On new complex soliton structures of the nonlinear partial differential equation describing the pulse narrowing nonlinear transmission lines

  • Dipankar Kumar
  • Aly R. Seadawy
  • Raju Chowdhury


The present paper studies the pulse narrowing nonlinear transmission lines equation, describing pulse narrowing in the field of communication engineering. More precisely, the pulse narrowing nonlinear transmission line equation is solved analytically using the recently developed techniques viz the modified Kudraysov method, the sine-Gordon equation expansion method and the extended sinh-Gordon equation expansion method. As a result, a wide range of dark, bright, dark–bright, singular or combined singular and optical soliton solutions for the pulse narrowing nonlinear transmission lines equation is formally obtained. All solutions have been verified back into its corresponding equation with the aid of maple package program.


Pulse narrowing nonlinear transmission lines Kirchhoffs current and Kirchhoffs voltage laws Mathematical methods Soliton and optical solutions 


  1. Afshari, E., Hajimiri, A.: Nonlinear transmission lines for pulse shaping in silicon. IEEE J. Solid State Circuits 40(3), 744–752 (2005)CrossRefGoogle Scholar
  2. Baskonus, H.M.: New acoustic wave behaviors to the Davey–Stewartson equation with power law nonlinearity arising in fluid dynamics. Nonlinear Dyn. 86(1), 177–183 (2016)MathSciNetCrossRefGoogle Scholar
  3. Baskonus, H.M., Bulut, H., Sulaiman, T.A.: Investigation of various travelling wave solutionsto the extended (2 + 1)-dimensional quantum ZK equation. Eur. Phys. J. Plus 132, 482 (2017)CrossRefGoogle Scholar
  4. Baskonus, H.M., Sulaiman, T.A., Bulut, H.: On the novel wave behaviors to the coupled nonlinear Maccaris system with complex structure. Optik 131, 1036–1043 (2017)ADSCrossRefGoogle Scholar
  5. Bulut, H., Sulaiman, T.A., Baskonus, H.M.: Dark, bright and other soliton solutions to the Heisenberg ferromagnetic spin chain equation. Superlattices Microstruct. (2017a).
  6. Bulut, H., Sulaiman, T.A., Baskonus, H.M.: On the new soliton and optical wave structures to some nonlinear evolution equations. Eur. Phys. J. Plus 132, 459 (2017b)CrossRefGoogle Scholar
  7. Bulut, H., Sulaiman, T.A., Baskonus, H.M., Akturk, T.: Complex acoustic gravity wave behaviors to some mathematical models arising in fluid dynamics and nonlinear dispersive media. Opt. Quant Electron. 50, 19 (2018)CrossRefGoogle Scholar
  8. 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(2), 767–773 (2017)CrossRefGoogle Scholar
  9. Hosseini, K., Bekir, A., Kaplan, M.: New exact traveling wave solutions of the Tzitzica-type evolution equations arising in non-linear optics. J. Mod. Opt. 64, 1688–1692 (2017a)ADSCrossRefGoogle Scholar
  10. Hosseini, K., Kumar, D., Kaplan, M., Bejarbaneh, E.Y.: New exact traveling wave solutions of the unstable nonlinear Schrdinger equations. Commun. Theor. Phys. 68(6), 761 (2017b)ADSCrossRefGoogle Scholar
  11. Hosseini, K., Samadani, F., Kumar, D., Faridi, M.: New optical solitons of cubic–quartic nonlinear Schrdinger equation. Optik 157, 1101–1105 (2018)ADSCrossRefGoogle Scholar
  12. Kengne, E., Lakhssassi, A.: Analytical studies of soliton pulses along two-dimensional coupled nonlinear transmission lines. Chaos Solitons Fract. 73, 191–201 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. Kengne, E., Malomed, B.A., Chui, S.T., Liu, W.M.: Solitary signals in electrical nonlinear transmission line. J. Math. Phys. 48, 013508 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. Khater, A.H., Callebaut, D.K., Seadawy, A.R.: General soliton solutions of an n-dimensional complex Ginzburg–Landau equation. Phys. Scr. 62, 353–357 (2000)ADSMathSciNetCrossRefGoogle Scholar
  15. Khater, A.H., Callebaut, D.K., Seadawy, A.R.: Nonlinear dispersive Kelvin–Helmholtz instabilities in magnetohydrodynamic flows. Phys. Scr. 67, 340–349 (2003)ADSCrossRefMATHGoogle Scholar
  16. Khater, M.M.A., Seadawy, A.R., Lu, D.: Dispersive optical soliton solutions for higher order nonlinear Sasa–Satsuma equation in mono mode fibers via new auxiliary equation method. Superlattices Microstruct. 113, 346–358 (2018)CrossRefGoogle Scholar
  17. Kumar, D., Hosseini, K., Samadani, F.: The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzica type equations in nonlinear optics. Optik 149, 439–446 (2017)ADSCrossRefGoogle Scholar
  18. Kumar, D., Seadawy, Aly R., Joardar, A.K.: Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology. Chin. J. Phys. 56(1), 75–85 (2018)MathSciNetCrossRefGoogle Scholar
  19. Lu, D., Seadawy, A., Arshad, M.: Applications of extended simple equation method on unstable nonlinear Schrdinger equations. Optik 140, 136–144 (2017)ADSCrossRefGoogle Scholar
