Abstract
In this paper, the perturbed nonlinear Schrödinger equation with dual-power law of nonlinearity is studied by adopting four mathematical methods namely the Modified Kudryashov method, the extended Tanh–Coth method, the modified simple equation method and soliton ansatz method. As a results, a set of various types of solitons that contains dark, singular, dark–singular, bright optical solitons and other form of optical soliton solutions are obtained. Firstly, we solve the perturbed nonlinear Schrödinger equation by considering the dual power law parameter using the first three integration methods and secondly we set this parameter equal to one in order, to solve the problem with the fourth method. The used methods in this paper present various applications in fields of nonlinear wave equations. Comparing our new results with well-known in literature are also given. Moreover, the graphical representations of the modulus of some optical soliton solutions and their 2-D profile are also depicted.
Similar content being viewed by others
References
Abbagari, S., et al.: N-rotating loop-soliton solution of the coupled integrable dispersionless equation. J. Appl. Math. Phys. 5, 1370–1379 (2014)
Bekir, A.: Application of the (G’/G)-expansion method for nonlinear evolution equations. Phys. Lett. A 372, 3400–3406 (2008a)
Bekir, A.: Applications of the extended tanh method for coupled nonlinear evolution equations. Commun. Nonlinear Sci. Numer. Simul. 13, 1748–1757 (2008b)
Bekir, A.: New solitons and periodic wave solutions for some nonlinear physical models by using the sine-cosine method. Phys. Scr. 77, 045008 (2008c)
Bekir, A., Boz, A.: Exact solutions for nonlinear evolution equations using exp-function method. Phys. Lett A 372, 1619–1625 (2008)
Biswas, A.: Temporal 1-soliton solution of the complex Ginzburg-Landau equation with power law nonlinearity. Progr. Electromagn. Res. 96, 1–7 (2009)
Biswas, A., Konar, S.: Introduction to Non-Kerr Law Optical Solitons. CRC Press, Boca Raton (2006)
Boudoue, M.H., et al.: Dispersive solitons in optical metamaterials having parabolic form of nonlinearity. Optik Int. J. Light Electron Opt. 179, 1009–1018 (2018)
Doka, S.Y., Mibaile, J., Gambo, B., Kofane, T.C.: Optical chirped soliton in metamaterials. Nonlinear Dyn. 90, 13–18 (2017)
Gambo, B., Bouetou, B., Kamgang, V.K., Kofane, T.C.: Dynamical survey of a generalized-Zakharov equation and its exact travelling wave solutions. Appl. Math. Comput. 217, 203–211 (2010)
Ghanbari, B., Osman, M.S., Baleanu, D.: Generalized exponential rational function method for extended Zakharov–Kuzetsov equation with conformable derivative. Mod. Phys. Lett. A 34, 1950155 (2019)
Houwe, A., et al.: Solitons solutions of nonlinear Schrödinger equation in the left-handed metamaterials by three different techniques. J. Phys. Commun. 3, 011002 (2019)
Javid, A., Raza, N., Osman, M.S.: Multi-solitons of thermophoretic motion equation depicting the wrinkle propagation in substrate-supported graphene sheets. Commun. Theor. Phys. 71, 362–366 (2019)
Jawad, A.J.M., Biswas, A., et al.: Hamiltonian perturbation of optical solitons with parabolic law nonlinearity using three integration methodologies. Optik 160, 248–254 (2018)
Kohl, R., Milovic, D., Zerrad, E., Biswas, A.: Soliton perturbation theory for dispersion-managed optical fibers. J. Nonlinear Opt. Phys. Mater. 18, 227–270 (2009)
Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 2248–2253 (2012)
Liu, J.G., Osman, M.S., et al.: Different complex wave structures described by the Hirota equation with variable coefficients in inhomogeneous optical fibers. Appl. Phys. B 125, 175 (2019)
Nestor, S., et al.: New solitary waves for the Klein–Gordon–Zakharov equations. Mod. Phys. Lett. B 34, 2050246 (2020a)
Nestor, S., Betchewe, G., Inc, M., Doka, S.Y.: Exact traveling wave solutions to the higher-order nonlinear Schrödinger equation having Kerr nonlinearity form using two strategic integrations. Eur. Phys. J. Plus 135, 380 (2020bb)
Njikue, R., Bogning, J.R., Kofane, T.C.: 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. J. Phys. Commun. 2, 025030 (2018)
Osman, M.S.: Nonlinear interaction of solitary waves described by multi-rational wave solutions of the (2 + 1)-dimensional Kadomtsev–Petviashvili equation with variable coefficients. Nonlinear Dyn. 87(2), 1209–1216 (2017)
