Skip to main content
SpringerLink
Log in

Investigations of bright, dark, kink-antikink optical and other soliton solutions and modulation instability analysis for the (1+1)-dimensional resonant nonlinear Schrödinger equation with dual-power law nonlinearity

  • Published:
Optical and Quantum Electronics Aims and scope Submit manuscript

Abstract

In this study, using the Kudryashov R function method and the generalized Kudryashov method, the \((1+1)\)-dimensional resonant nonlinear Schrödinger equation with dual-power law nonlinearity has been effectively examined for finding optical soliton solutions. The Kudryashov R function technique has numerous advantages that make symbolic computing considerably simpler, particularly while dealing with strongly dispersive nonlinear equations. The generalised Kudryashov method is noteworthy due to its capacity to address a wide range of complex nonlinear ordinary differential equations (NLODEs) observed in diverse engineering, scientific and mathematical fields. Newly generated nonlinear Schrödinger equation is demonstrated by the resonant nonlinear Schrödinger equation used to describe nonlinear optical phenomena. In order to achieve the goal, the governing model was first transformed into a NLODE, and then the solution sets and solution functions were derived based on the definitions of the suggested approaches. Solitons are localized wave forms that retain their shape and stability as they propagate across optical fiber. Optical solitons are characterized by their ability to maintain their shape and amplitude during propagation, even when encountering other solitons. Using the suggested approaches, the singular, dark, kink-antikink and bright soliton solutions from the governing equation have been extracted. Under the appropriate selection of parameter values, 3D, 2D and contour graphs are shown to illustrate the physical characteristics of the obtained results. All generated solutions are demonstrated to be analytically stable by the analysis of modulation instability, which also reveals the stability and movement of the waves. These solutions have extensive implications in the field of telecommunications and nonlinear fiber optics and assist in understanding the physical phenomena underlying the equation. These methods are new and standardised, and they can be applied to solve a variety of mathematical and physical problems.

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

Similar content being viewed by others

Availability of data and materials

No data were utilised in the study described in the article.

References

  • Abbagari, S., Houwe, A., Akinyemi, L., Inc, M., Doka, S.Y., Crepin, K.T.: Synchronized wave and modulation instability gain induce by the effects of higher-order dispersions in nonlinear optical fibers. Optical and Quantum Electronics 54(10), 642 (2022)

    Article  Google Scholar 

  • Abbagari, S., Houwe, A., Doka, S.Y., Inc, M., Bouetou, T.B.: Specific optical solitons solutions to the coupled Radhakrishnan-Kundu-Lakshmanan model and modulation instability gain spectra in birefringent fibers. Optical and Quantum Electronics 54, 1–25 (2022)

    Article  Google Scholar 

  • Ali, K.K., Zabihi, A., Rezazadeh, H., Ansari, R., Inc, M.: Optical soliton with Kudryashov’s equation via sine-Gordon expansion and Kudryashov methods. Optical and Quantum Electronics 53, 1–15 (2021)

    Article  Google Scholar 

  • Biswas, A., Rezazadeh, H., Mirzazadeh, M., Eslami, M., Ekici, M., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical soliton perturbation with Fokas-Lenells equation using three exotic and efficient integration schemes. Optik 165, 288–294 (2018)

    Article  ADS  Google Scholar 

  • Cakicioglu, H., Ozisik, M., Secer, A., Bayram, M.: Optical soliton solutions of Schrödinger-Hirota equation with parabolic law nonlinearity via generalized Kudryashov algorithm. Optical and Quantum Electronics 55(5), 407 (2023)

    Article  Google Scholar 

  • Che, W.-J., Liu, C., Akhmediev, N.: Fundamental and second-order dark soliton solutions of two-and three-component Manakov equations in the defocusing regime. Physical Review E 107(5), 054206 (2023)

    Article  ADS  MathSciNet  Google Scholar 

  • Chen, S.-J., Ma, W.-X., Lü, X.: Bäcklund transformation, exact solutions and interaction behaviour of the (3+ 1)-dimensional Hirota-Satsuma-Ito-like equation. Communications in Nonlinear Science and Numerical Simulation 83, 105135 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Das, N., Saha Ray, S.: Dispersive optical soliton wave solutions for the time-fractional perturbed nonlinear Schrödinger equation with truncated M-fractional conformable derivative in the nonlinear optical fibers. Optical and Quantum Electronics 54(9), 544 (2022)

    Article  Google Scholar 

  • Das, N., Saha Ray, S.: Exact traveling wave solutions and soliton solutions of conformable M-fractional modified nonlinear Schrödinger model. Optik 287, 171060 (2023)

