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 \(d\)).
Similar content being viewed by others
References
Madelung, E.: Quantum theory in hydrodynamical form. Z. Phys. 40, 332 (1926)
Korn, A.: Schrödingers Wllenmechanik und meine eigenen mechanischen Theorien. Berührungspunkte und Divergenzen Zeitschrift für Physik 44, 745 (1927)
Auletta, G.: Foundation and Interpretation of Quantum Mechanics. World Scientific, Singapore (2000)
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)
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)
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)
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)
Fedele, R.: Envelope solitons versus solitons. Phys. Scr. 65, 502 (2002)
Fedele, R., Schamel, H., Shukla, P.K.: Solitons in the Madelung’s fluid. Phys. Scr. T98, 18 (2002)
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)
Lü, X.: Solitary waves with the Madelung fluid description: a generalized derivative nonlinear Schrödinger equation. submitted (2015)
Wyller, T., Flå, T., Rasmussen, J.J.: Classification of kink type solutions to the extended derivative nonlinear Schrödinger equation. Phys. Scr. 57, 427 (1998)
Clarkson, P.A.: Dimensional reductions and exact solutions of a generalized nonlinear Schrödinger equation. Nonlinearity 5, 453 (1992)
Clarkson, P.A., Cosgrove, C.M.: Painlevé analysis of the non-linear Schrödinger family of equations. J. Phys. A 20, 2003 (1987)
Lü, X.: Soliton behavior for a generalized mixed nonlinear Schrödinger model with N-fold Darboux transformation. Chaos 23(033137), 1–8 (2013)
Kundu, A.: Landau–Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger-type equations. J. Math. Phys. 25, 3433 (1984)
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)
Hasegawa, A., Tappert, F.D.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers I. Anomalous dispersion. Appl. Phys. Lett. 23, 142 (1973)
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)
Agrawal, G.P.: Nonlinear Fiber Optics. Academic Press, New York (1995)
Hasegawa, A., Kodama, Y.: Solitons in Optical Communication. Oxford University Press, Oxford (1995)
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)
Fan, E.G.: Darboux transformation and soliton-like solutions for the Gerdjikov–Ivanov equation. J. Phys. A 33, 6925 (2000)
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)
Kaup, D.J., Newell, A.C.: An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys. 19, 798 (1978)
Chen, H.H., Lee, Y.C., Liu, C.S.: Integrability of nonlinear Hamiltonian systems by inverse scattering method. Phys. Scr. 20, 490 (1979)
Fan, E.G.: A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations. J. Math. Phys. 42, 4327 (2001)
Kakei, S., Sasa, N., Satsuma, J.: Bilinearization of a generalized derivative nonlinear Schrödinger equation. J. Phys. Soc. Jpn. 64, 1519 (1995)
Kundu, A.: Exact solutions to higher-order nonlinear equations through gauge transformation. Phys. D 25, 399 (1987)
Wadati, M., Konno, K., Ichikawa, Y.H.: A generalization of inverse scattering method. J. Phys. Soc. Jpn. 46, 1965 (1979)
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)
Geng, X.G., Tam, H.W.: Darboux transformation and soliton solutions for generalized nonlinear Schrödinger equations. J. Phys. Soc. Jpn. 68, 1508 (1999)
Geng, X.G.: A hierarchy of non-linear evolution equations its Hamiltonian structure and classical integrable system. Phys. A 180, 241 (1992)
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)
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)
Lü, X.: New bilinear Bäcklund transformation with multisoliton solutions for the (2+1)-dimensional Sawada-Kotera mode. Nonlinear Dyn. 76, 161 (2014)
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)
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
Corresponding author
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-015-1985-5
Keywords
- Generalized mixed nonlinear Schrödinger equation
- Madelung fluid description
- Solitary wave
- Envelope soliton