Abstract
In this work, we present a complete dynamical analysis of lump, breather, M-shaped, and other waveforms propagating in a nonlinear PDE governing nonlinear low-pass electrical transmission lines. We utilize the Hirota bilinear transformation approach with the help of Mathematica to report a number of wave solutions, including bright and dark lumps, solitons, breathers, and kink waves, along with their periodic and aperiodic forms. Energy distribution, wave interactions, and changes are presented in the form of 3D, contour, and 2D plots, which demonstrate the nonlinear characteristics that govern the dynamics. These results provide a better understanding of the propagation, stability, and interaction of waveforms which are useful in signal and energy transport and also in the construction of complex nonlinear electric circuits.
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The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
References
Munteanu, L., & Donescu, S. (2004). Introduction to soliton theory: applications to mechanics (Vol. 143). Springer Science & Business Media.
Zhou, Z., Wang, L. & Yan, Z. Deep neural networks learning forward and inverse problems of two-dimensional nonlinear wave equations with rational solitons. Comput. Math. Appl. 151, 164–171 (2023).
Belyaeva, T. L. & Serkin, V. N. Nonautonomous solitons: Applications from nonlinear optics to BEC and hydrodynamics. Hydrodyn. Adv. Top. 8, 51–76 (2011).
Faridi, W. A., Iqbal, M., Riaz, M. B., AlQahtani, S. A. & Wazwaz, A. M. The fractional soliton solutions of dynamical system arising in plasma physics: The comparative analysis. Alex. Eng. J. 95, 247–261 (2024).
Tan, M. et al. Integral order photonic RF signal processors based on a soliton crystal micro-comb source. J. Optics 23, 125701 (2021).
Uzdin, V. M., Thonig, D., Göbel, B. & Bessarab, P. F. Nucleation and stability of exotic solitons in condensed matter. Front. Phys. 11, 1275990 (2023).
Yin, X., Xu, L. & Yang, L. Evolution and interaction of soliton solutions of Rossby waves in geophysical fluid mechanics. Nonlinear Dyn. 111(13), 12433–12445 (2023).
Peyrard, M. Introduction to solitons and their applications in physics and biology. In Nonlinear Excitations in Biomolecules: Les Houches School, May 30 to June 4, 1994 11–25 (Berlin, Heidelberg, Springer, Berlin Heidelberg, 1995).
Zayed, E. M. E. & Al-Nowehy, A. G. The modified simple equation method, the exp-function method, and the method of soliton Ansatz for solving the long-short wave resonance equations. Zeitschrift für Naturforschung A 71(2), 103–112 (2016).
Javed, F., Rani, B., Chahlaoui, Y., Baskonus, H. M. & Raza, N. Nonlinear dynamics of wave structures for the Davey-Stewartson system: A truncated Painlevé approach. Nonlinear Dyn. 112(24), 22189–22200 (2024).
Shakeel, M., Attaullah, Shah & N. A., & Chung, J. D,. Modified exp-function method to find exact solutions of microtubules nonlinear dynamics models. Symmetry 15, 360 (2023).
Dubey, S. & Chakraverty, S. Application of modified extended tanh method in solving fractional order coupled wave equations. Math. Comput. Simul. 198, 509–520 (2022).
Shahzad, T. et al. Explicit solitary wave profiles and stability analysis of biomembranes and nerves. Mod. Phys. Lett. B 5, 2450305 (2024).
Tariq, M. M., Riaz, M. B. & Aziz-ur-Rehman, M. Investigation of space-time dynamics of Akbota equation using Sardar sub-equation and Khater methods: Unveiling bifurcation and chaotic structure. Int. J. Theor. Phys. 63(8), 1–24 (2024).
Zhang, Z. H. & Liu, J. G. A fourth-order nonlinear equation studied by using a multivariate bilinear neural network method. Nonlinear Dyn. 112(12), 10229–10237 (2024).
Boussaha, A. Some new optical waves interactions via modified Jacobi elliptic functions method. Utilitas Math. 120, 1963–1967 (2023).
Cai, G., Wang, Q. & Huang, J. A modified F-expansion method for solving breaking soliton equation. Int. J. Nonlinear Sci. 2(2), 122–128 (2006).
Ling, L., Zhao, L. C. & Guo, B. Darboux transformation and classification of solution for mixed coupled nonlinear Schrödinger equations. Commun. Nonlinear Sci. Numer. Simul. 32, 285–304 (2016).
