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Propagation of W-shaped and M-shaped solitons with multi-peak interaction for ultrashort light pulse in fibers

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Abstract

We study the propagation of M-shaped solitons and kink-periodic waves in an optical fiber medium exhibiting Kerr dispersion and quintic nonlinearity by applying the symbolic computation with ansatz functions technique and logarithmic transformation. We used seven different methods, namely, three waves technique, homoclinic breather technique, M-shaped solitons, single exponential form, double exponential form, kink cross-rational and trigonometric and hyperbolic ansatz approaches. The quintic derivative nonlinear Schrödinger equation (QD-NLSE) is applied to model for ultrashort light pulse in fibers and self-steepening processes. We compute two categories of M-shaped solitons and their dynamics are revealed in figures by appropriate values of parameters. Moreover two types of interaction of M-shaped solitons with kink wave are revealed. We attain multiwaves, kink cross-rational (KCR), Homoclinic breather, periodic cross-rational solutions (PCRs) for governing model. Finally, the stability property of all solutions is analytically studied under the finite selection of parameters.

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References

  • Abbagari, S., Alphonse, H., Akinyemi, L., Saliou, Y., Bouetou, T.: Modulation instability gain and discrete soliton interaction in gyrotropic molecular chain. Chaos, Solitons & Fractals 160, 112255 (2022)

    MathSciNet  MATH  Google Scholar 

  • Ahmed, S., Ashraf, R., Seadawy, A.R., Rizvi, S., Younis, M., Althobaiti, A., El-Shehawi, A.M.: Lump, multi-wave, kinky breathers, interactional solutions and stability analysis for general (2+ 1)-rth dispersionless dym equation. Results in Phys. 25, 104160 (2021)

    Google Scholar 

  • Ahmed, I., Seadawy, A.R., Lu, D.: Kinky breathers, w-shaped and multi-peak solitons interaction in (2+ 1)-dimensional nonlinear schrödinger equation with kerr law of nonlinearity. Eur. Phys. J. Plus 134(3), 1–10 (2019)

    Google Scholar 

  • Ahmed, I., Seadawy, A.R., Lu, D.: M-shaped rational solitons and their interaction with kink waves in the fokas-lenells equation. Phys. Scr. 94(5), 055205 (2019)

    ADS  Google Scholar 

  • Ahmed, S., Seadawy, A.R., Rizvi, S.T.: Study of breathers, rogue waves and lump solutions for the nonlinear chains of atoms. Opt. Quant. Electron. 54, 320 (2022)

    Google Scholar 

  • Akram, U., Seadawy, A.R., Rizvi, S., Younis, M., Althobaiti, S., Sayed, S.: Traveling wave solutions for the fractional wazwaz-benjamin-bona-mahony model in arising shallow water waves. Results in Phys. 20, 103725 (2021)

    Google Scholar 

  • Ali, K., Seadawy, A.R., Ahmed, S., Rizvi, S.T.: Discussion on rational solutions for nematicons in liquid crystals with kerr law. Chaos, Solitons & Fractals 160, 112218 (2022)

    MathSciNet  MATH  Google Scholar 

  • Bilal, M., Seadawy, A.R., Younis, M., Rizvi, S., El-Rashidy, K., Mahmoud, S.F.: Analytical wave structures in plasma physics modelled by gilson-pickering equation by two integration norms. Results in Phys. 23, 103959 (2021)

    Google Scholar 

  • Bilal, M., Seadawy, A.R., Younis, M., Rizvi, S., Zahed, H.: Dispersive of propagation wave solutions to unidirectional shallow water wave dullin-gottwald-holm system and modulation instability analysis. Math. Methods in the Appl. Sci. 44, 6 (2021)

    MathSciNet  MATH  Google Scholar 

  • Chen, J., Luan, Z., Zhou, Q., Alzahrani, A.K., Biswas, A., Liu, W.: Periodic soliton interactions for higher-order nonlinear schrödinger equation in optical fibers. Nonlinear Dyn. 100(3), 2817–2821 (2020)

    Google Scholar 

  • Daoui, A.K., Messouber, A., Triki, H., Zhou, Q., Biswas, A., Liu, W., ... Belic, M.R.: Propagation of chirped periodic and localized waves with higher-order effects through optical fibers. Chaos, Solitons & Fractals 146, 110873 (2021)

