Abstract
In this study, we investigate rogue wave dynamics and modulational instability using the Manakov system in a nonlinear electrical transmission line with second couplings. Using semi-discrete approximation, we demonstrate how the dynamics of rogue waves in this type of transmission line can be governed by the Manakov system. To study the dynamics of rogue waves in this structure via this approximation, we used the parameters of this transmission line and derived new forms of propagating rogue wave solutions. The solutions obtained are presented as new rogue waves of types I and II. In this work, we show that the dynamics of different types of rogue waves in different types of nonlinear electrical transmission lines can be studied using the Manakov system. Indeed, with the choice of small values of inductance \((L_{3})\) in the two types of rogue waves, the effects of the second coupling are clearly visible during the formation of these waves, namely at the level shapes, hollows, and amplitude. Additionally, it can be observed that the dispersion capacity \((C_{S})\) also affects the shapes, troughs, peaks, and widths of these rogue waves as the troughs gradually disappear, and the peak widths decrease when the dispersion capacity \((C_{S})\) increases. Finally, concerning the modulational instability in this structure, the essential information that we can retain is that these second couplings \((L_{3})\) would impact the zones of instability, which could gradually disappear along this line. To avoid overload, we limited ourselves to these major effects. The results obtained by this Manakov system show not only its efficiency and robustness, but also its potential applicability to other types of useful nonlinear electrical transmission lines, and that these new forms of rogue waves do indeed exist in nonlinear electrical transmission lines with second couplings. This feature has not been sufficiently addressed in this type of nonlinear electrical transmission line and will be useful in many branches of physics.
Similar content being viewed by others
Data Availability Statement
This manuscript has associated data in a data repository. [Authors’ comment: There is no additional new data associated with this article.]
References
S. Malik, H. Almusawa, S. Kumar, A.-M. Wazwaz, M.S. Osman, A (2+ 1)-dimensional Kadomtsev–Petviashvili equation with competing dispersion effect: Painlevé analysis, dynamical behavior and invariant solutions. Results Phys. 23, 104043 (2021)
S. Djennadi, N. Shawagfeh, M.S. Osman, J.F. Gomez-Aguilar, O.A. Arqub, The Tikhonov regularization method for the inverse source problem of time fractional heat equation in the view of ABC-fractional technique. Phys. Scr. 96(9), 094006 (2021)
H.F. Ismael, H. Bulut, C. Park, M.S. Osman, M-lump, N-soliton solutions, and the collision phenomena for the (2+ 1)-dimensional Date-Jimbo–Kashiwara–Miwa equation. Results Phys. 19, 103329 (2020)
K.K. Ali, S. Abd El, A. Mohamed, E.M.H. Mohamed, B. Samet, S. Kumar, M.S. Osman, Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series. Adv. Differ. Equ. 2020(1), 1–23 (2020)
S. Kumar, M. Niwas, M.S. Osman, M.A. Abdou, Abundant different types of exact soliton solution to the (4+ 1)-dimensional Fokas and (2+ 1)-dimensional breaking soliton equations. Commun. Theor. Phys. 73(10), 105007 (2021)
R.U. Rahman, M.M.M. Qousini, A. Alshehri, S.M. Eldin, K. El-Rashidy, M.S. Osman, Evaluation of the performance of fractional evolution equations based on fractional operators and sensitivity assessment. Results Phys. 49, 106537 (2023)
