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A generalized Kundu--Eckhaus equation with an extra-dispersion: pulses configuration

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Abstract

Raman effect is due to self-phase modulation (SPM), which is embedded in Kundu--Eckhaus equation KEE. Here, the objective of this work is to present a generalized KEE by accounting for an extra dispersion which may be induced by Raman scattering. Also, attention is focused to study the effects of the extra dispersion.Which are investigated via obtaining the exact solutions of the new model equation. These solutions are found by the unified method and by introducing a new transformation that ispects soliton- periodic wave collision. We aim to show that a variety of shapes of optical pulses OPs propagation in optical fibers occurs. Waves of multiple geometric structures are observed. Among these waves, hybrid lumps, soliton, cascade, complex chirped, hybrid w-shaped, rhombus (diamond) waves and soliton self phase modulation.The characteristics of the pulses; intensity, frequency, wavelength, polarization, and spectral content are identified. The results found here are of great interest in experimenting the effects of the induced dispersion on pulses configurations. Further, the colliding dynamics are inspected and as it is observed that no rogue or sharp waves formation hold, so the collision is elastic.

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References

  • Abdel-Gawad H.I., Elazab, N. S., Osman, M.: ”Exact solutions of space dependent Korteweg–de Vries equation by the extended unified method.” J. Phys. Soc. Jpn. 82, 044004 (2013)

  • Abdel-Gawad, H.I.: Towards a unified method for exact solutions of evolution equations. An application to reaction diffusion equations with finite memory transport. J. Stat. Phys. 147, 506 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  • Abdel-Gawad, H.I.: Solutions of the generalized transient stimulated Raman scattering equation. Optical pulses compression. Optik 230, 166314 (2021)

    Article  ADS  Google Scholar 

  • Abdel-Gawad, H.I., Biswas, A.: Multi-soliton solutions based on interactions of basic traveling waves with an applications to the non local Boussinesq equation. Acta Phys. Pol. B 47, 1101–1112 (2016)

    Article  ADS  Google Scholar 

  • Abdel-Gawad, H.I., Tantawy, M.: A novel model for lasing cavities in the presence of population inversion: Bifurcation and stability analysis. Chaos Solitons Fractals 144, 110693 (2021)

    Article  MathSciNet  Google Scholar 

  • Ablowitz, M., Musslimani, Z.H.: Integrable nonlocal nonlinear Schrodinger equation. Phys. Rev. Lett. 110, 064105 (2013)

    Article  ADS  Google Scholar 

  • Alfano, R.R., Shapiro, S.L.: Observation of Self-Phase Modulation and Small-Scale Filaments in Crystals and Glasses. Phys. Rev. Lett. 24, 592 (1970)

    Article  ADS  Google Scholar 

  • Anderson, D., Lisak, M.: Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical wave guides. Phys. Rev. A 27, 1393 (1983)

    Article  ADS  Google Scholar 

  • Andresen E. R., Dudley, J. M., Oron, D., Finot, C.E, and H. Rigneault, H.: Transform-limited spectral compression by self-phase modulation of amplitude-shaped pulses with negative chirp. Opt. Lett. 36 (2011) No. 5

  • Aslan, E.C., Tchier, F., Inc, M.: On optical solitons of the Schrodinger-Hirota equation with power law nonlinearity in optical fibers. Superlattices Microstruct. 105, 48–55 (2017)

    Article  ADS  Google Scholar 

  • Baleanu, D., Inc, M., Aliyu, A.I., Yusuf, A.D.: Dark optical solitons and conservation laws to the resonance nonlinear Shrodinger’s equation with Kerr law nonlinearity. Optik 147, 248–255 (2017)

    Article  ADS  Google Scholar 

  • Baleanu, D., Inc, M., Yusuf, A., Aliyuc, A.I.: Optical solitons, nonlinear self-adjointness and conservation laws for Kundu--Eckhaus equation. Chin. J. Phys. 55, 2341–2355 (2017)

    Article  MathSciNet  Google Scholar 

  • Baleanua, D., Inc, M., Abdullahi, Yusuf A., Aliyu, A.I.: Optical solitons, nonlinear self-adjointness and conservation laws for Kundu--Eckhaus equation. Chin. J. Phys. 55, 2341–2355 (2017)