Behera, S. (2024). Optical solitons for the Hirota-Ramani equation via improved \(\frac{G^{\prime }}{G}\)-expansion method. Modern Physics Letters B, 2450403.
Shahzad, T., Baber, M. Z., Sulaiman, T. A., Ahmad, M. O. & Yasin, M. W. Optical wave profiles for the higher order cubic-quartic Bragg-gratings with anti-cubic nonlinear form. Optical Quantum Electron. 56(1), 67 (2024).
Manafian, J., Ilhan, O. A. & Avazpour, L. The extended auxiliary equation mapping method to determine novel exact solitary wave solutions of the nonlinear fractional PDEs. Int. J. Nonlinear Sci. Numer. Simul. 22(1), 69–82 (2021).
El-Borai, M. M., El-Owaidy, H. M., Ahmed, H. M. & Arnous, A. H. Exact and soliton solutions to nonlinear transmission line model. Nonlinear Dyn. 87, 767–773 (2017).
Kengne, E., Malomed, B. A., Chui, S. T. & Liu, W. M. Solitary signals in electrical nonlinear transmission line. J. Math. Phys. 48, 24 (2007).
Zayed, E. M. E. & Alurrfi, K. A. E. A new Jacobi elliptic function expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines. Chaos Solitons Fractals 78, 148–155 (2015).
Kumar, D., Seadawy, A. R. & Chowdhury, R. On new complex soliton structures of the nonlinear partial differential equation describing the pulse narrowing nonlinear transmission lines. Optical Quantum Electron. 50, 1–14 (2018).
Malwe, B. H., Betchewe, G., Doka, S. Y. & Kofane, T. C. Travelling wave solutions and soliton solutions for the nonlinear transmission line using the generalized Riccati equation mapping method. Nonlinear Dyn. 84, 171–177 (2016).
El-Borai, M. M., El-Owaidy, H. M., Ahmed, H. M. & Arnous, A. H. Exact and soliton solutions to nonlinear transmission line model. Nonlinear Dyn. 87, 767–773 (2017).
Mostafa, S. I. Analytical study for the ability of nonlinear transmission lines to generate solitons. Chaos Solitons Fractals 39(5), 2125–2132 (2009).
Abdoulkary, S., Beda, T., Dafounamssou, O., Tafo, E. W. & Mohamadou, A. Dynamics of solitary pulses in the nonlinear low-pass electrical transmission lines through the auxiliary equation method. J. Mod. Phys. Appl 2, 69–87 (2013).
Iqbal, M. et al. Dynamical analysis of optical soliton structures for wave propagation in nonlinear low-pass electrical transmission lines under effective approach. Optical Quantum Electron. 56(6), 1036 (2024).
Kumar, D., Seadawy, A. R. & Haque, M. R. Multiple soliton solutions of the nonlinear partial differential equations describing the wave propagation in nonlinear low–pass electrical transmission lines. Chaos Solitons Fractals 115, 62–76 (2018).
Iqbal, M., Seadawy, A. R., Lu, D. & Zhang, Z. Multiple optical soliton solutions for wave propagation in nonlinear low-pass electrical transmission lines under analytical approach. Optical Quantum Electron. 56(1), 35 (2024).
Islam, M. E., Barman, H. K., & Akbar, M. A. (2021). Stable soliton solutions to the nonlinear low-pass electrical transmission lines and the Cahn-Allen equation. Heliyon, 7(5).
Kayum, M. A., Ara, S., Barman, H. K. & Akbar, M. A. Soliton solutions to voltage analysis in nonlinear electrical transmission lines and electric signals in telegraph lines. Results Phys. 18, 103269 (2020).
Aslan, I. Exact solutions for a local fractional DDE associated with a nonlinear transmission line. Commun. Theor Phys. 66(3), 315 (2016).
El-Ganaini, S. & Kumar, H. A variety of new traveling and localized solitary wave solutions of a nonlinear model describing the nonlinear low-pass electrical transmission lines. Chaos Solitons Fractals 140, 110218 (2020).
Motcheyo, A. T., Tchawoua, C., Siewe, M. S. & Tchameu, J. T. Supratransmission phenomenon in a discrete electrical lattice with nonlinear dispersion. Commun. Nonlinear Sci. Numer. Simul. 18(4), 946–952 (2013).
