Skip to main content
SpringerLink
Log in

Solitary Wave Solutions for \((1+2)\)-Dimensional Nonlinear Schrödinger Equation with Dual Power Law Nonlinearity

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

Here, \(\tan \left( \frac{\phi (\xi )}{2}\right) \)-expansion method is being applied on \((1+2)\)-dimensional nonlinear Schrödinger equation (NLSE) with dual power law nonlinearity. Spatial solitons and optial nonlinearities in materials like photovoltaic–photorefractive, polymer and organic can be identified by seeking help from NLSE with dual power law nonlinearity. Abundant exact traveling wave solutions consisting free parameters are established in terms of exponential functions. Various arbitrary constants obtained in the solutions help us to discuss the graphical behavior of solutions and also grants flexibility to form a link with large variety of physical phenomena. Moreover, graphical representation of solutions are shown vigorously in order to visualize the behavior of the solutions acquired for the equation.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non linear media. Sov. J. Exp. Theor. Phys. 34, 62–69 (1972)

    Google Scholar 

  2. Hefter, E.F.: Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics. Phys. Rev. A 32, 1201–1204 (1985)

    Google Scholar 

  3. Akhmediev, N., Ankiewicz, A., Crespo, J.M.S.: Does the nonlinear Schrödinger equation correctly describe beam propagation? Opt. Lett. 18, 411–413 (1993)

    Google Scholar 

  4. Bergé, L.: Wave collapse in physics: principles and applications to light and plasma waves. Phys. Rep. 303, 259–370 (1998)

    MathSciNet  Google Scholar 

  5. Berloff, N.G.: Nonlinear dynamics of secondary protein folding. Phys. Lett. A 337, 391–396 (2005)

    MATH  Google Scholar 

  6. Filikhin, I., Deyneka, E., Melikyan, H., Vlahovic, B.: Electron states of semiconductor quantum ring with geometry and size variations. Mol. Simul. 31, 779–785 (2005)

    Google Scholar 

  7. Vitanov, N.K., Chabchoub, A., Hoffmann, N.: Deep-water waves: on the nonlinear Schrödinger equation and its solutions. J. Theor. App. Mech. 43, 43–54 (2013)

    MATH  Google Scholar 

  8. Christiansen, P.L., Sorensen, M.P., Scott, A.C.: Nonlinear science at the dawn of the 21st century. Springer, Berlin (2000)

    MATH  Google Scholar 

  9. Tian, B., Gao, Y.T.: Symbolic-computation study of the perturbed nonlinear Schrödinger model in inhomogeneous optical fibers. Phys. Lett. A 342, 228–236 (2005)

    MATH  Google Scholar 

  10. Anderson, D.: Variational approach to nonlinear pulse propagation in optical fibers. Phys. Rev. A 27, 3135–3145 (1983)

    Google Scholar 

  11. Liu, X.Q., Yan, Z.L.: Some exact solutions of the variable coefficient Schrödinger equation. Commun. Nonlinear Sci. 12, 1355–1359 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Liu, W.J., Tian, B., Zhang, H.Q., Xu, T., Li, H.: Solitary wave pulses in optical fibers with normal dispersion and higher-order effects. Phys. Rev. A 79, 063810-1–063810-8 (2009)

    Google Scholar 

  13. Biswas, A., Milovic, D.: Bright and dark solitons of the generalized nonlinear Schrödinger’s equation. Commun. Nonlinear Sci. 15, 1473–1484 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Feng, Y.J., Gao, Y.T., Sun, Z.Y., Zuo, D.W., Shen, Y.J., Sun, Y.H., Xue, L., Yu, X.: Anti-dark solitons for a variable-coefficient higher-order nonlinear Schrödinger equation in an inhomogeneous optical fiber. Phys. Scr. 90, 045201-1–045201-9 (2015)

    Google Scholar 

  15. Bulut, H., Sulaiman, T.A., Baskonus, H.M., Aktürk, T.: On the bright and singular optical solitons to the \((2+1)\)-dimensional NLS and the Hirota equations. Opt. Quantum Electron. 50, 134-1–134-12 (2018)

    Google Scholar 

  16. Biswas, A., Konar, S.: Introduction to Non-Kerr Law Optical Solitons. CRC Press, New York (2006)

    MATH  Google Scholar 

  17. Wazwaz, A.M.: Exact solutions for the fourth order nonlinear Schrödinger equations with cubic and power law nonlinearities. Math. Comput. Model. 43, 802–808 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Biswas, A.: 1-Soliton solution of \((1+2)\)-dimensional nonlinear Schrödinger’s equation in dual power law media. Phys. Lett. A 372, 5941–5943 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Zhang, Z.Y., Liu, Z.H., Miao, X.J., Chen, Y.Z.: New exact solutions to the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity. Appl. Math. Comput. 216, 3064–3072 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Zhou, Q., Yao, D., Chen, F.: Analytical study of optical solitons in media with Kerr and parabolic-law nonlinearities. J. Mod. Opt. 60, 1652–1657 (2013)

    Google Scholar 

  21. Biswas, A., Ekici, M., Sonmezoglu, A., Arshed, S., Belic, M.: Optical soliton perturbation with full nonlinearity by extended trial function method. Opt. Quantum Electron. 50, 449-1–449-58 (2018)

    Google Scholar 

  22. Biswas, A., Milovic, D.: Optical solitons in \(1+2\) dimensions with non-Kerr law nonlinearity. Eur. Phys. J. Spec. Top. 173, 81–86 (2009)

    Google Scholar 

  23. Bulut, H., Pandir, Y., Demiray, S.T.: Exact solutions of nonlinear Schrödinger’s equation with dual power-law nonlinearity by extended trial equation method. Wave. Random Complex 24, 439–451 (2014)

