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Investigation of Exact Solutions of Perturbed Nonlinear Schrödinger's Equation by $\exp\left(-\Phi\left(\xi\right)\right)$-Expansion Method

Year 2017, Volume: 21 Issue: 2, 373 - 379, 03.07.2017

Mehmet Ekici

Abstract

This paper presents an analytic study on optical solitons of a perturbed nonlinear Schr\"{o}dinger's equation (NLSE). An integration tool that is the $\exp\left(-\Phi\left(\xi\right\right)$-expansion approach is used to find exact solutions. As a consequence, hyperbolic, trigonometric and rational function solutions are extracted by this approach.

Keywords

Solitons Perturbed nonlinear Schrödinger's equation; The $\exp\left[-\Phi\left(\xi\right)\right]$-expansion approach

References

  • [1] Zhou, Q., Liu, L., Liu, Y., Yu, H., Yao, P., Wei, C., Zhang, H., 2015. Exact optical solitons in metamaterials with cubic-quintic nonlinearity and third-order dispersion. Nonlinear Dynamics 80(3), 1365-1371.
  • [2] Zhou, Q., Mirzazadeh, M., Ekici, M., Sonmezoglu, A., 2016. Analytical study of solitons in non-Kerr nonlinear negative-index materials. Nonlinear Dynamics 86, 623-638.
  • [3] Biswas, A., Khan, K.R., Mahmood, M.F., 2014. Bright and dark solitons in optical metamaterials. Optik 125(3), 3299-3302.
  • [4] Xu, Y., Savescu, M., Khan, K.R., Mahmood, M.F., Biswas, A., Belic, M., 2016. Soliton propagation through nanoscale waveguides in optical metamaterials. Optics and Laser Technology 77, 177-186.
  • [5] Saha, M., Sarma, A.K., 2013. Modulation instability in nonlinear metamaterials induced by cubic-quintic nonlinearities and higher order dispersive effects. Optics Communications 291, 321-325.
  • [6] Yang, R., Zhang, Y., 2011. Exact combined solitary wave solutions in nonlinear metamaterials. Journal of the Optical Society of America B 28(1), 123-127.
  • [7] Yomba, E., 2005. Construction of new solutions to the fully nonlinear generalized Camassa-Holm equations by an indirect F-function method. Journal of Mathematical Physics 46, 123504-123512.
  • [8] He, J.H., Wu, X.H., 2006. Exp-function method for nonlinear wave equations. Chaos, Solitons and Fractals 30, 700-708.
  • [9] Hirota, R., 1973. Exact N-soliton of the wave equation of long waves in shallow water and in nonlinear lattices. Journal of Mathematical Physics 14, 810-814.
  • [10] He, J.H., 2005. Homotopy perturbation method for bifurcation of nonlinear problems. International Journal of Nonlinear Sciences and Numerical Simulation 6, 207-208.
  • [11] Abdou, M.A., Soliman, A.A., 2005. New applications of variational iteration method. Physica D 211, 1-8.
  • [12] He, J.H., 2004. Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos, Solitons and Fractals 19, 847-851.
  • [13] Abassy, T.A., El-Tawil, M.A., Saleh, H.K., 2004. The solution of KdV and mKdV equations using Adomian Pade approximation. International Journal of Nonlinear Sciences and Numerical Simulation 5, 327-340.
  • [14] Antonova, M., Biswas, A., 2009. Adiabatic parameter dynamics of perturbed solitary waves. Communications in Nonlinear Science and Numerical Simulation 14, 734-748.
  • [15] Wang, M.L., 1995. Solitary wave solutions for variant Boussinesq equations. Physics Letters A 199, 169-172.
  • [16] Ablowitz, M.J., Clarkson, P.A., 1991. Solitons: Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge.
  • [17] Liu, S.K., Fu, Z.T., Liu, S.D., Zhao, Q., 2001. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Physics Letters A 289, 69-74.
  • [18] Tascan, F., Bekir, A., 2009. Analytic solutions of the (2+1)-dimensional nonlinear evolution equations using the sine-cosine method. Applied Mathematics and Computation 215, 3134-3139.
  • [19] Ozis, T., Aslan, I., 2009. Symbolic computation and exact and explicit solutions of some nonlinear evolution equation in mathematical physics. Communications in Theoretical Physics 51, 577-580.
  • [20] Manafian, J., Lakestani, M., Bekir, A., 2016. Study of the analytical treatment of the (2+1)-dimensional Zoomeron, the Duffing and the SRLW equations via a new analytical approach. International Journal of Applied and Computational Mathematics 2(2), 243-268.
  • [21] Khan, K., Akbar, M.A., 2013. Application of exp(F(x))-expansion method to find the exact solutions of modified Benjamin-Bona-Mahony equation. World Applied Sciences Journal 24(10), 1373-1377.
  • [22] Roshid, H.O., Kabir, M.R., Bhowmik, R.C., Datta, B.K., 2014. Investigation of solitary wave solutions for Vakhnenko-Parkes equation via exp-function and exp(F(x ))-expansion method. SpringerPlus 3:692.
  • [23] Kaplan, M., Bekir, A., 2016. A novel analytical method for time-fractional differential equations. Optik 127, 8209-8214.