S. Qureshi, M.A. Akanbi, A.A. Shaikh, A.S. Wusu, O.M. Ogunlaran, W. Mahmoud, M.S. Osman, A new adaptive nonlinear numerical method for singular and stiff differential problems. Alex. Eng. J. 74, 585–597 (2023)
F. Tasnim, M.A. Akbar, M.S. Osman, The extended direct algebraic method for extracting analytical solitons solutions to the cubic nonlinear Schrödinger equation involving beta derivatives in space and time. Fractal Fractional 7(6), 426 (2023)
H.F. Ismael, T. Abdulkadir Sulaiman, H.R. Nabi, W. Mahmoud, M.S. Osman, Geometrical patterns of time variable Kadomtsev–Petviashvili (I) equation that models dynamics of waves in thin films with high surface tension. Nonlinear Dyn. 111(10), 9457–9466 (2023)
A. Tripathy, S. Sahoo, H. Rezazadeh, Z.P. Izgi, M.S. Osman, Dynamics of damped and undamped wave natures in ferromagnetic materials. Optik 281, 170817 (2023)
L. Akinyemi, A. Houwe, S. Abbagari, A.M. Wazwaz, H.M. Alshehri, M.S. Osman, Effects of the higher-order dispersion on solitary waves and modulation instability in a monomode fiber. Optik 288, 171202 (2023)
L. Akinyemi, A. Houwe, S. Abbagari, A.M. Wazwaz, H.M. Alshehri, M.S. Osman, A study on stochastic longitudinal wave equation in a magneto-electro-elastic annular bar to find the analytical solutions. Commun. Theor. Phys. 2, 23 (2023)
M.A. Chowdhury, M.M. Miah, M.A. Iqbal, H.M. Alshehri, D. Baleanu, M.S. Osman, Advanced exact solutions to the nano-ionic currents equation through MTs and the soliton equation containing the RLC transmission line. Eur. Phys. J. Plus 138(6), 1–11 (2023)
L. Draper, Freak ocean. Mar (1965)
P. Müller, C. Garrett, A. Osborne, Rogue waves. Oceanography 18(3), 66 (2005)
D.R. Solli, C. Ropers, P. Koonath, B. Jalali, Optical rogue waves. Nature 450(7172), 1054–1057 (2007)
B. Frisquet, B. Kibler, G. Millot, Collision of Akhmediev breathers in nonlinear fiber optics. Phys. Rev. X 3(4), 041032 (2013)
W.M. Moslem, R. Sabry, S.K. El-Labany, P.K. Shukla, Dust-acoustic rogue waves in a nonextensive plasma. Phys. Rev. E 84(6), 066402 (2011)
L. Stenflo, M. Marklund, Rogue waves in the atmosphere. J. Plasma Phys. 76(3–4), 293–295 (2010)
A.N. Ganshin, V.B. Efimov, G.V. Kolmakov, L.P. Mezhov-Deglin, P.V.E. McClintock, Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium. Phys. Rev. Lett. 101(6), 065303 (2008)
M. Shats, H. Punzmann, H. Xia, Capillary rogue waves. Phys. Rev. Lett. 104(10), 104503 (2010)
K. Manikandan, P. Muruganandam, M. Senthilvelan, M. Lakshmanan, Manipulating matter rogue waves and breathers in Bose–Einstein condensates. Phys. Rev. E 90(6), 062905 (2014)
Z.-Y. Yan, Financial rogue waves. Commun. Theor. Phys. 54(5), 947 (2010)
E. Kengne, W.M. Liu, Transmission of rogue wave signals through a modified Noguchi electrical transmission network. Phys. Rev. E 99(6), 062222 (2019)
F.I.I. Ndzana, G. Djelah, A. Mohamadou, Solitonic rogue waves dynamics in a nonlinear electrical transmission line with the next nearest neighbor couplings. Chin. J. Phys. 77, 1927–1945 (2022)
E. Kengne, W.M. Liu, L.Q. English, B.A. Malomed, Ginzburg–Landau models of nonlinear electric transmission networks. Phys. Rep. 982, 1–124 (2022)
T.B. Benjamin, Instability of periodic wavetrains in nonlinear dispersive systems. Proc. Roy. Soc. Lond. Ser. A Math. Phys. Sci. 299(1456), 59–76 (1967)
N.N. Akhmediev, V.I. Korneev, Modulation instability and periodic solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 69(2), 1089–1093 (1986)
N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Rogue waves and rational solutions of the nonlinear Schrödinger equation. Phys. Rev. E 80(2), 026601 (2009)
A. Ankiewicz, J.M. Soto-Crespo, N. Akhmediev, Rogue waves and rational solutions of the Hirota equation. Phys. Rev. E 81(4), 046602 (2010)
