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Systematic exploration of solitary wave characteristics for the high-order dispersive extended nonlinear Schrödinger model

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Abstract

Nonlinear dispersive waves influenced by higher–order dispersion effects play a pivotal role across various scientific disciplines. While numerical simulations offer valuable approximations, analytical solutions provide a comprehensive mathematical characterization. In this study, exact solitary wave solutions for the high-order Dispersive Extended Nonlinear Schrödinger (\(\mathbb {DENLS}\)) equation were systematically derived using two recent computational techniques. The \(\mathbb {DENLS}\) model incorporates third- and fifth-order dispersion terms, extending beyond the standard nonlinear Schrödinger equation and rendering it non–integrable. This model delineates the mathematical and physical characteristics of nonlinear dispersive waves, with applications spanning optics and plasma physics. The derived solutions significantly advance our understanding of high-order nonlinear wave behaviors in such systems. Multiple solution types, including periodic, rational, and hyperbolic solitary waveforms, were obtained employing the Khater II method and generalized rational methods. Notably, the periodic solution unveiled the emergence of secondary peaks, highlighting the profound impact of higher-order dispersion on the envelope structure. Additionally, the localized rational and hyperbolic solutions depicted robust nonlinear excitation. Validation of the solutions was conducted through consistency checks, stability analyses, examination of conserved quantities such as momentum, and comparisons with numerical simulations. Both the Khater II and generalized rational methodologies proved effective in deriving closed-form representations of waves governed by the non–integrable high-order \(\mathbb {DENLS}\) model. The analytical wave–forms presented herein contribute to an enriched mathematical depiction of nonlinear dispersive phenomena, facilitating quantitative parameter extraction. This study enhances our understanding of complex nonlinear wave propagation in domains characterized by higher–order chromatic effects, such as ultrafast fiber optics, plasma physics, and Bose–Einstein condensates. The formulations introduced in this study advance theoretical tools with broad applicability in the realm of nonlinear sciences.

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References

  • Abbagari, S., Houwe, A., Akinyemi, L., Bouetou, T.B.: Modulated wave patterns brought by higher-order dispersion and cubic-quintic nonlinearity in monoatomic chains with anharmonic potential. Wave Motion 123, 103220 (2023)

    MathSciNet  Google Scholar 

  • Ahmad, J., Akram, S., Noor, K., Nadeem, M., Bucur, A., Alsayaad, Y.: Soliton solutions of fractional extended nonlinear Schrödinger equation arising in plasma physics and nonlinear optical fiber. Sci. Rep. 13, 10877 (2023)

    ADS  Google Scholar 

  • Alam, B.E., Javid, A.: Optical dual-waves to a new dual-mode extension of a third order dispersive nonlinear Schrödinger’s equation. Phys. Lett. A 480, 128954 (2023)

    Google Scholar 

  • Ali, K.K., Yusuf, A., Alquran, M., Tarla, S.: New physical structures and patterns to the optical solutions of the nonlinear Schrödinger equation with a higher dimension. Commun. Theor. Phys. 75(8), 085003 (2023)

    ADS  Google Scholar 

  • Ashraf, R., Hussain, S., Ashraf, F., Akgül, A., El Din, S.M.: The extended Fan’s sub-equation method and its application to nonlinear Schrödinger equation with saturable nonlinearity. Results Phys. 52, 106755 (2023)

    Google Scholar 

  • Bai, C.-L., Cai, Y.-J., Luo, Q.-L.: Breathers and rogue waves derived from an extended multi-dimensional N-coupled higher-order nonlinear Schrödinger equation in optical communication systems. Commun. Theor. Phys. 70(3), 255 (2018)

    ADS  MathSciNet  Google Scholar 

  • Brun, E., Li, G., Liu, R., Zine, Y.: Global well-posedness of one-dimensional cubic fractional nonlinear Schrödinger equations in negative Sobolev spaces. arXiv:2311.13370, (2023)

