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Quasi-periodic, chaotic and travelling wave structures of modified Gardner equation

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  • Jhangeer, Adil
  • Hussain, Amjad
  • Junaid-U-Rehman, M.
  • Baleanu, Dumitru
  • Riaz, Muhammad Bilal

Abstract

In this paper, the nonlinear modified Gardner (mG) equation is under consideration which represents the super nonlinear proliferation of the ion-acoustic waves and quantum electron-positronion magneto plasmas. The considered model is investigated with the help of Lie group analysis. Lie point symmetries are computed under the invariance criteria of Lie groups and symmetry group for each generator is reported. Furthermore, the one-dimensional optimal system of subalgebras is developed by adjoint technique and then we compute the similarity reductions corresponding to each vector field present in the optimal system, with the help of similarity reduction method we have to convert the PDE into the ODE. Some exact explicit solutions of obtained ordinary differential equations were constructed by the power series technique. With the aid of the Galilean transformation, the model is transformed into a planer dynamical system and the bifurcation behaviour is recorded. All practicable types of phase portraits with regard to the parameters of the problem considered are plotted. Meantime, sensitivity is observed by utilizing sensitivity analysis. In addition, the influence of physical parameters is studied by the application of an extrinsic periodic power. With additional perturbed term, quasi-periodic and quasi-periodic-chaotic behaviours is reported.

Suggested Citation

  • Jhangeer, Adil & Hussain, Amjad & Junaid-U-Rehman, M. & Baleanu, Dumitru & Riaz, Muhammad Bilal, 2021. "Quasi-periodic, chaotic and travelling wave structures of modified Gardner equation," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    • Handle: RePEc:eee:chsofr:v:143:y:2021:i:c:s0960077920309693
    DOI: 10.1016/j.chaos.2020.110578
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    References listed on IDEAS

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    1. Wazwaz, Abdul-Majid, 2005. "The tanh method: solitons and periodic solutions for the Dodd–Bullough–Mikhailov and the Tzitzeica–Dodd–Bullough equations," Chaos, Solitons & Fractals, Elsevier, vol. 25(1), pages 55-63.
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    Cited by:

    1. Kudryashov, Nikolay A. & Kutukov, Aleksandr A. & Biswas, Anjan & Zhou, Qin & Yıldırım, Yakup & Alshomrani, Ali Saleh, 2023. "Optical solitons for the concatenation model: Power-law nonlinearity," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    2. Weifang Yan & Linlin Wang & Min Zhang, 2024. "Existence of Kink and Antikink Wave Solutions of Singularly Perturbed Modified Gardner Equation," Mathematics, MDPI, vol. 12(6), pages 1-9, March.
    3. Hassan Almusawa & Adil Jhangeer, 2024. "Exploring Wave Interactions and Conserved Quantities of KdV–Caudrey–Dodd–Gibbon Equation Using Lie Theory," Mathematics, MDPI, vol. 12(14), pages 1-12, July.

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