Togueu Motcheyo, A. B., Fezeu, G. J., Siewe Siewe, M., Buckjohn, Nono Dueyou & C., & Tchawoua, C,. Travelling waves in discrete electrical lattice with nonlinear symmetric capacitor. J. Comput. Electron. 22, 41–49 (2023).
Simo Motcheyo, P. A., Woulaché, R. L., Tchomgo Felenou, E., Togueu Motcheyo, A. B. & Kofane, T. C. Propagating discrete breather due to plane wave in electrical transmission lattice with imaginary resistance. Phys. Scripta 100(6), 065243 (2025).
Rizvi, S. T., Seadawy, A. R. & Ahmed, S. Bell and Kink type, Weierstrass and Jacobi elliptic, multiwave, kinky breather, M-shaped and periodic-kink-cross rational solutions for Einstein’s vacuum field model. Optical Quantum Electron. 56(3), 456 (2024).
Ma, Y. L., Wazwaz, A. M. & Li, B. Q. Soliton resonances, soliton molecules, soliton oscillations and heterotypic solitons for the nonlinear Maccari system. Nonlinear Dyn. 111(19), 18331–18344 (2023).
Ma, Y. L. & Li, B. Q. Bifurcation solitons and breathers for the nonlocal Boussinesq equations. Appl. Math. Lett. 124, 107677 (2022).
Khan, M. H., & Wazwaz, A. M. (2020). Lump, multi-lump, cross kinky-lump and manifold periodic-soliton solutions for the (2+ 1)-D Calogero–Bogoyavlenskii–Schiff equation. Heliyon, 6(4).
Ahmed, S., Seadawy, A. R., Rizvi, S. T. & Mubaraki, A. M. Homoclinic breathers and soliton propagations for the nonlinear (3+ 1)-dimensional Geng dynamical equation. Results Phys. 52, 106822 (2023).
Ceesay, B. et al. Modelling symmetric ion-acoustic wave structures for the BBMPB equation in fluid ions using Hirota’s bilinear technique. Symmetry 15(9), 1682 (2023).
Wang, M. et al. Lump, lumpoff, rogue wave, breather wave and periodic lump solutions for a (3+ 1)-dimensional generalized Kadomtsev-Petviashvili equation in fluid mechanics and plasma physics. Int. J. Comput. Math. 97(12), 2474–2486 (2020).
Yan, X. W., Tian, S. F., Dong, M. J., Zhou, L. & Zhang, T. T. Characteristics of solitary wave, homoclinic breather wave and rogue wave solutions in a (2+ 1)-dimensional generalized breaking soliton equation. Comput. Math. Appl. 76(1), 179–186 (2018).
Kumar, S. & Mohan, B. A novel and efficient method for obtaining Hirota’s bilinear form for the nonlinear evolution equation in (n+ 1) dimensions. Partial Differ. Equ. Appl. Math. 5, 100274 (2022).
Alsallami, S. A., Rizvi, S. T. & Seadawy, A. R. Study of stochastic–fractional Drinfel’d–Sokolov–Wilson equation for M-shaped rational, homoclinic breather, periodic and Kink-Cross rational solutions. Mathematics 11(6), 1504 (2023).
Remoissenet, M. (2013). Waves called solitons: concepts and experiments. Springer Science & Business Media.
Ozsahin, D. U. et al. Multiwaves, breathers, lump and other solutions for the Heimburg model in biomembranes and nerves. Sci. Rep. 14, 10180 (2024).
Ceesay, B., Ahmed, N., Baber, M. Z. & Akgül, A. Breather, lump, M-shape and other interaction for the Poisson–Nernst–Planck equation in biological membranes. Optical Quantum Electron. 56(5), 853 (2024).
Ceesay, B., Ahmed, N. & Macías-Díaz, J. E. Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota’s bilinear method. Open Phys. 22(1), 20240057 (2024).
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M.Z.B Writing original draft, Software, Formal analysis, Methodology A.S Validation, Visualization, Resources, Supervision B.C Writing original draft, Software, Investigation, Methodology N.A Writing review and editing, Supervision, Conceptualization
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Baber, M.Z., Shafee, A., Ceesay, B. et al. Dynamical analysis of lump, breather, M-shaped and other wave profiles propagating in a nonlinear PDE describing the nonlinear low-pass electrical transmission lines. Sci Rep (2026). https://doi.org/10.1038/s41598-026-45214-9
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DOI: https://doi.org/10.1038/s41598-026-45214-9