    Article  ADS  Google Scholar 

  • Das, N., Saha Ray, S.: Highly dispersive optical solitons and solitary wave solutions for the (2+ 1)-dimensional Mel’nikov equation in modelling interaction of long waves with short wave packets in two dimensions, Journal of Nonlinear Optical Physics and Materials (2023)

  • Das, N., Saha Ray, S.: Bright and singular optical soliton solutions of modified nonlinear Schrödinger equation with conformable fractional derivative in deep water waves, 2023 International Conference on Fractional Differentiation and Its Applications (ICFDA), IEEE, pp. 1–5 (2023)

  • Dikwa, J., Houwe, A., Abbagari, S., Akinyemi, L., Inc, M.: Modulated waves patterns in the photovoltaic photorefractive crystal. Optical and Quantum Electronics 54(12), 842 (2022)

    Article  Google Scholar 

  • Durur, H.: Different types analytic solutions of the (1+ 1)-dimensional resonant nonlinear Schrödinger’s equation using \(\left(G^{\prime }/G\right)\)-expansion method. Modern Physics Letters B 34(03), 2050036 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  • Durur, H., Yokuş, A., Duran, S.: Investigation of exact soliton solutions of nematicons in liquid crystals according to nonlinearity conditions. International Journal of Modern Physics B, p. 2450054 (2023)

  • Ekici, M., Sonmezoglu, A., Biswas, A., Belic, M.R.: Optical solitons in (2+ 1)-Dimensions with Kundu-Mukherjee-Naskar equation by extended trial function scheme. Chinese Journal of Physics 57, 72–77 (2019)

    Article  ADS  Google Scholar 

  • El-Ganaini, S. Kumar, S.: Symbolic computation to construct new soliton solutions and dynamical behaviors of various wave structures for two different extended and generalized nonlinear Schrödinger equations using the new improved modified generalized sub-ODE proposed method, Mathematics and Computers in Simulation (2023)

  • Eldidamony, H.A., Ahmed, H.M., Zaghrout, A.S., Ali, Y.S., Arnous, A.H.: Optical solitons with Kudryashov’s quintuple power law nonlinearity having nonlinear chromatic dispersion using modified extended direct algebraic method. Optik 262, 169235 (2022)

    Article  ADS  Google Scholar 

  • Farahat, S., Shazly, E.E., El-Kalla, I., Kader, A.A.: Bright, dark and kink exact soliton solutions for perturbed Gerdjikov-Ivanov equation with full nonlinearity. Optik 277, 170688 (2023)

    Article  ADS  Google Scholar 

  • Ghanbari, B., Baleanu, D.: A novel technique to construct exact solutions for nonlinear partial differential equations. The European Physical Journal Plus 134(10), 506 (2019)

    Article  ADS  Google Scholar 

  • Houwe, A., Abbagari, S., Akinyemi, L., Rezazadeh, H., Doka, S.Y.: Peculiar optical solitons and modulated waves patterns in anti-cubic nonlinear media with cubic-quintic nonlinearity. Optical and Quantum Electronics 55(8), 719 (2023)

    Article  Google Scholar 

  • Inan, I.E., Inc, M., Rezazadeh, H., Akinyemi, L.: Optical solitons of (3+ 1) dimensional and coupled nonlinear Schrodinger equations. Optical and Quantum Electronics 54(4), 261 (2022)

    Article  Google Scholar 

  • Inc, M., Aliyu, A.I., Yusuf, A., Baleanu, D.: Gray optical soliton, linear stability analysis and conservation laws via multipliers to the cubic nonlinear Schrödinger equation. Optik 164, 472–478 (2018)

    Article  ADS  Google Scholar 

  • Iqbal, M.A., Baleanu, D., Miah, M.M., Ali, H.S., Alshehri, H.M., Osman, M.: New soliton solutions of the mZK equation and the Gerdjikov-Ivanov equation by employing the double \((G/G, 1/G)\)-expansion method. Results in Physics 47, 106391 (2023)

    Article  Google Scholar 

  • Jiang, D., Zha, Q.-L.: Solitons, breathers and periodic rogue waves for the variable-coefficient seventh-order nonlinear Schrödinger equation, http://iopscience.iop.org/article/10.1088/1402-4896/acdeb4Physica Scripta (2023)

  • Kaplan, M., Bekir, A., Akbulut, A.: A generalized Kudryashov method to some nonlinear evolution equations in mathematical physics. Nonlinear Dynamics 85, 2843–2850 (2016)

    Article  MathSciNet  Google Scholar 

  • Khan, A., Saifullah, S., Ahmad, S., Khan, J., Baleanu, D.: Multiple bifurcation solitons, lumps and rogue waves solutions of a generalized perturbed KdV equation. Nonlinear Dynamics 111(6), 5743–5756 (2023)