  • Cinar, M., Cakicioglu, H., Secer, A., Ozisik, M., Bayram, M.: Optical solitons of improved perturbed nonlinear Schrödinger equation with cubic-quintic-septic and triple-power laws in optical metamaterials. Phys. Scripta 98(7), 075220 (2023)

    ADS  Google Scholar 

  • Fang, H., Tang, L., Lin, P.: Bragg scattering of nonlinear surface waves by sinusoidal sandbars. J. Fluid Mech. 979, A13 (2024)

    ADS  MathSciNet  Google Scholar 

  • Figueira, R.O., Panthee, M.: Decay of the radius of spatial analyticity for the modified KdV equation and the nonlinear Schrödinger equation with third order dispersion. arXiv:2307.09096, (2023)

  • Günhan Ay, N., Yaşar, E.: Novel dispersive soliton solutions to a fractional nonlinear Schrödinger equation related with ultrashort pulses. Pramana 97(3), 106 (2023)

    ADS  Google Scholar 

  • Kengne, E.: Mathematical modeling of chirped modulated waves along a multi-coupled nonlinear electrical transmission line with dispersive elements. Wave Motion 123, 103221 (2023)

    MathSciNet  Google Scholar 

  • Khater, M.M.A.: Hybrid accurate simulations for constructing some novel analytical and numerical solutions of three-order GNLS equation. Int. J. Geom. Methods Mod. Phys. 20(9), 2350159 (2023)

    MathSciNet  Google Scholar 

  • Khater, M.M.A.: Computational and numerical wave solutions of the Caudrey-Dodd-Gibbon equation. Heliyon 9, e13511 (2023)

    Google Scholar 

  • Khater, M.M.A.: In solid physics equations, accurate and novel soliton wave structures for heating a single crystal of sodium fluoride. Int. J. Mod. Phys. B 37(7), 2350068–139 (2023)

    ADS  Google Scholar 

  • Khater, M.M.A.: Prorogation of waves in shallow water through unidirectional Dullin-Gottwald-Holm model; computational simulations. Int. J. Mod. Phys. B 37(8), 2350071 (2023)

    ADS  Google Scholar 

  • Khater, M.M.A.: Novel computational simulation of the propagation of pulses in optical fibers regarding the dispersion effect. Int. J. Mod. Phys. B 37(9), 2350083 (2023)

    ADS  MathSciNet  Google Scholar 

  • Khater, M.M.A.: In surface tension; gravity-capillary, magneto-acoustic, and shallow water waves’ propagation. Eur. Phys. J. Plus 138(4), 320 (2023)

    Google Scholar 

  • Khater, M.M.A.: Long waves with a small amplitude on the surface of the water behave dynamically in nonlinear lattices on a non-dimensional grid. Int. J. Mod. Phys. B 37(19), 2350188 (2023)

    ADS  Google Scholar 

  • Khater, M.M.A.: Abundant and accurate computational wave structures of the nonlinear fractional biological population model. Int. J. Mod. Phys. B 37(18), 2350176 (2023)

    ADS  Google Scholar 

  • Khater, M.M.A.: Advancements in computational techniques for precise solitary wave solutions in the (1 + 1)-dimensional Mikhailov-Novikov-Wang equation. Int. J. Theor. Phys. 62(7), 152 (2023)

    MathSciNet  Google Scholar 

  • Khater, M.M.A.: Numerous accurate and stable solitary wave solutions to the generalized modified equal-width equation. Int. J. Theor. Phys. 62(7), 151 (2023)

    ADS  MathSciNet  Google Scholar 

  • Khater, M.M.A.: Horizontal stratification of fluids and the behavior of long waves. Eur. Phys. J. Plus 138(8), 715 (2023)

    Google Scholar 

  • Khater, M.M.A.: Soliton propagation under diffusive and nonlinear effects in physical systems; (1+1)-dimensional MNW integrable equation. Phys. Lett. A 480, 128945 (2023)

    MathSciNet  Google Scholar 

  • Khater, M.M.A.: Physical and dynamic characteristics of high-amplitude ultrasonic wave propagation in nonlinear and dissipative media. Mod. Phys. Lett. B 37(36), 2350210 (2023)

