Higher-Order Nonlinear Schrödinger Equation Family in Optical Fiber and Solitary Wave Solutions

Rodrique Njikue; Jean Roger Bogning; Timoleon Crépin Kofané

doi:doi:10.11648/j.ajop.20180603.12

| Peer-Reviewed

Higher-Order Nonlinear Schrödinger Equation Family in Optical Fiber and Solitary Wave Solutions

Rodrique Njikue, Jean Roger Bogning, Timoleon Crépin Kofané

Published in American Journal of Optics and Photonics (Volume 6, Issue 3)

Received: 19 August 2018 Accepted: 6 September 2018 Published: 14 December 2018

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In this paper, we modify with an appropriate analytical technique, the characteristics of the optical fiber through the modification of the coefficients of the highly nonlinear partial differential equation, which initially governs the dynamics of the propagation in such a wave guide. The procedure consists to assign arbitrary coefficients to the various terms of the established nonlinear partial differential equation, such as the one that embodies the propagation dynamics in a strongly nonlinear optical fiber and subsequently establishing the constraint equations linking these coefficients and thus the analys is makes it possible to enumerate the criteria for which obtaining the desired solutions is possible. These coefficients are like indicators which characterize the various modifications made in this medium of transmission. The nonlinear evolution equation that served as mathematical model for this study is the higher-order nonlinear Schrödinger equation which better describes the propagation of an ultrafast pulse in an optical fiber. The use of the Bogning-Djeumen Tchaho-Kofané method enabled not only to establish the constraint relations, but also the solitary wave solutions and plane wave solutions. We want through the results obtained in this article to give the specialists of the manufacture of transmission media such as optical fiber, to consider the modification of the properties of this wave guide during manufacture, depending on the type of signal that one wants to propagate in this case notably the solitary wave.

Published in	American Journal of Optics and Photonics (Volume 6, Issue 3)
DOI	10.11648/j.ajop.20180603.12
Page(s)	31-41
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Schrödinger Equation, Higher-Order Nonlinear Effects, Solitary Wave Solution, Periodic Travelling Wave Solutions, Bogning–Djeumen–Kofané Method

