How the coherent instabilities of an intense high-energy charged-particle beam in the presence of nonlocal effects can be explained within the context of the Madelung fluid description. Eur. Phys. J. B 49, 275 (2006)

    Article  Google Scholar 

  5. Shukla, P.K., Fedele, R., Onorato, M., Tsintsadze, N.L.: Envelope solitons induced by high-order effects of light-plasma interaction. Eur. Phys. J. B 29, 613 (2002)

    Article  Google Scholar 

  6. Fedele, R., Schamel, H., Karpman, V.I., Shukla, P.K.: Envelope solitons of nonlinear Schrödinger equation with an anti-cubic nonlinearity. J. Phys. A 36, 1169 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Fedele, R., Schamel, H.: Solitary waves in the Madelung’s fluid: connection between the nonlinear Schrödinger equation and the Korteweg-de Vries equation. Eur. Phys. J. B 27, 313 (2002)

    Article  Google Scholar 

  8. Fedele, R.: Envelope solitons versus solitons. Phys. Scr. 65, 502 (2002)

    Article  MATH  Google Scholar 

  9. Fedele, R., Schamel, H., Shukla, P.K.: Solitons in the Madelung’s fluid. Phys. Scr. T98, 18 (2002)

    Article  Google Scholar 

  10. Grecu, D., Grecu, A.T., Visinescu, A., Fedele, R., Nicola, S.de: Solitary waves in a Madelung fluid description of derivative NLS equations. J. Non-Linear Math. Phys. 15, 209 (2008)

    Article  Google Scholar 

  11. Lü, X.: Solitary waves with the Madelung fluid description: a generalized derivative nonlinear Schrödinger equation. submitted (2015)

  12. Wyller, T., Flå, T., Rasmussen, J.J.: Classification of kink type solutions to the extended derivative nonlinear Schrödinger equation. Phys. Scr. 57, 427 (1998)

    Article  Google Scholar 

  13. Clarkson, P.A.: Dimensional reductions and exact solutions of a generalized nonlinear Schrödinger equation. Nonlinearity 5, 453 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. Clarkson, P.A., Cosgrove, C.M.: Painlevé analysis of the non-linear Schrödinger family of equations. J. Phys. A 20, 2003 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lü, X.: Soliton behavior for a generalized mixed nonlinear Schrödinger model with N-fold Darboux transformation. Chaos 23(033137), 1–8 (2013)

    Article  Google Scholar 

  16. Kundu, A.: Landau–Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger-type equations. J. Math. Phys. 25, 3433 (1984)

    Article  MathSciNet  Google Scholar 

  17. Lü, X., Peng, M.S.: Systematic construction of infinitely many conservation laws for certain nonlinear evolution equations in mathematical physics. Commun. Nonlinear Sci. Numer. Simulat. 18, 2304 (2013)

    Article  MATH  Google Scholar 

  18. Hasegawa, A., Tappert, F.D.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers I. Anomalous dispersion. Appl. Phys. Lett. 23, 142 (1973)

  19. Mollenauer, L.F., Stolen, R.H., Gordon, J.P.: Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys. Rev. Lett. 45, 1095 (1980)

    Article  Google Scholar 

  20. Agrawal, G.P.: Nonlinear Fiber Optics. Academic Press, New York (1995)

    Google Scholar 

  21. Hasegawa, A., Kodama, Y.: Solitons in Optical Communication. Oxford University Press, Oxford (1995)

    Google Scholar 

  22. Gerdjikov, V.S., Ivanov, M.I.: The quadratic bundle of general form and the nonlinear evolution equations. II. Hierarchies of Hamiltonian structures. Bulg. J. Phys. 10, 130 (1983)

    MathSciNet  Google Scholar 

  23. Fan, E.G.: Darboux transformation and soliton-like solutions for the Gerdjikov–Ivanov equation. J. Phys. A 33, 6925 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Fan, E.G.: Integrable evolution systems based on Gerdjikov–Ivanov equations, bi-Hamiltonian structure, finite-dimensional integrable systems and N-fold Darboux transformation. J. Math. Phys. 41, 7769 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kaup, D.J., Newell, A.C.: An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys. 19, 798 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  26. Chen, H.H., Lee, Y.C., Liu, C.S.: Integrability of nonlinear Hamiltonian systems by inverse scattering method. Phys. Scr. 20, 490 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  27. Fan, E.G.: A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations. J. Math. Phys. 42, 4327 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kakei, S., Sasa, N., Satsuma, J.: Bilinearization of a generalized derivative nonlinear Schrödinger equation. J. Phys. Soc. Jpn. 64, 1519 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  29. Kundu, A.: Exact solutions to higher-order nonlinear equations through gauge transformation. Phys. D 25, 399 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  30. Wadati, M., Konno, K., Ichikawa, Y.H.: A generalization of inverse scattering method. J. Phys. Soc. Jpn. 46, 1965 (1979)

    Article  MathSciNet  Google Scholar 

  31. Xia, T.C., Chen, X.H., Chen, D.Y.: Darboux transformation and soliton-like solutions of nonlinear Schrödinger equations. Chaos Solitons Fractals 26, 889 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  32. Geng, X.G., Tam, H.W.: Darboux transformation and soliton solutions for generalized nonlinear Schrödinger equations. J. Phys. Soc. Jpn. 68, 1508 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  33. Geng, X.G.: A hierarchy of non-linear evolution equations its Hamiltonian structure and classical integrable system. Phys. A 180, 241 (1992)

    Article  MathSciNet  Google Scholar 

  34. Dai, C.Q., Wang, X.G., Zhou, G.Q.: Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials. Phys. Rev. A 89, 013834 (2014)

    Article  Google Scholar 

  35. Dai, C.Q., Wang, Y.Y., Zhang, X.F.: Controllable Akhmediev breather and Kuznetsov-Ma soliton trains in PT-symmetric coupled waveguides. Opt. Express 22, 29862 (2014)

    Article  Google Scholar 

  36. Lü, X.: New bilinear Bäcklund transformation with multisoliton solutions for the (2+1)-dimensional Sawada-Kotera mode. Nonlinear Dyn. 76, 161 (2014)

    Article  Google Scholar 

  37. Lü, X., Li, J.: Integrability with symbolic computation on the Bogoyavlensky–Konoplechenko model: bell-polynomial manipulation, bilinear representation, and Wronskian solution. Nonlinear Dyn. 77, 2135 (2014)

    Google Scholar 

Download references

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant No. 61308018, China Postdoctoral Science Foundation under Grant No. 2014T70031, and the Fundamental Research Funds for the Central Universities of China (2014RC019 and 2015JBM111).

Author information

Authors and Affiliations

  1. Department of Mathematics, Beijing Jiao Tong University, Beijing, 100044, People’s Republic of China

    Xing Lü

  2. Department of Mathematics and Statistics, University of South Florida, Tampa, FL, 33620-5700, USA

    Xing Lü

Authors
  1. Xing Lü
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xing Lü.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lü, X. Madelung fluid description on a generalized mixed nonlinear Schrödinger equation. Nonlinear Dyn 81, 239–247 (2015). https://doi.org/10.1007/s11071-015-1985-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-015-1985-5

Keywords

Mathematics Subject Classification

Search

Navigation