    ADS  MathSciNet  Google Scholar 

  • Khater, M.M.A.: Analyzing pulse behavior in optical fiber: novel solitary wave solutions of the perturbed Chen-Lee-Liu equation. Mod. Phys. Lett. B 37(34), 2350177 (2023)

    ADS  MathSciNet  Google Scholar 

  • Khater, M.M.: Waves in motion: unraveling nonlinear behavior through the Gilson-Pickering equation. Eur. Phys. J. Plus 138(12), 1138 (2023)

    Google Scholar 

  • Khater, M.M.A.: Physics of crystal lattices and plasma; analytical and numerical simulations of the Gilson-Pickering equation. Results Phys. 44, 106193 (2023)

    Google Scholar 

  • Khater, M.M.A.: Multi-vector with nonlocal and non-singular kernel ultrashort optical solitons pulses waves in birefringent fibers. Chaos, Solitons Fractals 167, 113098 (2023)

    MathSciNet  Google Scholar 

  • Khater, M.M.A.: A hybrid analytical and numerical analysis of ultra-short pulse phase shifts. Chaos Solitons Fractals 169, 113232 (2023)

    MathSciNet  Google Scholar 

  • Khater, M.M.A.: Characterizing shallow water waves in channels with variable width and depth; computational and numerical simulations. Chaos Solitons Fractals 173, 113652 (2023)

    MathSciNet  Google Scholar 

  • Khater, M.M.A.: Computational simulations of propagation of a tsunami wave across the ocean. Chaos Solitons Fractals 174, 113806 (2023)

    MathSciNet  Google Scholar 

  • Khater, M.M.: Advanced computational techniques for solving the modified KdV-KP equation and modeling nonlinear waves. Opt. Quant. Electron. 56(1), 6 (2024)

    ADS  Google Scholar 

  • Khater, M.M.: Novel constructed dark, bright and rogue waves of three models of the well-known nonlinear Schrödinger equation. Int. J. Mod. Phys. B 38(03), 2450023 (2024)

    ADS  Google Scholar 

  • Khater, M.M.: Exploring the rich solution landscape of the generalized Kawahara equation: insights from analytical techniques. Eur. Phys. J. Plus 139(2), 184 (2024)

    Google Scholar 

  • Khater, M.M.: Wave propagation and evolution in a (1+ 1)-dimensional spatial-temporal domain: a comprehensive study. Mod. Phys. Lett. B 38(05), 2350235 (2024)

    ADS  Google Scholar 

  • Li, Y., Falessi, M.V., Lauber, P., Li, Y., Qiu, Z., Wei, G., Zonca, F.: Physics of drift Alfvén instabilities and energetic particles in fusion plasmas. Plasma Phys. Controll. Fusion 65(8), 084001 (2023)

    ADS  Google Scholar 

  • Liu, S., Zhang, Y., Virally, S., Karimi, E., Malomed, B.A., Seletskiy, D.V.: Observation of the spectral bifurcation in the fractional nonlinear Schrödinger equation. arXiv:2311.15150, (2023)

  • Mathanaranjan, T., Hashemi, M.S., Rezazadeh, H., Akinyemi, L., Bekir, A.: Chirped optical solitons and stability analysis of the nonlinear Schrödinger equation with nonlinear chromatic dispersion. Commun. Theor. Phys. 75(8), 085005 (2023)

    ADS  Google Scholar 

  • Ndoungalah, S., Roger Deffo, G., Djine, A., Bruno Yamgoué, S.: Dissipation and amplification management in an electrical model of microtubules: hybrid behavior network. Chin. Phys. B 32(11), 110505 (2023)

    ADS  Google Scholar 

  • Pal, T., Dhar, A.K.: Current modified higher-order Schrödinger equation of broader bandwidth capillary-gravity waves. Phys. Fluids 35(12), 127104 (2023)