References

[1]	D. Anderson, (1983). Variational approach to nonlinear pulse propagation in optical fibers. Physical review A, 27, 3135.
[2]	P. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, A. Barthelemy, (1987). Picosecond steps and dark pulses through nonlinear single mode fibers. Optics Communications, 62, 374-379.
[3]	B. Tian, W. R. Shan, C. Y. Zhang, G. M. Wei, Y. T. Gao, (2005). Transformations for a generalized variable-coefficient nonlinear Schrödinger model from plasma physics, arterial mechanics and optical fibers with symbolic computation. The European Physical Journal B-Condensed Matter and Complex Systems, 47, 329-332.
[4]	C. A. Jones, P. H. Roberts, (1982). Motions in a Bose condensate. IV. Axisymmetric solitary waves. Journal of Physics A: Mathematical and General, 15, 2599.
[5]	P. Ruprecht, M. J. Holland, K. Burnett, M. Edwards, (1995). Time-dependent solution of the nonlinear Schrödinger equation for Bose-condensed trapped neutral atoms. Physical Review A, 51, 4704.
[6]	H. B. Thacker, (1981). Exact integrability in quantum field theory and statistical systems, Reviews of Modern Physics, 53,253.
[7]	M. Peyrard, (2004).Nonlinear dynamics and statistical physics of DNA. Nonlinearity, 17, 2 R1.
[8]	C. Nore, M. E. Brachet, S. Fauve, (1993). Numerical study of hydrodynamics using the nonlinear Schrödinger equation. Physica D: Nonlinear Phenomena, 65,154-162.
[9]	A. Hasegawa, F. Tappert, (1973). Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Applied Physics Letters, 23,142-144.
[10]	A. Hasegawa, F. Tappert, (1973).Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion. Applied Physics Letters, 23, 171-172.
[11]	Z. Xu, L. Li, Z. Li, Z, G. Zhou, (2002). Soliton interaction under the influence of higher-order effects. Optics Communications, 210, 375-384.
[12]	Z. Li, L. Li, H. Tian, G. Zhou, (2000). New types of solitary wave solutions for the higher order nonlinear Schrödinger equation. Physical review letters, 84, 4096.
[13]	K. Porsezian, K. Nakkeeran, (1996). Optical solitons in presence of Kerr dispersion and self-frequency shift. Physical review letters, 76, 3955.
[14]	C. Sien, C. W. Chang, S. Wen, (1994). Femtosecond soliton propagation in erbium-doped fiber amplifiers: the equivalence of two different models. Optics communications, 106,193-196.
[15]	S. L. Palacios, A. Guinea, J. M. Fernandez-Diaz, R. D. Crespo, (1999). Dark solitary waves in the nonlinear Schrödinger equation with third order dispersion, self-steepening, and self-frequency shift. Physical Review E, 60, R45
[16]	D. Mihalache, N. Truta, L-C. Crasovan, (1997). Painlevé analysis and bright solitary waves of the higher-order nonlinear Schrödinger equation containing third-order dispersion and self-steepening term. Physical Review E, 56, 1064.
[17]	J. R. Bogning, (2014). Solitary Wave Solutions of the High-order Nonlinear Schrödinger Equation in Dispersive Single Mode Optical Fibers. American Journal of Computational and Applied Mathematics, 4, 45-50.
[18]	N. Sasa, J. Satsuma, (1991). New-type of soliton solutions for a higher-order nonlinear Schrödinger equation. Journal of the Physical Society of Japan, 60, 409-417.
[19]	V. I. Karpman, (2004). The extended third-order nonlinear Schrödinger equation and Galilean transformation. The European Physical Journal B-Condensed Matter and Complex Systems, 39,341-350.
[20]	S. Ghosh, A. Kundu, S. Nandy, (1999). Soliton solutions, Liouville integrability and gauge equivalence of Sasa Satsuma equation. Journal of Mathematical Physics, 40, 1993-2000.
[21]	J. Xu, F. Engui, (2013). The unified transform method for the Sasa–Satsuma equation on the half-line, Proc. R. Soc. A, 469,20130068.
[22]	Wang, Mingliang, L. Xiangzheng, Jinliang Zhang, (2007).Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation. Physics Letters A, 363, 96-101.
[23]	Yan, Zhenya, (2003). Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres. Chaos, Solitons & Fractals, 16,759-766.
[24]	Hong, Baojian, L. Dianchen, (2009). New Jacobi elliptic functions solutions for the higher-order nonlinear Schrodinger equation, International journal of nonlinear science, 7, 360-367.
[25]	T. C. T. Djeumen, J. R. Bogning, T. C. Kofane, (2011).Multi-Soliton solutions of the modified Kuramoto-Sivashinsky’s equation by the BDK method. Far East Journal of dynamical systems, 15, 83-98.
[26]	T. C. T. Djeumen, J. R. Bogning, T. C. Kofané, (2012).Modulated Soliton Solution of the Modified Kuramoto-Sivashinsky's Equation. American Journal of Computational and Applied Mathematics, 2, 218-224.
[27]	J. R. Bogning, T. C. T. Djeumen, T. C. Kofané, (2012). Construction of the soliton solutions of the Ginzburg–Landau equations by the new Bogning–Djeumen Tchaho–Kofané method. Physica Scripta, 85, 025013.
[28]	J. R. Bogning, (2012). Generalization of the Bogning-Djeumen Tchaho-Kofané method for the construction of the solitary waves and the survey of the instabilities. Far East J. Dyn. Sys, 20,101-119.
[29]	J. R. Bogning, T. C. T. Djeumen, T. C. Kofané, (2013).Solitary wave solutions of the modified Sasa-Satsuma nonlinear partial differential equation. American Journal of Computational and Applied Mathematics, 3, 97-107.
[30]	J. R. Bogning, (2013). Pulse soliton solutions of the modified KdV and Born-Infeld equations. International Journal of Modern Nonlinear Theory and Application, 2, 135.
[31]	J. R. Bogning, (2015). N th Order Pulse Solitary Wave Solution and Modulational Instability in the Boussinesq Equation. American Journal of Computational and Applied Mathematics, 5,182-188.
[32]	J. R. Bogning, C. T. Djeumen, H. M. Omanda, (2016).Combined Solitary Wave Solutions in Higher-order Effects Optical Fibers. British Journal of Mathematics & Computer Science, 13, 1-12.
[33]	H. Kumar, C. Fakir, (2013).Dark and bright solitary wave solutions of the higher order nonlinear Schrödinger equation with self-steepening and self-frequency shift effects. Journal of Nonlinear Optical Physics & Materials, 22, 1350001.
[34]	R. Njikue, J. R. Bogning, T. C. Kofane, (2018).Exact bright and dark solitary wave solutions of the generalized higher-order nonlinear Schrödinger equation describing the propagation of ultra-short pulse in optical fiber. Journal of Physics Communications, 2, 025030.
[35]	A. R. Seadawy, L. Dianchen, (2017).Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Schrödinger equation and its stability. Results in physics, 7, 43-48.
[36]	K. Porsezian, K. Senthilnathan, S. Devipriya, (2005). Modulational instability in fiber Bragg grating with non-Kerr nonlinearity. IEEE journal of quantum electroics, 41,789-796.