S. Xu, J. He, L. Wang, The Darboux transformation of the derivative nonlinear Schrödinger equation. J. Phys. A Math. Theor. 44(30), 305203 (2011)
U. Bandelow, N. Akhmediev, Sasa–Satsuma equation: soliton on a background and its limiting cases. Phys. Rev. E 86(2), 026606 (2012)
L. Liu, B. Tian, Y.-Q. Yuan, Z. Du, Dark-bright solitons and semirational rogue waves for the coupled Sasa-Satsuma equations. Phys. Rev. E 97(5), 052217 (2018)
X. Wang, B. Yang, Y. Chen, Y. Yang, Higher-order rogue wave solutions of the Kundu–Eckhaus equation. Phys. Scr. 89(9), 095210 (2014)
L.-C. Zhao, C. Liu, Z.-Y. Yang, The rogue waves with quintic nonlinearity and nonlinear dispersion effects in nonlinear optical fibers. Commun. Nonlinear Sci. Numer. Simul. 20(1), 9–13 (2015)
P. Gaillard, Families of quasi-rational solutions of the NLS equation and multi-rogue waves. J. Phys. A Math. Theor. 44(43), 435204 (2011)
V.B. Matveev, M.A. Salle et al., Darboux Transformations and Solitons (Springer, Berlin, 1991), p.17
N.N. Akhmediev, N.V. Mitzkevich, Extremely high degree of N-soliton pulse compression in an optical fiber. IEEE J. Quant. Electron. 27(3), 849–857 (1991)
B. Yang, W.-G. Zhang, H.-Q. Zhang, S.-B. Pei, Generalized Darboux transformation and rogue wave solutions for the higher-order dispersive nonlinear Schrödinger equation. Phys. Scr. 88(6), 065004 (2013)
J.S. He, H.R. Zhang, L.H. Wang, K. Porsezian, A.S. Fokas, Generating mechanism for higher-order rogue waves. Phys. Rev. E 87(5), 052914 (2013)
E. Kengne, W.M. Liu, Engineering rogue waves with quintic nonlinearity and nonlinear dispersion effects in a modified Nogochi nonlinear electric transmission network. Phys. Rev. E 102(1), 012203 (2020)
D. Ahmadou, H. Alphonse, M. Justin, G. Betchewe, D.Y. Serge, K.T. Crepin, M. Inc, New coupled rogue waves propagating backward and forward and modulation instability in a composite nonlinear right-and left-handed transmission line. Eur. Phys. J. Plus 136, 1–26 (2021)
G. Djelah, F. Ndzana, S. Abdoulkary, A. Mohamadou, First and second order rogue waves dynamics in a nonlinear electrical transmission line with the next nearest neighbor couplings. Chaos Solitons Fractals 167, 113087 (2023)
G.P. Veldes, J. Cuevas, P.G. Kevrekidis, D.J. Frantzeskakis, Quasidiscrete microwave solitons in a split-ring-resonator-based left-handed coplanar waveguide. Phys. Rev. E 83(4), 046608 (2011)
M. Remoissenet, M. Remoissenet, Solitons in nonlinear transmission lines. Waves Called Solitons Concepts Exp. 8, 37–64 (1996)
E. Kengne, A. Lakhssassi, W.M. Liu, Dynamics of modulated waves in a lossy modified Noguchi electrical transmission line. Phys. Rev. E 91(6), 062915 (2015)
T. Taniuti, N. Yajima, Perturbation method for a nonlinear wave modulation I. J. Math. Phys. 10(8), 1369–1372 (1969)
A.I. Dyachenko, V.E. Zakharov, Modulation instability of Stokes wave\(\rightarrow \) freak wave. J. Exp. Theor. Phys. Lett. 81(6), 255–259 (2005)
J.K. Duan, B.Y. Long, Q. Wei, M.H. Fan, Super rogue waves in coupled electric transmission lines. Indian J. Phys. 94, 879–883 (2020)
G.P. Veldes, J. Cuevas, P.G. Kevrekidis, D.J. Frantzeskakis, Coupled backward-and forward-propagating solitons in a composite right-and left-handed transmission line. Phys. Rev. E 88(1), 013203 (2013)
D. Wen-Shan, H. Xue-Ren, S. Yu-Ren, L. Ke-Pu, S. Jian-An, Weakly two-dimensional solitary waves on coupled nonlinear transmission lines. Chin. Phys. Lett. 19(9), 1231 (2002)
W.-S. Duan, Nonlinear waves propagating in the electrical transmission line. Europhys. Lett. 66(2), 192 (2004)
J.K. Duan, Y.L. Bai, Rogue wave in coupled electric transmission line. Indian J. Phys. 92(3), 369–375 (2018)