    MATH  Google Scholar 

  24. Salathiel, Y., Betchewe, G., Doka, S.Y., Crepin, K.T.: Exact traveling wave solutions to the fourth-order dispersive nonlinear Schrödinger equation with dual-power law nonlinearity. Math. Methods Appl. Sci. 39, 1135–1143 (2016)

    MathSciNet  MATH  Google Scholar 

  25. Ali, A., Seadawy, A.R., Lu, D.: Soliton solutions of the nonlinear Schrödinger equation with the dual power law nonlinearity and resonant nonlinear Schrödinger equation and their modulation instability analysis. Optik 145, 79–88 (2017)

    Google Scholar 

  26. He, J.H.: Variational iteration method-some recent results and new interpretations. J. Comput. Appl. Math. 207, 3–17 (2007)

    MathSciNet  MATH  Google Scholar 

  27. Wazwaz, A.M.: The Hirota’s direct method for multiple-soliton solutions for three model equations of shallow water waves. Appl. Math. Comput. 201, 489–503 (2008)

    MathSciNet  MATH  Google Scholar 

  28. Wang, M., Li, X., Zhang, J.: The \(\left(\frac{G^{\prime }}{G}\right)\)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372, 417–423 (2008)

    MathSciNet  Google Scholar 

  29. Rady, A.S.A., Osman, E.S., Khalfallah, M.: The homogeneous balance method and its application to the Benjamin–Bona–Mahoney (BBM) equation. Appl. Math. Comput. 217, 1385–1390 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Hosseini, K., Ansari, R., Gholamin, P.: Exact solutions of some nonlinear systems of partial differential equations by using the first integral method. J. Math. Anal. Appl. 387, 807–814 (2012)

    MathSciNet  MATH  Google Scholar 

  31. Ma, W.X., Zhu, Z.: Solving the \((3+1)\)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl. Math. Comput. 218, 11871–11879 (2012)

    MathSciNet  MATH  Google Scholar 

  32. Gurefe, Y., Misirli, E., Sonmezoglu, A., Ekici, M.: Extended trial equation method to generalized nonlinear partial differential equations. Appl. Math. Comput. 219, 5253–5260 (2013)

    MathSciNet  MATH  Google Scholar 

  33. Hafez, M.G., Alam, M.N., Akbar, M.A.: Traveling wave solutions for some important coupled nonlinear physical models via the coupled Higgs equation and the Maccari system. J. King Saud Univ. Sci. 27, 105–112 (2015)

    Google Scholar 

  34. Zahran, E.H.M., Khater, M.M.A.: Modified extended tanh-function method and its applications to the Bogoyavlenskii equation. Appl. Math. Model. 40, 1769–1775 (2016)

    MathSciNet  Google Scholar 

  35. Raslan, K.R., Danaf, T.S.E., Ali, K.K.: New exact solution of coupled general equal width wave equation using sine–cosine function method. J. Egypt. Math. Soc. 25, 350–354 (2017)

    MathSciNet  MATH  Google Scholar 

  36. Arshad, M., Seadawy, A.R., Lu, D.: Optical soliton solutions of the generalized higher-order nonlinear Schrödinger equations and their applications. Opt. Quantum Electron. 49, 421-1–421-16 (2017)

    Google Scholar 

  37. Arora, R., Chauhan, A.: Lie symmetry analysis and some exact solutions of \((2+1)\)-dimensional KdV–Burgers equation. Int. J. Appl. Comput. Math. 5, 15-1–15-13 (2019)

    MathSciNet  MATH  Google Scholar 

  38. Aghdaei, M.F., Manafian, J., Zadahmad, M.: Analytic study of sixth-order thin film equation by \(\tan \left(\frac{\phi (\xi )}{2}\right)\)-expansion method. Opt. Quantum Electron. 48, 410-1–410-14 (2016)

    Google Scholar 

  39. Manafian, J., Lakestani, M., Bekir, A.: Study of the analytical treatment of the \((2+1)\)-dimensional Zoomeron, the Duffing and the SRLW equations via a new analytical approach. Int. J. Appl. Comput. Math. 2, 243–268 (2016)

    MathSciNet  MATH  Google Scholar 

  40. Manafian, J., Lakestani, M.: Abundant soliton solutions for the Kundu–Eckhaus equation via \(\tan \left(\frac{\phi (\xi )}{2}\right)\)-expansion method. Optik 127, 5543–5551 (2016)

    Google Scholar 

  41. Ahmed, N., Irshad, A., Din, S.T.M., Khan, U.: Exact solutions of perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity by improved \(\tan \left(\frac{\phi (\xi )}{2}\right)\)-expansion method. Opt. Quantum Electron. 50, 45-1–45-27 (2018)

    Google Scholar 

  42. Galati, L., Zheng, S.: Nonlinear Schrödinger equations for Bose–Einstein condensates. AIP Conf. Proc. 1562, 50–64 (2013)

    Google Scholar 

Download references

Acknowledgements

We are grateful to the reviewers for their encouraging comments that were helpful in improving this paper further.

Author information

Authors and Affiliations

  1. Department of Mathematics, Jaypee Institute of Information Technology, Noida, U.P., India

    Pallavi Verma & Lakhveer Kaur

Authors
  1. Pallavi Verma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lakhveer Kaur
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pallavi Verma.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Verma, P., Kaur, L. Solitary Wave Solutions for \((1+2)\)-Dimensional Nonlinear Schrödinger Equation with Dual Power Law Nonlinearity. Int. J. Appl. Comput. Math 5, 128 (2019). https://doi.org/10.1007/s40819-019-0711-2

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s40819-019-0711-2

Keywords

Search

Navigation