    Article  Google Scholar 

  • Kivshar, Y. S., Agrawal, G. P.: Optical solitons: from fibers to photonic crystals, Academic press (2003)

  • Kudryashov, N.A.: Method for finding highly dispersive optical solitons of nonlinear differential equations. Optik 206, 163550 (2020)

    Article  ADS  Google Scholar 

  • Liu, R., Zhang, H.-Q., Wei, Y.-C., Zhang, Y.: Multi-breather and high-order rogue waves for the quintic nonlinear Schrödinger equation on the elliptic function background. Communications in Nonlinear Science and Numerical Simulation 124, 107314 (2023)

    Article  MATH  Google Scholar 

  • Liu, J.-G., Zhou, L., He, Y.: Multiple soliton solutions for the new (2+ 1)-dimensional Korteweg-de Vries equation by multiple exp-function method. Applied Mathematics Letters 80, 71–78 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • Malik, S., Kumar, S.: Pure-cubic optical soliton perturbation with full nonlinearity by a new generalized approach. Optik 258, 168865 (2022)

    Article  ADS  Google Scholar 

  • Malik, S., Kumar, S., Biswas, A., Yıldırım, Y., Moraru, L., Moldovanu, S., Iticescu, C., Moshokoa, S.P., Bibicu, D., Alotaibi, A.: Gap solitons in fiber bragg gratings having polynomial law of nonlinear refractive index and cubic-quartic dispersive reflectivity by lie symmetry. Symmetry 15(5), 963 (2023)

    Article  ADS  Google Scholar 

  • Ozisik, M., Secer, A., Bayram, M., Cinar, M., Ozdemir, N., Esen, H., and Onder, I.: Investigation of optical soliton solutions of higher-order nonlinear Schrödinger equation having Kudryashov nonlinear refractive index, Optik, p. 170548 (2023)

  • Rehman, S., Bilal, M., Inc, M., Younas, U., Rezazadeh, H., Younis, M., Mirhosseini-Alizamini, S.: Investigation of pure-cubic optical solitons in nonlinear optics. Optical and Quantum Electronics 54(7), 400 (2022)

    Article  Google Scholar 

  • Rehman, H.U., Jafar, S., Javed, A., Hussain, S., Tahir, M.: New optical solitons of Biswas-Arshed equation using different techniques. Optik 206, 163670 (2020)

    Article  ADS  Google Scholar 

  • Rehman, H.U., Ullah, N., Imran, M.: Highly dispersive optical solitons using Kudryashov’s method. Optik 199, 163349 (2019)

    Article  ADS  Google Scholar 

  • Rehman, H. U., Inc, M., Asjad, M. I., Habib, A., Munir, Q.: New soliton solutions for the space-time fractional modified third order Korteweg–de Vries equation, Journal of Ocean Engineering and Science (2022)

  • Rezazadeh, H., Davodi, A.G., Gholami, D.: Combined formal periodic wave-like and soliton-like solutions of the conformable Schrödinger-KdV equation using the G/G-expansion technique. Results in Physics 47, 106352 (2023)

    Article  Google Scholar 

  • Rezazadeh, H., Korkmaz, A., Eslami, M., Vahidi, J., Asghari, R.: Traveling wave solution of conformable fractional generalized reaction Duffing model by generalized projective Riccati equation method. Optical and Quantum Electronics 50, 1–13 (2018)

    Article  Google Scholar 

  • Rizvi, S.T., Seadawy, A.R., Ahmed, S., Bashir, A.: Optical soliton solutions and various breathers lump interaction solutions with periodic wave for nonlinear Schrödinger equation with quadratic nonlinear susceptibility. Optical and Quantum Electronics 55(3), 286 (2023)

    Article  Google Scholar 

  • Roubíček, T.: Nonlinear partial differential equations with applications, Vol. 153, Springer Science & Business Media (2013)

  • 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. The European Physical Journal Plus 132, 1–19 (2017)

    Article  Google Scholar 

  • Tamilselvan, K., Kanna, T., Khare, A.: Nonparaxial elliptic waves and solitary waves in coupled nonlinear Helmholtz equations. Communications in Nonlinear Science and Numerical Simulation 39, 134–148 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Tozar, A., Tasbozan, O., Kurt, A.: Optical soliton solutions for the (1+ 1)-dimensional resonant nonlinear Schröndinger’s equation arising in optical fibers. Optical and Quantum Electronics 53(6), 316 (2021)

    Article  Google Scholar 

  • Ullah, N., Asjad, M.I., Ur Rehman, H., Akgül, A.: Construction of optical solitons of Radhakrishnan-Kundu-Lakshmanan equation in birefringent fibers. Nonlinear Engineering 11(1), 80–91 (2022)