  • Ghaffar, A., Ali, A., Ahmed, S., Akram, S., Junjua, M.-u-D., Baleanu, D., Nisar, K.S.: A novel analytical technique to obtain the solitary solutions for nonlinear evolution equation of fractional order. Adv. Difference Equ. 2020, 308 (2020)

    MathSciNet  MATH  Google Scholar 

  • Houwe, A., Djorwe, P., Abbagari, S., Doka, S., Nana, E.S.: Discrete solitons in nonlinear optomechanical array. Chaos, Solitons & Fractals 154, 111593 (2022)

    MathSciNet  Google Scholar 

  • Houwe, A., Rezazadeh, H., Bekir, A., Doka, S.: Traveling-wave solutions of the klein-gordon equations with m-fractional derivative. Pramana 96, 26 (2022)

    ADS  Google Scholar 

  • Houwe, A., Saliou, Y., Djorwe, P., Abbagari, S., Akinyemi, L., Doka, S.: Modulation instability gain and modulated wave shape incited by the acoustic longitudinal vibrations in molecular chain model. Phys. Scr. 97, 085206 (2022)

    ADS  Google Scholar 

  • Jia, T.T., Gao, Y.T., Deng, G.F., Hu, L.: Quintic time-dependent-coefficient derivative nonlinear schrödinger equation in hydrodynamics or fiber optics: bilinear forms and dark/anti-dark/gray solitons. Nonlinear Dyn. 98(1), 269–282 (2019)

    MATH  Google Scholar 

  • Jia, T.T., Gao, Y.T., Feng, Y.J., Hu, L., Su, J.J., Li, L.Q., Ding, C.C.: On the quintic time-dependent coefficient derivative nonlinear schrödinger equation in hydrodynamics or fiber optics. Nonlinear Dyn. 96(1), 229–241 (2019)

    MATH  Google Scholar 

  • Jiang, C., Cui, J., Qian, X., Song, S.: High-order linearly implicit structure-preserving exponential integrators for the nonlinear schrödinger equation. J. Sci. Comput. 90(1), 1–27 (2022)

    MATH  Google Scholar 

  • Korpinar, Z., Inc, M., Bayram, M.: Some new exact solutions for derivative nonlinear schrödinger equation with the quintic non-kerr nonlinearity. Mod. Phys. Lett. B 34(06), 2050079 (2020)

    ADS  Google Scholar 

  • Kudryashov, N.A.: Almost general solution of the reduced higher-order nonlinear schrödinger equation. Optik 230, 166347 (2021)

    ADS  Google Scholar 

  • Kudryashov, N.A.: Optical solitons of the resonant nonlinear schrödinger equation with arbitrary index. Optik 235, 166626 (2021)

    ADS  Google Scholar 

  • Liu, Y., Li, B., Wazwaz, A.M.: Novel high-order breathers and rogue waves in the boussinesq equation via determinants. Math. Methods in the Appl. Sci. 43(6), 3701–3715 (2020)

    ADS  MathSciNet  MATH  Google Scholar 

  • Ma, G., Zhao, J., Zhou, Q., Biswas, A., Liu, W.: Soliton interaction control through dispersion and nonlinear effects for the fifth-order nonlinear schrödinger equation. Nonlinear Dyn. 106(3), 2479–2484 (2021)

    Google Scholar 

  • Malomed: Multidimensional solitons in nonlocal media in multidimensional solitons. AIP Publishing LLC., Melville, New York (2022). https://doi.org/10.1063/9780735425118-013

  • Manafian, J., Mohammadi Ivatloo, B., Abapour, M.: Breather wave, periodic, and cross-kink solutions to the generalized bogoyavlensky-konopelchenko equation. Math. Methods in the Appl. Sci. 43(4), 1753–1774 (2020)

    ADS  MathSciNet  MATH  Google Scholar 

  • Mihalache, D.: Localized structures in optical and matter-wave media: a selection of recent studies. Rom. Rep. Phys. 73, 403 (2021)

    Google Scholar 

  • Mo, Y., Ling, L., Zeng, D.: Data-driven vector soliton solutions of coupled nonlinear schrödinger equation using a deep learning algorithm. Phys. Lett. A 421, 127739 (2022)

    MATH  Google Scholar 

  • Osman, M.S.: Multi-soliton rational solutions for some nonlinear evolution equations. Open Phys. 14(1), 26–36 (2016)