Osman, M.S.: One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada–Kotera equation. Nonlinear Dyn. 96, 1491–1496 (2019a)
Osman, M.S.: New analytical study of water waves described by coupled fractional variant Boussinesq equation in fluid dynamics. Pramana 93, 26 (2019b)
Osman, M.S., Wazwaz, A.M.: A general bilinear form to generate different wave structures of solitons for a (3+1)-dimensional Boiti-Leon–Manna–Pempinelli equation. Math. Methods Appl. Sci. 42, 6277–6283 (2019)
Osman, M.S., Rezazadeh, H., Eslami, M.: Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov–Kuznetsov equation with power law nonlinearity. Nonlinear Eng. 8, 559–567 (2019a)
Osman, M.S., Lu, D., Khater, M.M.A.: A study of optical wave propagation in the nonautonomous Schrödinger-Hirota equation with power-law nonlinearity. Results Phys. 13, 102157 (2019b)
Rezazadeh, H.: New solitons solutions of the complex Ginzburg–Landau equation with Kerr law nonlinearity. Optik 167, 218–227 (2018)
Rezazadeh, H., et al.: New optical solitons of nonlinear conformable fractional Schrödinger-Hirota equation. Optik 172, 545–553 (2018a)
Rezazadeh, H., et al.: Mitigating internet bottleneck with fractional temporal evolution of optical solitons having quadratic-cubic nonlinearity. Optik 164, 84–92 (2018b)
Rezazadeh, H., Manafian, J., Khodadad, F.S., Nazari, F.: Traveling wave solutions for density-dependent conformable fractional diffusion-reaction equation by the first integral method and the improved tan(1/2())-expansion method. Opt. Quant. Electron. 50(3), 121 (2018c)
Rezazadeh, H., Osman, M.S., et al.: Hyperbolic rational solutions to a variety of conformable fractional Boussinesq-Like equations. Nonlinear Eng. 8, 224–230 (2019)
Savaissou, N., et al.: New Jacobi elliptic solutions and other solutions with quadratic-cubic nonlinearity using two mathematical methods. Asian Eur. J. Math. 13, 2050043 (2020)
Seadawy, A.R.: Two-dimensional interaction of a shear flow with a free surface in a stratified fluid and its solitary-wave solutions via mathematical methods. Eur. Phys. J. Plus 132, 518 (2017a)
Seadawy, A.R.: Travelling-wave solutions of a weakly nonlinear two-dimensional higher-order Kadomtsev–Petviashvili dynamical equation for dispersive shallow-water waves. Eur. Phys. J. Plus 132, 29 (2017b)
Seadawy, A.R., Arshad, M., Lu, D.: Stability analysis of new exact traveling-wave solutions of new coupled KdV and new coupled Zakharov–Kuznetsov systems. Eur. Phys. J. Plus 132, 162 (2017)
Wang, M., Li, X., Zhang, J.L.: Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation. Phys. Lett. A 363, 96–101 (2007)
Wazwaz, A.M.: The extended tanh method for abundant solitary wave solutions of nonlinear wave equations. Appl. Math. Comput. 187, 1131–1142 (2007)
Yepez-Martinez, H., Rezazadeh, H., et al.: The extended modified method applied to optical solitons solutions in birefringent fibers with weak nonlocal nonlinearity and four wave mixing. Chin. J. Phys. 58, 137–150 (2019)
Zayed, E.M.E., Abdelaziz, M.A.M.: The tanh function method using a generalized wave transformation for nonlinear equations. Int. J. Nonlinear Sci. Numer. Simul. 11, 595–601 (2010)
Zayed, E.M., Al-Nowehy, A.G.: Many new exact solutions to the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms using three different techniques. Optik 143, 84–103 (2017)
Zhang, J.L., Wang, M., Li, X.: The subsidiary ordinary differential equations and the exact solutions of the higher order dispersive nonlinear Schrödinger equation. Phys. Lett. A 357, 188–195 (2006)
Zhou, Q., Yao, D.Z., Chen, F., et al.: Optical solitons with nonlinear dispersion in polynomial law medium. J. Optoelectron. Adv. Mater. 17, 82–86 (2015)
Acknowledgements
The five authors of this work wish to thank the referees for their comments on this paper.
Funding
This research received no external funding.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Savaissou, N., Gambo, B., Rezazadeh, H. et al. Exact optical solitons to the perturbed nonlinear Schrödinger equation with dual-power law of nonlinearity. Opt Quant Electron 52, 318 (2020). https://doi.org/10.1007/s11082-020-02412-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-020-02412-7