    Article  ADS  Google Scholar 

  • Wang, Y.-Y., Dai, C.-Q., Xu, Y.-Q., Zheng, J., Fan, Y.: Dynamics of nonlocal and localized spatiotemporal solitons for a partially nonlocal nonlinear Schrödinger equation. Nonlinear Dynamics 92, 1261–1269 (2018)

    Article  Google Scholar 

  • Wazwaz, A.-M.: Multiple-soliton solutions for extended (3+ 1)-dimensional Jimbo-Miwa equations. Applied Mathematics Letters 64, 21–26 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Whitham, G.B.: Linear and nonlinear waves. John Wiley and Sons, Hoboken, New Jersey, U.S. (2011)

    MATH  Google Scholar 

  • Wu, Y., Yao, C., Wang, C., Yang, L., Wang, X., Ren, G., and Li, P.: High-Energy 1.96–2.4 \(\mu\)m Tunable Raman Soliton Generation in All-Silica-Fiber Tm-Doped Amplifier, IEEE Photonics Technology Letters, 35(7), pp. 361–364 (2023)

  • Xu, X., Yang, Y.: Breather and nondegenerate solitons in the two-component modified Korteweg-de Vries equation. Applied Mathematics Letters 144, 108695 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  • Yadav, R., Malik, S., Kumar, S., Sharma, R., Biswas, A., Yıldırım, Y., González-Gaxiola, O., Moraru, L., Alghamdi, A.A.: Highly dispersive W-shaped and other optical solitons with quadratic-cubic nonlinearity: Symmetry analysis and new Kudryashov’s method. Chaos, Solitons and Fractals 173, 113675 (2023)

    Article  MathSciNet  Google Scholar 

  • Yang, J.: Nonlinear waves in integrable and nonintegrable systems, SIAM (2010)

  • Younis, M., Iftikhar, M., Rehman, H.U.: Exact solutions to the nonlinear Schrödinger and Eckhaus equations by modified simple equation method. Journal of Advanced Physics 3(1), 77–79 (2014)

    Article  Google Scholar 

  • Yépez-Martínez, H., Rezazadeh, H., Inc, M., Akinlar, M.A., Gomez-Aguilar, J.F.: Analytical solutions to the fractional Lakshmanan-Porsezian-Daniel model. Optical and Quantum Electronics 54, 1–41 (2022)

    Article  Google Scholar 

  • Zayed, E., Alurrfi, K.: On solving two higher-order nonlinear PDEs describing the propagation of optical pulses in optic fibers using the \(\left(G^{\prime }/G,1/G\right)\)-expansion method. Ricerche di Matematica 64, 167–194 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Zayed, E., Alurrfi, K.: Extended auxiliary equation method and its applications for finding the exact solutions for a class of nonlinear Schrödinger-type equations. Applied Mathematics and Computation 289, 111–131 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, X., Jia, Z., Huang, Y., Wu, J., Wang, T., Chen, Y., Yu, Y., Yang, Y., Xiao, J., Qin, W., et al.: Flat-top soliton frequency comb generation through intra-cavity dispersion engineering in a brillouin laser cavity. Journal of Lightwave Technology 41(6), 1820–1833 (2023)

    Article  ADS  Google Scholar 

  • Zhang, R.-F., Li, M.-C., Cherraf, A., Vadyala, S.R.: The interference wave and the bright and dark soliton for two integro-differential equation by using BNNM. Nonlinear Dynamics 111(9), 8637–8646 (2023)

    Article  Google Scholar 

Download references

Funding

The “Council of Scientific and Industrial Research (CSIR)" fellowship scheme, which provided financial support under Grant No. 09/983(0043)/2019-EMR-I, is gratefully acknowledged by the first author.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Rourkela, Rourkela, Odisha, 769008, India

    Nilkanta Das & S. Saha Ray

Authors
  1. Nilkanta Das
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Saha Ray
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Nilkanta Das: Writing-Original draft, Conceptualization, Methodology, Investigation,. S. Saha Ray: Supervision, Writing-review and editing.

Corresponding author

Correspondence to S. Saha Ray.

Ethics declarations

Ethical Approval

Not applicable.

Conflict of interest

The authors declare they have no competing interests in this manuscript.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Das, N., Saha Ray, S. Investigations of bright, dark, kink-antikink optical and other soliton solutions and modulation instability analysis for the (1+1)-dimensional resonant nonlinear Schrödinger equation with dual-power law nonlinearity. Opt Quant Electron 55, 1071 (2023). https://doi.org/10.1007/s11082-023-05341-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-023-05341-3

Keywords

Mathematics Subject Classification

Search

Navigation