Year 2017, Volume: 21 Issue: 2, 373 - 379, 03.07.2017

Mehmet Ekici

Abstract

References

  • [1] Zhou, Q., Liu, L., Liu, Y., Yu, H., Yao, P., Wei, C., Zhang, H., 2015. Exact optical solitons in metamaterials with cubic-quintic nonlinearity and third-order dispersion. Nonlinear Dynamics 80(3), 1365-1371.
  • [2] Zhou, Q., Mirzazadeh, M., Ekici, M., Sonmezoglu, A., 2016. Analytical study of solitons in non-Kerr nonlinear negative-index materials. Nonlinear Dynamics 86, 623-638.
  • [3] Biswas, A., Khan, K.R., Mahmood, M.F., 2014. Bright and dark solitons in optical metamaterials. Optik 125(3), 3299-3302.
  • [4] Xu, Y., Savescu, M., Khan, K.R., Mahmood, M.F., Biswas, A., Belic, M., 2016. Soliton propagation through nanoscale waveguides in optical metamaterials. Optics and Laser Technology 77, 177-186.
  • [5] Saha, M., Sarma, A.K., 2013. Modulation instability in nonlinear metamaterials induced by cubic-quintic nonlinearities and higher order dispersive effects. Optics Communications 291, 321-325.
  • [6] Yang, R., Zhang, Y., 2011. Exact combined solitary wave solutions in nonlinear metamaterials. Journal of the Optical Society of America B 28(1), 123-127.
  • [7] Yomba, E., 2005. Construction of new solutions to the fully nonlinear generalized Camassa-Holm equations by an indirect F-function method. Journal of Mathematical Physics 46, 123504-123512.
  • [8] He, J.H., Wu, X.H., 2006. Exp-function method for nonlinear wave equations. Chaos, Solitons and Fractals 30, 700-708.
  • [9] Hirota, R., 1973. Exact N-soliton of the wave equation of long waves in shallow water and in nonlinear lattices. Journal of Mathematical Physics 14, 810-814.
  • [10] He, J.H., 2005. Homotopy perturbation method for bifurcation of nonlinear problems. International Journal of Nonlinear Sciences and Numerical Simulation 6, 207-208.
  • [11] Abdou, M.A., Soliman, A.A., 2005. New applications of variational iteration method. Physica D 211, 1-8.
  • [12] He, J.H., 2004. Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos, Solitons and Fractals 19, 847-851.
  • [13] Abassy, T.A., El-Tawil, M.A., Saleh, H.K., 2004. The solution of KdV and mKdV equations using Adomian Pade approximation. International Journal of Nonlinear Sciences and Numerical Simulation 5, 327-340.
  • [14] Antonova, M., Biswas, A., 2009. Adiabatic parameter dynamics of perturbed solitary waves. Communications in Nonlinear Science and Numerical Simulation 14, 734-748.
  • [15] Wang, M.L., 1995. Solitary wave solutions for variant Boussinesq equations. Physics Letters A 199, 169-172.
  • [16] Ablowitz, M.J., Clarkson, P.A., 1991. Solitons: Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge.
  • [17] Liu, S.K., Fu, Z.T., Liu, S.D., Zhao, Q., 2001. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Physics Letters A 289, 69-74.
  • [18] Tascan, F., Bekir, A., 2009. Analytic solutions of the (2+1)-dimensional nonlinear evolution equations using the sine-cosine method. Applied Mathematics and Computation 215, 3134-3139.
  • [19] Ozis, T., Aslan, I., 2009. Symbolic computation and exact and explicit solutions of some nonlinear evolution equation in mathematical physics. Communications in Theoretical Physics 51, 577-580.
  • [20] Manafian, J., Lakestani, M., Bekir, A., 2016. Study of the analytical treatment of the (2+1)-dimensional Zoomeron, the Duffing and the SRLW equations via a new analytical approach. International Journal of Applied and Computational Mathematics 2(2), 243-268.
  • [21] Khan, K., Akbar, M.A., 2013. Application of exp(F(x))-expansion method to find the exact solutions of modified Benjamin-Bona-Mahony equation. World Applied Sciences Journal 24(10), 1373-1377.
  • [22] Roshid, H.O., Kabir, M.R., Bhowmik, R.C., Datta, B.K., 2014. Investigation of solitary wave solutions for Vakhnenko-Parkes equation via exp-function and exp(F(x ))-expansion method. SpringerPlus 3:692.
  • [23] Kaplan, M., Bekir, A., 2016. A novel analytical method for time-fractional differential equations. Optik 127, 8209-8214.
There are 23 citations in total.