  20. Lu, D., Seadawy, A.R., Arshad, M.: Brightdark solitary wave and elliptic function solutions of unstable nonlinear Schrdinger equation and their applications. Opt. Quantum Electron. 50(1), 23 (2018)CrossRefGoogle Scholar
  21. Ma, W.-X., Yong, X., Zhang, H.-Q.: Diversity of interaction solutions to the (2 + 1)-dimensional Ito equation. Comput. Math. Appl. (2017).
  22. Ma, W.X., Fuchssteiner, B.: Explicit and exact solutions to a Kolmogorov–Petrovskii–Piskunov equation. Int. J. Non-Linear Mech. 31, 329–338 (1996)MathSciNetCrossRefMATHGoogle Scholar
  23. Ma, W.-X., Lee, J.-H.: A transformed rational function method and exact solutions to the 3 + 1 dimensional Jimbo Miwa equation. Chaos Solitons Fractals 42, 1356–1363 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. Ma, W.-X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264(4), 2633–2659 (2018)MathSciNetCrossRefMATHGoogle Scholar
  25. Ma, W.-X., Zhu, Z.: Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl. Math. Comput. 218, 11871–11879 (2012)MathSciNetMATHGoogle Scholar
  26. Malwe, B.H., Betchewe, G., Doka, S.Y., Kofane, T.C.: Soliton wave solutions for the nonlinear transmission line using the Kudryashov method and the \(\frac{G^{\prime }}{G}\)-expansion method. Appl. Math. Comput. 239, 299–309 (2014)MathSciNetMATHGoogle Scholar
  27. 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(1), 171–177 (2016)MathSciNetCrossRefMATHGoogle Scholar
  28. Pelap, F.B., Faye, M.: Soliton-like excitations in a one dimensional electrical transmission line. J. Math. Phys. 46, 033502-1 (2005)ADSCrossRefMATHGoogle Scholar
  29. Seadawy, A.R.: Three-dimensional nonlinear modified Zakharov–Kuznetsov equation of ion-acoustic waves in a magnetized plasma. Comput. Math. Appl. 71, 201–212 (2016)MathSciNetCrossRefGoogle Scholar
  30. Seadawy, A.: The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrodinger equation and its solutions. Optik Int. J. Light Electron Opt. 139, 31–43 (2017a)CrossRefGoogle Scholar
  31. Seadawy, A.: Modulation instability analysis for the generalized derivative higher order nonlinear Schrödinger equation and its the bright and dark soliton solutions. J. Electromagn. Waves Appl. 31(14), 1353–1362 (2017b)MathSciNetCrossRefGoogle Scholar
  32. Seadawy, A.R., Lu, D.: Ion acoustic solitary wave solutions of three-dimensional nonlinear extended Zakharov–Kuznetsov dynamical equation in a magnetized two-ion-temperature dusty plasma. Results Phys. 6, 590–593 (2016)ADSCrossRefGoogle Scholar
  33. 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. Eur. Phys. J. Plus 132, 162 (2017)CrossRefGoogle Scholar
  34. Sekulic, D.L., Satoric, M.V., Zivanov, M.B., Bajic, J.S.: Soliton-like pulses along electrical nonlinear transmission line. Elecron. Electr. Eng. 121, 53–58 (2012)Google Scholar
  35. Yan, Z., Zhang, H.: New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water. Phys. Lett. A 285(5), 355–362 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  36. Yang, J.-Y., Ma, W.-X., Qin, Z.: Lump and lump-soliton solutions to the (2 + 1)-dimensional Ito equation. Anal. Math. Phys. (2017).
  37. Younis, M., Ali, S.: Solitary wave and shock wave solitons to the transmission line model for nano-ionic currents along microtubules. Appl. Math. Comput. 246, 460–463 (2014)MathSciNetCrossRefMATHGoogle Scholar
  38. Younis, M., Rizvi, S.T.R., Ali, S.: Analytical and soliton solutions: nonlinear model of nanobioelectronics transmission lines. Appl. Math. Comput. 265, 994–1002 (2015)MathSciNetGoogle Scholar
  39. Zayed, E.M.E., Alurrfi, K.A.E.: A new Jacobi elliptic function expansion method for solving a nonlinear PDE describing pulse narrowing nonlinear transmission lines. J. Partial Differ. Equ. 28, 128–138 (2015)MathSciNetCrossRefMATHGoogle Scholar
  40. Zayed, E.M.E., Alurrfi, K.A.E.: The generalized projective Riccati equations method and its applications to nonlinear PDEs describing nonlinear transmission Lines. Commun. Appl. Electron. 3(4), 1–8 (2015)CrossRefGoogle Scholar
  41. Zhang, J., Ma, W.-X.: Mixed lump-kink solutions to the BKP equation. Comput. Math. Appl. 74, 591–596 (2017)MathSciNetCrossRefGoogle Scholar
  42. Zhao, H., Ma, W.-X.: Mixed lumpkink solutions to the KP equation. Comput. Math. Appl. 74, 1399–1405 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Dipankar Kumar
    • 1
    • 2
  • Aly R. Seadawy
    • 3
    • 4
  • Raju Chowdhury
    • 5
  1. 1.Graduate School of Systems and Information EngineeringUniversity of TsukubaTsukubaJapan
  2. 2.Department of PhysicsBangabandhu Sheikh Mujibur Rahman Science and Technology UniversityGopalganjBangladesh
  3. 3.Mathematics Department, Faculty of ScienceTaibah UniversityAl-Madinah Al-MunawarahSaudi Arabia
  4. 4.Mathematics Department, Faculty of ScienceBeni–Suef UniversityBeni SuefEgypt
  5. 5.Department of Natural ScienceStamford University BangladeshDhakaBangladesh

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