K. Manikandan, M. Senthilvelan, R.A. Kraenkel, On the characterization of vector rogue waves in two-dimensional two coupled nonlinear Schrödinger equations with distributed coefficients. Eur. Phys. J. B 89(10), 1–11 (2016)
L. Wang, J. He, H. Xu, J. Wang, K. Porsezian, Generation of higher-order rogue waves from multibreathers by double degeneracy in an optical fiber. Phys. Rev. E 95(4), 042217 (2017)
Z. Rahman, M. Zulfikar Ali, H.-O. Roshid, Closed form soliton solutions of three nonlinear fractional models through proposed improved Kudryashov method. Chin. Phys. B 30(5), 050202 (2021)
K. Tai, A. Hasegawa, A. Tomita, Observation of modulational instability in optical fibers. Phys. Rev. Lett. 56(2), 135 (1986)
T.B. Benjamin, J.E. Feir, The disintegration of wave trains on deep water. J. Fluid Mech. 27(3), 417–430 (1967)
D.H. Peregrine, Interaction of water waves and currents. Adv. Appl. Mech. 16, 9–117 (1976)
L. Salasnich, A. Parola, L. Reatto, Modulational instability and complex dynamics of confined matter-wave solitons. Phys. Rev. Lett. 91(8), 080405 (2003)
T. Taniuti, H. Washimi, Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma. Phys. Rev. Lett. 21(4), 209 (1968)
S. Watanabe, Self-modulation of a nonlinear ion wave packet. J. Plasma Phys. 17(3), 487–501 (1977)
H. Bailung, Y. Nakamura, Observation of modulational instability in a multi-component plasma with negative ions. J. Plasma Phys. 50(2), 231–242 (1993)
L.-C. Zhao, L. Ling, Quantitative relations between modulational instability and several well-known nonlinear excitations. JOSA B 33(5), 850–856 (2016)
L.-C. Zhao, G.-G. Xin, Z.-Y. Yang, Rogue-wave pattern transition induced by relative frequency. Phys. Rev. E 90(2), 022918 (2014)
M.S. Ullah, M. Mostafa, M. Zulfikar Ali, H.O. Roshid, M. Akter, Soliton solutions for the Zoomeron model applying three analytical techniques. Plos One 18(7), e0283594 (2023)
M.S. Ullah, D. Baleanu, M. Zulfikar Ali et al., Novel dynamics of the Zoomeron model via different analytical methods. Chaos Solitons Fractals 174, 113856 (2023)
E. Kengne, W.M. Liu, Solitonlike pulses along a modified Noguchi nonlinear electrical network with second-neighbor interactions: analytical studies. Phys. Rev. E 97(5), 052205 (2018)
X.-L. Chen, S. Abdoulkary, P.G. Kevrekidis, L.Q. English, Resonant localized modes in electrical lattices with second-neighbor coupling. Phys. Rev. E 98(5), 052201 (2018)
G. Dematteis, T. Grafke, M. Onorato, E. Vanden-Eijnden, Experimental evidence of hydrodynamic instantons: the universal route to rogue waves. Phys. Rev. X 9(4), 041057 (2019)
A. Tikan, F. Bonnefoy, G. Roberti, G. El, A. Tovbis, G. Ducrozet, A. Cazaubiel, G. Prabhudesai, G. Michel, Prediction and manipulation of hydrodynamic rogue waves via nonlinear spectral engineering. Phys. Rev. Fluids 7(5), 054401 (2022)
A. Romero-Ros, G.C. Katsimiga, S.I. Mistakidis, B. Prinari, G. Biondini, P. Schmelcher, P.G. Kevrekidis, Theoretical and numerical evidence for the potential realization of the Peregrine soliton in repulsive two-component Bose-Einstein condensates. Phys. Rev. A 105(5), 053306 (2022)
S. Coulibaly, M. Taki, A. Bendahmane, G. Millot, B. Kibler, M.G. Clerc, Turbulence-induced rogue waves in Kerr resonators. Phys. Rev. X 9(1), 011054 (2019)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ahmadou, D., Alphonse, H., Justin, M. et al. Dynamics of rogue waves and modulational instability with the Manakov system in a nonlinear electric transmission line with second couplings. Eur. Phys. J. Plus 138, 1113 (2023). https://doi.org/10.1140/epjp/s13360-023-04773-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-023-04773-w