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Journal Section Articles
Authors

Mehmet Ekici This is me

Publication Date July 3, 2017
Published in Issue Year 2017 Volume: 21 Issue: 2

Cite

APA Ekici, M. (2017). Investigation of Exact Solutions of Perturbed Nonlinear Schrödinger’s Equation by $\exp\left(-\Phi\left(\xi\right)\right)$-Expansion Method. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 21(2), 373-379. https://doi.org/10.19113/sdufbed.70927
AMA Ekici M. Investigation of Exact Solutions of Perturbed Nonlinear Schrödinger’s Equation by $\exp\left(-\Phi\left(\xi\right)\right)$-Expansion Method. J. Nat. Appl. Sci. August 2017;21(2):373-379. doi:10.19113/sdufbed.70927
Chicago Ekici, Mehmet. “Investigation of Exact Solutions of Perturbed Nonlinear Schrödinger’s Equation by $\exp\left(-\Phi\left(\xi\right)\right)$-Expansion Method”. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 21, no. 2 (August 2017): 373-79. https://doi.org/10.19113/sdufbed.70927.
EndNote Ekici M (August 1, 2017) Investigation of Exact Solutions of Perturbed Nonlinear Schrödinger’s Equation by $\exp\left(-\Phi\left(\xi\right)\right)$-Expansion Method. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 21 2 373–379.
IEEE M. Ekici, “Investigation of Exact Solutions of Perturbed Nonlinear Schrödinger’s Equation by $\exp\left(-\Phi\left(\xi\right)\right)$-Expansion Method”, J. Nat. Appl. Sci., vol. 21, no. 2, pp. 373–379, 2017, doi: 10.19113/sdufbed.70927.
ISNAD Ekici, Mehmet. “Investigation of Exact Solutions of Perturbed Nonlinear Schrödinger’s Equation by $\exp\left(-\Phi\left(\xi\right)\right)$-Expansion Method”. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 21/2 (August 2017), 373-379. https://doi.org/10.19113/sdufbed.70927.
JAMA Ekici M. Investigation of Exact Solutions of Perturbed Nonlinear Schrödinger’s Equation by $\exp\left(-\Phi\left(\xi\right)\right)$-Expansion Method. J. Nat. Appl. Sci. 2017;21:373–379.
MLA Ekici, Mehmet. “Investigation of Exact Solutions of Perturbed Nonlinear Schrödinger’s Equation by $\exp\left(-\Phi\left(\xi\right)\right)$-Expansion Method”. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 21, no. 2, 2017, pp. 373-9, doi:10.19113/sdufbed.70927.
Vancouver Ekici M. Investigation of Exact Solutions of Perturbed Nonlinear Schrödinger’s Equation by $\exp\left(-\Phi\left(\xi\right)\right)$-Expansion Method. J. Nat. Appl. Sci. 2017;21(2):373-9.
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