Cite This Article

Plain Text BibTeX RIS

APA Style

Rodrique Njikue, Jean Roger Bogning, Timoleon Crépin Kofané. (2018). Higher-Order Nonlinear Schrödinger Equation Family in Optical Fiber and Solitary Wave Solutions. American Journal of Optics and Photonics, 6(3), 31-41. https://doi.org/10.11648/j.ajop.20180603.12

Copy | Download

ACS Style

Rodrique Njikue; Jean Roger Bogning; Timoleon Crépin Kofané. Higher-Order Nonlinear Schrödinger Equation Family in Optical Fiber and Solitary Wave Solutions. Am. J. Opt. Photonics 2018, 6(3), 31-41. doi: 10.11648/j.ajop.20180603.12

Copy | Download

AMA Style

Rodrique Njikue, Jean Roger Bogning, Timoleon Crépin Kofané. Higher-Order Nonlinear Schrödinger Equation Family in Optical Fiber and Solitary Wave Solutions. Am J Opt Photonics. 2018;6(3):31-41. doi: 10.11648/j.ajop.20180603.12

Copy | Download

@article{10.11648/j.ajop.20180603.12,
  author = {Rodrique Njikue and Jean Roger Bogning and Timoleon Crépin Kofané},
  title = {Higher-Order Nonlinear Schrödinger Equation Family in Optical Fiber and Solitary Wave Solutions},
  journal = {American Journal of Optics and Photonics},
  volume = {6},
  number = {3},
  pages = {31-41},
  doi = {10.11648/j.ajop.20180603.12},
  url = {https://doi.org/10.11648/j.ajop.20180603.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajop.20180603.12},
  abstract = {In this paper, we modify with an appropriate analytical technique, the characteristics of the optical fiber through the modification of the coefficients of the highly nonlinear partial differential equation, which initially governs the dynamics of the propagation in such a wave guide. The procedure consists to assign arbitrary coefficients to the various terms of the established nonlinear partial differential equation, such as the one that embodies the propagation dynamics in a strongly nonlinear optical fiber and subsequently establishing the constraint equations linking these coefficients and thus the analys is makes it possible to enumerate the criteria for which obtaining the desired solutions is possible. These coefficients are like indicators which characterize the various modifications made in this medium of transmission. The nonlinear evolution equation that served as mathematical model for this study is the higher-order nonlinear Schrödinger equation which better describes the propagation of an ultrafast pulse in an optical fiber. The use of the Bogning-Djeumen Tchaho-Kofané method enabled not only to establish the constraint relations, but also the solitary wave solutions and plane wave solutions. We want through the results obtained in this article to give the specialists of the manufacture of transmission media such as optical fiber, to consider the modification of the properties of this wave guide during manufacture, depending on the type of signal that one wants to propagate in this case notably the solitary wave.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Higher-Order Nonlinear Schrödinger Equation Family in Optical Fiber and Solitary Wave Solutions
AU  - Rodrique Njikue
AU  - Jean Roger Bogning
AU  - Timoleon Crépin Kofané
Y1  - 2018/12/14
PY  - 2018
N1  - https://doi.org/10.11648/j.ajop.20180603.12
DO  - 10.11648/j.ajop.20180603.12
T2  - American Journal of Optics and Photonics
JF  - American Journal of Optics and Photonics
JO  - American Journal of Optics and Photonics
SP  - 31
EP  - 41
PB  - Science Publishing Group
SN  - 2330-8494
UR  - https://doi.org/10.11648/j.ajop.20180603.12
AB  - In this paper, we modify with an appropriate analytical technique, the characteristics of the optical fiber through the modification of the coefficients of the highly nonlinear partial differential equation, which initially governs the dynamics of the propagation in such a wave guide. The procedure consists to assign arbitrary coefficients to the various terms of the established nonlinear partial differential equation, such as the one that embodies the propagation dynamics in a strongly nonlinear optical fiber and subsequently establishing the constraint equations linking these coefficients and thus the analys is makes it possible to enumerate the criteria for which obtaining the desired solutions is possible. These coefficients are like indicators which characterize the various modifications made in this medium of transmission. The nonlinear evolution equation that served as mathematical model for this study is the higher-order nonlinear Schrödinger equation which better describes the propagation of an ultrafast pulse in an optical fiber. The use of the Bogning-Djeumen Tchaho-Kofané method enabled not only to establish the constraint relations, but also the solitary wave solutions and plane wave solutions. We want through the results obtained in this article to give the specialists of the manufacture of transmission media such as optical fiber, to consider the modification of the properties of this wave guide during manufacture, depending on the type of signal that one wants to propagate in this case notably the solitary wave.
VL  - 6
IS  - 3
ER  -

Copy | Download

Author Information

Rodrique Njikue

Department of Physics, Faculty of Science, University of Yaounde I, Yaoundé, Cameroon
Jean Roger Bogning

Department of Physics, Higher Teacher Training College, University of Bamenda, Bamenda, Cameroon
Timoleon Crépin Kofané