    ADS  Google Scholar 

  • Rizvi, S.T., Seadawy, A.R., Farah, N., Ahmad, S.: Application of hirota operators for controlling soliton interactions for bose-einstien condensate and quintic derivative nonlinear schrödinger equation. Chaos, Solitons & Fractals 159, 112128 (2022)

    MATH  Google Scholar 

  • Rizvi, S., Seadawy, A.R., Younis, M., Ali, I., Althobaiti, S., Mahmoud, S.F.: Soliton solutions, painleve analysis and conservation laws for a nonlinear evolution equation. Results in Phys. 23, 103999 (2021)

    Google Scholar 

  • Rizvi, S., Seadawy, A.R., Younis, M., Iqbal, S., Althobaiti, S., El-Shehawi, A.M.: Various optical soliton for a weak fractional nonlinear schrödinger equation with parabolic law. Results in Phys. 23, 103998 (2021)

    Google Scholar 

  • Rogers, C., Chow, K.: Localized pulses for the quintic derivative nonlinear schrödinger equation on a continuous-wave background. Phys. Rev. E 86(3), 037601 (2012)

    ADS  Google Scholar 

  • Rogers, C., Malomed, B., Hua Li, J., Wing Chow, K.: Propagating wave patterns in a derivative nonlinear schrödinger system with quintic nonlinearity. J. Phys. Soc. Jpn. 81(9), 094005 (2012)

    ADS  Google Scholar 

  • Savaissou, N., Gambo, B., Rezazadeh, H., Bekir, A., Doka, S.Y.: Exact optical solitons to the perturbed nonlinear schrödinger equation with dual-power law of nonlinearity. Opt. Quant. Electron. 52(6), 1–16 (2020)

    Google Scholar 

  • Seadawy, A.R.: The generalized nonlinear higher order of kdv equations from the higher order nonlinear schrödinger equation and its solutions. Optik 139, 31–43 (2017)

    ADS  Google Scholar 

  • Seadawy, A.R., Ahmed, S., Rizvi, S.T., Ali, K.: Lumps, breathers, interactions and rogue wave solutions for a stochastic gene evolution in double chain deoxyribonucleic acid system. Chaos, Solitons & Fractals 161, 112307 (2022a)

    MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R., Ahmed, S., Rizvi, S.T., Ali, K.: Various forms of lumps and interaction solutions to generalized vakhnenko parkes equation arising from high-frequency wave propagation in electromagnetic physics. J. Geom. Phys. 176, 104507 (2022b)

    MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R., Ahmed, S., Rizvi, S.T., Nazar, K.: Applications for mixed chen-lee-liu derivative nonlinear schrödinger equation in water wave flumes and optical fibers. Opt. Quant. Electron. 55(34), 1–22 (2023)

    Google Scholar 

  • Seadawy, A.R., Bilal, M., Younis, M., Rizvi, S., Althobaiti, S., Makhlouf, M.: Analytical mathematical approaches for the double-chain model of dna by a novel computational technique. Chaos, Solitons & Fractals 144, 110669 (2021)

    MathSciNet  Google Scholar 

  • Seadawy, A.R., Rehman, S., Younis, M., Rizvi, S., Althobaiti, S., Makhlouf, M.: Modulation instability analysis and longitudinal wave propagation in an elastic cylindrical rod modelled with pochhammer-chree equation. Phys. Scr. 96, 045202 (2021)

    ADS  Google Scholar 

  • Seadawy, A.R., Rizvi, S.T., Ahmed, S.: Multiple lump, generalized breathers, akhmediev breather, manifold periodic and rogue wave solutions for generalized fitzhugh-nagumo equation: Applications in nuclear reactor theory. Chaos, Solitons & Fractals 161, 112326 (2022c)

    MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R., Rizvi, S.T., Ahmed, S.: Weierstrass and jacobi elliptic, bell and kink type, lumps, ma and kuznetsov breathers with rogue wave solutions to the dissipative nonlinear schrödinger equation. Chaos, Solitons & Fractals 160, 112258 (2022d)

    MATH  Google Scholar 

  • Seadawy, A.R., Rizvi, S.T., Ali, I., Younis, M., Ali, K., Makhlouf, M., Althobaiti, A.: Conservation laws, optical molecules, modulation instability and painlevé analysis for the chen-lee-liu model. Opt. Quant. Electron. 53, 172 (2021)