    Article  MathSciNet  Google Scholar 

  • Baskonous, H.M., Bulut, H.: On the complex structures of Kundu--Eckhaus equation via improved Bernoulli sub-equation function method. Waves Random Complex Media 25, 720–728 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  • Bayindi, C.: Rogue waves of the Kundu--Eckhaus equation in a chaotic wave field. Phys. Rev. E 93, 032201 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  • Bayındı, C.: Rogue wave spectra of the Kundu--Eckhaus equation. Phys. Rev. E 93, 062215 (2016)

    Article  ADS  Google Scholar 

  • Carr, L.D., Charles, W.C., Reinhardt, W.P.: Stationary solutions of the one-dimensional nonlinear Schrodinger equation. II. Case of attractive nonlinearity. Phys. Rev. A 62, 063611 (2000)

    Article  ADS  Google Scholar 

  • Chao, K., Dhakal, S., Qin, J., Kim, M., Peng, Y.A.: Dispersive Raman Spectral Imaging System for Food Safety and Quality Evaluation. Appl. Sci. 8, 431 (2018)

    Article  Google Scholar 

  • Chatterjee, S.K., Khan, S.N.: Designing tunable narrowband parametric source in Chalcogenide-based dynamic fiber geometry. J. Opt. 23, 015503 (2021)

    Article  ADS  Google Scholar 

  • d’Avenia, P.: Non-radially symmetric solutions of nonlinear Schrodinger equation coupled with Maxwell equations. Adv. Nonlinear. Stud. 2, 2 (2016)

    MathSciNet  Google Scholar 

  • Ghanbari, B., Gomez-Aguilar, J.F.: Optical soliton solutions for the nonlinear Radhakrishnan–Kundu–Lakshmanan equation. Mod. Phys. Lett. B 33(32), 1950402 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  • Gomez-Aguilar, J.F., Osman, M.S., Raza, N., Asad, Zubair A., Arshed, S., Ghoneim, M.E., Mahmoud, E.E., Abdel-Aty, A.-H.: Optical solitons in birefringent fibers with quadratic-cubic nonlinearity using three integration architectures. AIP Adv. 11(2), 025121 (2021)

    Article  ADS  Google Scholar 

  • Guo, B., Ling, L., Liu, Q.: P: Nonlinear Schrodinger equation: Generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85, 026607 (2012)

    Article  ADS  Google Scholar 

  • Hosseini, K., Aligoli, M., Mirzazadeh, M., Eslami, M., Gomez-Aguilar, J.F.: Dynamics of rational solutions in a new generalized Kadomtsev–Petviashvili equation. Mod. Phys. Lett. B 33(35), 1950437 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  • Inc, M., Ates, E., Tchier, F.: Optical solitons of the coupled nonlinear Schrodinger equation with spatiotemporal dispersion. Nonlinear Dyn. 85, 1319–1329 (2016)

    Article  MathSciNet  Google Scholar 

  • Kilic, B., Inc, M.: On optical solitons of the resonant Schrodinger’s equation in optical fibers with dual-power law nonlinearity and time-dependent coefficients. Waves Random Complex Media 25, 245–251 (2015)

    Article  MathSciNet  Google Scholar 

  • Kilic, B., Inc, M.: Soliton solutions for the Kundu--Eckhaus equation with the aid of unified algebraic and auxiliary equation expansion methods. J. Electromag Waves Appl. 30, 871–879 (2016)

    Article  Google Scholar 

  • Kundu, A.: Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrodinger-type equations. J. Math. Phys. 25, 3433–3438 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  • Manafian, J., Murad, M.A.S., Alizadeh, A., Jafarmadar, S.: M-lump, interaction between lumps and stripe solitons solutions to the (2+1)-dimensional KP-BBM equation. Eur. Phys. J. Plus 135, 167 (2020)

  • Ma, Y.L.: Interaction and energy transition between the breather and rogue wave for a generalized nonlinear Schrodinger system with two higher-order dispersion operators in optical fibers. Nonlinear Dyn. 97, 95 (2019)

    Article  Google Scholar 

  • Ma, W.X.: Inverse scattering for nonlocal reverse-time nonlinear Schrodinger equations. Appl. Mathe. Lett. 102, 106161 (2020)