Department of Physics, Faculty of Science, University of Yaounde I, Yaoundé, Cameroon

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Rodrique Njikue, Jean Roger Bogning, Timoleon Crépin Kofané. (2018). Higher-Order Nonlinear Schrödinger Equation Family in Optical Fiber and Solitary Wave Solutions. American Journal of Optics and Photonics, 6(3), 31-41. https://doi.org/10.11648/j.ajop.20180603.12

Copy | Download

ACS Style

Rodrique Njikue; Jean Roger Bogning; Timoleon Crépin Kofané. Higher-Order Nonlinear Schrödinger Equation Family in Optical Fiber and Solitary Wave Solutions. Am. J. Opt. Photonics 2018, 6(3), 31-41. doi: 10.11648/j.ajop.20180603.12

Copy | Download

AMA Style

Rodrique Njikue, Jean Roger Bogning, Timoleon Crépin Kofané. Higher-Order Nonlinear Schrödinger Equation Family in Optical Fiber and Solitary Wave Solutions. Am J Opt Photonics. 2018;6(3):31-41. doi: 10.11648/j.ajop.20180603.12

Copy | Download

@article{10.11648/j.ajop.20180603.12,
  author = {Rodrique Njikue and Jean Roger Bogning and Timoleon Crépin Kofané},
  title = {Higher-Order Nonlinear Schrödinger Equation Family in Optical Fiber and Solitary Wave Solutions},
  journal = {American Journal of Optics and Photonics},
  volume = {6},
  number = {3},
  pages = {31-41},
  doi = {10.11648/j.ajop.20180603.12},
  url = {https://doi.org/10.11648/j.ajop.20180603.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajop.20180603.12},
  abstract = {In this paper, we modify with an appropriate analytical technique, the characteristics of the optical fiber through the modification of the coefficients of the highly nonlinear partial differential equation, which initially governs the dynamics of the propagation in such a wave guide. The procedure consists to assign arbitrary coefficients to the various terms of the established nonlinear partial differential equation, such as the one that embodies the propagation dynamics in a strongly nonlinear optical fiber and subsequently establishing the constraint equations linking these coefficients and thus the analys is makes it possible to enumerate the criteria for which obtaining the desired solutions is possible. These coefficients are like indicators which characterize the various modifications made in this medium of transmission. The nonlinear evolution equation that served as mathematical model for this study is the higher-order nonlinear Schrödinger equation which better describes the propagation of an ultrafast pulse in an optical fiber. The use of the Bogning-Djeumen Tchaho-Kofané method enabled not only to establish the constraint relations, but also the solitary wave solutions and plane wave solutions. We want through the results obtained in this article to give the specialists of the manufacture of transmission media such as optical fiber, to consider the modification of the properties of this wave guide during manufacture, depending on the type of signal that one wants to propagate in this case notably the solitary wave.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Higher-Order Nonlinear Schrödinger Equation Family in Optical Fiber and Solitary Wave Solutions
AU  - Rodrique Njikue
AU  - Jean Roger Bogning
AU  - Timoleon Crépin Kofané
Y1  - 2018/12/14
PY  - 2018
N1  - https://doi.org/10.11648/j.ajop.20180603.12
DO  - 10.11648/j.ajop.20180603.12
T2  - American Journal of Optics and Photonics
JF  - American Journal of Optics and Photonics
JO  - American Journal of Optics and Photonics
SP  - 31
EP  - 41
PB  - Science Publishing Group
SN  - 2330-8494
UR  - https://doi.org/10.11648/j.ajop.20180603.12
AB  - In this paper, we modify with an appropriate analytical technique, the characteristics of the optical fiber through the modification of the coefficients of the highly nonlinear partial differential equation, which initially governs the dynamics of the propagation in such a wave guide. The procedure consists to assign arbitrary coefficients to the various terms of the established nonlinear partial differential equation, such as the one that embodies the propagation dynamics in a strongly nonlinear optical fiber and subsequently establishing the constraint equations linking these coefficients and thus the analys is makes it possible to enumerate the criteria for which obtaining the desired solutions is possible. These coefficients are like indicators which characterize the various modifications made in this medium of transmission. The nonlinear evolution equation that served as mathematical model for this study is the higher-order nonlinear Schrödinger equation which better describes the propagation of an ultrafast pulse in an optical fiber. The use of the Bogning-Djeumen Tchaho-Kofané method enabled not only to establish the constraint relations, but also the solitary wave solutions and plane wave solutions. We want through the results obtained in this article to give the specialists of the manufacture of transmission media such as optical fiber, to consider the modification of the properties of this wave guide during manufacture, depending on the type of signal that one wants to propagate in this case notably the solitary wave.
VL  - 6
IS  - 3
ER  -

Copy | Download