    Google Scholar 

  • Seadawy, A.R., Rizvi, S.T., Ashraf, M.A., Younis, M., Hanif, M.: Rational solutions and their interactions with kink and periodic waves for a nonlinear dynamical phenomenon. Int. J. Mod. Phys. B 35(23), 2150236 (2021)

    ADS  MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R., Rizvi, S.T.R., Ahmad, S., Younis, M., Baleanu, D.: Lump, lump-one stripe, multiwave and breather solutions for the hunter-saxton equation. Open Phys. 19, 1–10 (2021). https://doi.org/10.1515/phys-2020-0224

  • Taghizadeh, N., Mirzazadeh, M., Farahrooz, F.: Exact solutions of the nonlinear schrödinger equation by the first integral method. J. Math. Anal. Appl. 374(2), 549–553 (2011)

    MathSciNet  MATH  Google Scholar 

  • Tariq, K.U., Zainab, H., Seadawy, A.R., Younis, M., Rizvi, S., Mousa, A.A.A.: On some novel optical wave solutions to the paraxial m-fractional nonlinear schrödinger dynamical equation. Opt. Quant. Electron. 53, 219 (2021)

    Google Scholar 

  • Triki, H., Biswas, A., Milović, D., Belić, M.: Chirped femtosecond pulses in the higher-order nonlinear schrödinger equation with non-kerr nonlinear terms and cubic-quintic-septic nonlinearities. Opt. Commun. 366, 362–369 (2016)

    ADS  Google Scholar 

  • Wang, K.J., Wang, G.D.: Variational theory and new abundant solutions to the (1+ 2)-dimensional chiral nonlinear schrödinger equation in optics. Phys. Lett. A 412, 127588 (2021)

    MATH  Google Scholar 

  • Wazwaz, A.M.: Higher dimensional nonlinear schrödinger equations in anomalous dispersion and normal dispersive regimes: Bright and dark optical solitons. Optik 222, 165327 (2020)

    ADS  Google Scholar 

  • Wazwaz, A.M.: Bright and dark optical solitons of the (2+ 1)-dimensional perturbed nonlinear schrödinger equation in nonlinear optical fibers. Optik 251, 168334 (2022)

    ADS  Google Scholar 

  • Wazwaz, A.M., Mehanna, M.: Bright and dark optical solitons for a new (3+ 1)-dimensional nonlinear schrödinger equation. Optik 241, 166985 (2021)

    ADS  Google Scholar 

  • Weng, W., Zhang, G., Zhang, M., Zhou, Z., Yan, Z.: Semi-rational vector rogon-soliton solutions and asymptotic analysis for any n-component nonlinear schrödinger equation with mixed boundary conditions. Physica D 432, 133150 (2022)

    MATH  Google Scholar 

  • Younas, U., Younis, M., Seadawy, A.R., Rizvi, S., Althobaiti, S., Sayed, S.: Diverse exact solutions for modified nonlinear schrödinger equation with conformable fractional derivative. Results in Phys. 20, 103766 (2021)

    Google Scholar 

  • Younis, M., Seadawy, A., Bilal, M., Rizvi, S., Althobaiti, S., Alkafafy, M.: Nonlinear dynamical wave structures to the date-jimbo-kashiwara-miwa equation and its modulation instability analysis. Mod. Phys. Lett. B 35(18), 2150300 (2021)

    ADS  MathSciNet  Google Scholar 

  • Younis, M., Younas, U., Bilal, M., Rehman, S., Rizvi, S.: Investigation of optical solitons with chen-lee-liu equation of monomode fibers by five free parameters. Indian J. Phys. 96(5), 1539–1546 (2022)

    ADS  Google Scholar 

  • Yue, C., Seadawy, A., Lu, D.: Stability analysis of the soliton solutions for the generalized quintic derivative nonlinear schrödinger equation. Results in Phys. 6, 911–916 (2016)

    ADS  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, Faculty of Science, Taibah University, 41411, Al-Madinah Al-Munawarah, Kingdom of Saudi Arabia

    Aly R. Seadawy

  2. Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan

    Syed T. R. Rizvi, Sarfaraz Ahmed & Tahira Batool

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  1. Aly R. Seadawy
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Seadawy, A.R., Rizvi, S.T.R., Ahmed, S. et al. Propagation of W-shaped and M-shaped solitons with multi-peak interaction for ultrashort light pulse in fibers. Opt Quant Electron 55, 221 (2023). https://doi.org/10.1007/s11082-022-04478-x

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