    Article  MathSciNet  Google Scholar 

  • Ma, W.X., Chen, M.: Direct search for exact solutions to the nonlinear Schrodinger equation. Appl. Math. Comput. 215, 2835–2842 (2009)

    MathSciNet  MATH  Google Scholar 

  • Manafian, J., Akestani, M.: Abundant soliton solutions for the Kundu--Eckhaus equation via tan(\(\phi \)(\(\xi \)))-expansion method. Optik 127, 5543–5551 (2016)

    Article  ADS  Google Scholar 

  • Okawachi, Y., Sharping, J.E., Xu, C., Alexander, L.G.: Large tunable optical delays via self-phase modulation and dispersion. Nonlinear Opt. Fibers 14, 12022–12027 (2006)

    Google Scholar 

  • Osman, M.S., Ali, K.K., Gomez-Aguilar, J.F.: A variety of new optical soliton solutions related to the nonlinear Schrodinger equation with time-dependent coefficients. Optik 222, 165389 (2020)

    Article  ADS  Google Scholar 

  • Perry, M.D., Ditmire, T., Stuart, B.C.: Self-phase modulation in chirped-pulse amplification. Optics. Lett. 19, 2149–2152 (1994)

    Article  ADS  Google Scholar 

  • Qiu, D., J. He, J., Hang, Y., Porsezian, K.: The Darboux transformation of the Kundu–Eckhaus equation, Proc. R. Soc. A 471 (2015) 20150236

  • Qiu, D., Cheng, W.: N-fold Darboux transformation of the two-component Kundu--Eckhaus equations and non-symmetric doubly localized rogue waves. Eur. Phys. J. Plus 13, 135 (2020)

    Google Scholar 

  • Sedeeg, A.K.H., Nuruddeen, R.I., Gomez-Aguilar, J.F.: Generalized optical soliton solutions to the (3+1)-dimensional resonant nonlinear Schrodinger equation with Kerr and parabolic law nonlinearities. Opt. Quant. Electr. 51(6), 173 (2019)

    Article  Google Scholar 

  • Serkin, V.N., Hasegawa, A.: Novel soliton solutions of the nonlinear Schrodinger equation Model. Phys. Rev. Lett. 85, 21 (2000)

    Article  Google Scholar 

  • Tedesco, J.M., Davis, K. L.: Calibration of dispersive Raman process analyzers. Proc. SPIE 3537,(1999)

  • Tzoar, N., Jain, M.: Self-phase modulation in long-geometry optical wave guides. Phys. Rev. A 23, 1266 (1981)

    Article  ADS  Google Scholar 

  • Ui, W.-T, Wazwaz, A.-M., Zhou,Q., LIiu W.-J.: Interaction and oscillation of three soliton solutions in the variable coefficients Kundu--Eckhaus equation for dispersion management systems, Rom. J. Phys. 64 , 203 (2019)

  • Wang, X., Yang, B., Chen, Y., Yang, Y.: Higher-order rogue wave solutions of the Kundu--Eckhaus equation. Phys. Scr. 89, 095210 (2014)

    Article  ADS  Google Scholar 

  • Wang, H., Boraeya, M.A., Williams, L., Ballesteros, D.L., Vehring, R.: Low-frequency shift dispersive Raman spectroscopy for the analysis of respirable dosage forms, I. J. Pharma. 469, 197–205 (2014)

    Google Scholar 

  • Wanga, D.-S., Guob, B., Wang, X.: Long-time asymptotics of the focusing Kundu--Eckhaus equation with nonzero boundary conditions. J. Diff. Equ. 266, 5209–5253 (2019)

    Article  ADS  MathSciNet  Google Scholar 

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  1. Mathematics Department Faculty of Science, Cairo University, Giza, Egypt

    H. I. Abdel-Gawad

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  1. H. I. Abdel-Gawad
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Topical Collection on Optical and Quantum Sciences in Africa, Guest edited by Salah Obayya, Alex Quandt, Andrew Forbes, Malik Maaza, Abdelmajid Belafhal and Mohamed Farhat.

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Abdel-Gawad, H.I. A generalized Kundu--Eckhaus equation with an extra-dispersion: pulses configuration. Opt Quant Electron 53, 705 (2021). https://doi.org/10.1007/s11082-021-03224-z

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