    ADS  Google Scholar 

  • Poli, E., Bottino, A., Korger, D., Maj, O., Palermo, F., Weber, H.: Wave beams, packets and pulses in inhomogeneous non-Hermitian media with dispersive gain or damping. New J. Phys. 26(1), 013016 (2024)

    ADS  MathSciNet  Google Scholar 

  • Riaz, H.W.A., Lin, J.: Quasi-Gramian solution of a noncommutative extension of the higher-order nonlinear Schrödinger equation. arXiv:2311.05841, (2023)

  • Seadawy, A.R., Cheemaa, N.: Applications of extended modified auxiliary equation mapping method for high-order dispersive extended nonlinear schrödinger equation in nonlinear optics. Mod. Phys. Lett. B 33(18), 1950203 (2019)

    ADS  Google Scholar 

  • Shehata, M.S.M., Bekir, A.: New perceptions for the bright and dark soliton solutions to the modified nonlinear Schrödinger equation. Int. J. Mod. Phys. B 37(21), 2350204 (2023)

    ADS  Google Scholar 

  • Tang, Y., Rezazadeh, H.: On logarithmic transformation-based approaches for retrieving traveling wave solutions in nonlinear optics. Results Phys. 51, 106672 (2023)

    Google Scholar 

  • Wang, S., Zhou, R., Duan, Z., Hu, G., Yuan, S., Li, L., Nakkeeran, K.: On the spectral sidebands’ evolution of mode-locked fiber lasers. Opt. Fiber Technol. 80, 103479 (2023)

    Google Scholar 

  • Zhang, J.-R., Zhu, F.-Y., Li, W.-P., Shen, Y.-J.: Solitons in fourth-order Schrödinger equation with parity-time-symmetric extended Rosen-Morse potentials. Phys. Scripta 98(8), 085217 (2023)

    ADS  Google Scholar 

  • Zhu, K., Sun, J., Qin, J., Li, Y., Tan, J., Miao, M., Wang, M.: Evolution of inter-pulse non-frequency shift components in multi-pulse pumped supercontinuum generation with normal dispersion. In: Q. He, D.-S. Kim, C.-F. Li (eds) Society of photo-optical instrumentation engineers (SPIE) conference series, Vol. 12775, p. 127751F (2023)

  • Ziani, Z., Bunel, T., Perego, A.M., Mussot, A., Conforti, M.: Theory of modulation instability in Kerr Fabry-Perot resonators beyond the mean field limit. arXiv:2307.13488, (2023)

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Acknowledgements

This work was partially supported by Science and Technology General Project of Jiangxi Provincial Department of Education (No. GJJ2203204). Additionally, the authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Large Groups Project under grant number (RGP. 2/554/44). All authors read and approved the final manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Nanchang Normal College of Applied Technology, Nanchang, 330036, Jiangxi, China

    Chanyuan Wang

  2. School of Medical Informatics and Engineering, Xuzhou Medical University, 209 Tongshan Road, Xuzhou, 221004, Jiangsu Province, PR China

    Raghda A. M. Attia & Mostafa M. A. Khater

  3. Department of Mathematics, Faculty of Science and Arts in Mahayil Asir, King Khalid University, Abha, Saudi Arabia

    Suleman H. Alfalqi & Jameel F. Alzaidi

  4. Department of Basic Science, Obour High Institute for Engineering and Technology, Cairo, 11828, Egypt

    Mostafa M. A. Khater

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  1. Chanyuan Wang
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  2. Raghda A. M. Attia
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Contributions

RAMA and MMAK conceived and designed the experiments, as well as performed the experiments. CW, SHA, and JFA analyzed and interpreted the data, contributed reagents, materials, analysis tools, or data, and wrote the paper.

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Correspondence to Mostafa M. A. Khater.

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Wang, C., Attia, R.A.M., Alfalqi, S.H. et al. Systematic exploration of solitary wave characteristics for the high-order dispersive extended nonlinear Schrödinger model. Opt Quant Electron 56, 892 (2024). https://doi.org/10.1007/s11082-024-06817-6

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