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Investigation of internal dynamics of soliton with the help of traveling wave soliton solution of Hamilton amplitude equation

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Abstract

The \(\varphi ^{6}\)-model expansion technique is used in this work to obtain dark, bright, periodic, singular, dark-bright, combined singular soliton and rational solutions to the Hamiltonian amplitude equation. This equation is used to analyze the stability of the exact solutions as well as to determine modulated wave train instabilities. The obtained results are mostly applicable in the study of nonlinear waves in plasma in the setting of a non-magnetized fluid-type plasma, the dispersion of the Langmuir (electrostatic) wave when it generates some low-frequency acoustic waves, such as the ion-acoustic wave, and other fields. The behavior of the traveling wave is analyzed using the frequency values, which is one of the internal dynamics of the dark soliton. This discussion is supported by graphs and the effects on wave behavior of different densities are simulated.

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References

  • Ali, K.K., Yilmazer, R.: M-lump solutions and interactions phenomena for the (2+ 1)-dimensional KdV equation with constant and time-dependent coefficients. Chin. J. Phys. 77, 2189–2200 (2022)

    Article  MathSciNet  Google Scholar 

  • Ali, K.K., Yilmazer, R., Osman, M.S.: Extended Calogero-Bogoyavlenskii-Schiff equation and its dynamical behaviors. Phys. Scripta 96(12), 125249 (2021)

    Article  ADS  Google Scholar 

  • Arnous, A.H., et al.: Optical solitons with complex Ginzburg-Landau equation by modified simple equation method. Optik 144, 475–480 (2017)

    Article  ADS  Google Scholar 

  • Asif, Y., Sulaiman, T.A., Bulut, H.: On the analytical and numerical solutions of the Benjamin-Bona-Mahony equation. Opt. Quantum Electr. 50(1), 31 (2018)

    Article  Google Scholar 

  • Doğan, K., Yokus, A.: A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations. Math. Comput. Simul. 60(6), 507–512 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Duran, S.: Dynamic interaction of behaviors of time-fractional shallow water wave equation system. Modern Phys. Lett. B 35(22), 2150353 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  • Duran, S., Karabulut, B.: Nematicons in liquid crystals with Kerr Law by sub-equation method. Alex. Eng. J. 61(2), 1695–1700 (2022)

    Article  Google Scholar 

  • Duran, S., Kaya, D.: Breaking analysis of solitary waves for the shallow water wave system in fluid dynamics. Eur. Phys. J. Plus 136(9), 1–12 (2021)

    Article  Google Scholar 

  • Durur, H., Yokuş, A.: Discussions on diffraction and the dispersion for traveling wave solutions of the (2+ 1)-dimensional paraxial wave equation. Math. Sci., pp. 1–11 (2021)

  • Eslami, M., Mirzazadeh, M.: The simplest equation method for solving some important nonlinear partial differential equations. Acta Univ. Apulensis 33, 117–130 (2013)

    MathSciNet  MATH  Google Scholar 

  • Feng, S.-Z., Li, Y.-G., Tian, L.-N., Zhou, Y.-B.: Periodic wave solutions for a new Hamiltonian amplitude equation. J. Lanzhou Univ. 43, 111–116 (2007)

    MathSciNet  MATH  Google Scholar 

  • Ghanbari, B., Dumitru, B.: A novel technique to construct exact solutions for nonlinear partial differential equations. Eur. Phys. J. Plus 134(10), 506 (2019)

    Article  Google Scholar 

  • Hulya, D.: Energy-carrying wave simulation of the Lonngren-wave equation in semiconductor materials. Int. J. Modern Phys. B 35(21), 2150213 (2021)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Jalil, M., Heidari, S.: Periodic and singular kink solutions of the Hamiltonian amplitude equation. Adv. Math. Models Appl. 4(2), 134–149 (2019)

    Google Scholar 

  • Kaya, D., Yokuş, A., Demiroğlu, U.: Comparison of exact and numerical solutions for the Sharma-Tasso-Olver equation, pp. 53–65. Cham, Numerical Solutions of Realistic Nonlinear Phenomena. Springer, Berlin (2020)

  • Krishnan, E.V., Yan, Z.Y.: Jacobian elliptic function solutions using Sinh-Gordon equation expansion method. Int. J. Appl. Math. Mech. 2, 1–10 (2006)

    Google Scholar 

  • Kumar, S., Singh, K., Gupta, R.K.: Coupled higgs field equation and hamiltonian amplitude equation: Lie classical approach and (G’ / G) -expansion method. Pramana J. Phys. 79, 41–60 (2012)

    Article  ADS  Google Scholar 

  • Latifi, A., Leon, J.: On the interaction of Langmuir waves with acoustic waves in plasmas. Phys. Lett. A 152(3–4), 171–177 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  • Liu, D., Ju, X., Ilhan, O.A., Manafian, J., Ismael, H.F.: Multi-waves, breathers, periodic and cross-kink solutions to the (2+ 1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation. J. Ocean Univ. China 20(1), 35–44 (2021)

    Article  ADS  Google Scholar 

  • Mirzazadeh, M.: Modified simple equation method and its applications to nonlinear partial differential equations. Inf. Sci. Lett. 3, 1–9 (2014)

    Article  Google Scholar 

  • Mirzazadeh, M.: Topological and non-topological soliton solutions of hamiltonian amplitude equation by He’s Semi-inverse method and ansatz approach. J. Egypt. Math. Soc. 23, 292–296 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Morozov, I.V., Norman, G.E.: Collisions and Langmuir waves in nonideal plasmas. J. Exp. Theor. Phys. 100(2), 370–384 (2005)

    Article  ADS  Google Scholar 

  • Peng, Y.: Exact periodic solutions to a new Hamiltonian amplitude equation. J. Phys. Soc. Jpn. 72, 1356–1359 (2003)

    Article  ADS  MATH  Google Scholar 

  • Sachin, K., Singh, K., Gupta, R.K.: Coupled Higgs field equation and Hamiltonian amplitude equation: Lie classical approach and (G’/G)-expansion method. Pramana 79(1), 41–60 (2012)

    Article  ADS  Google Scholar 

  • Serbay, D., Yokuş, A., Hülya, D., Kaya, D.: Refraction simulation of internal solitary waves for the fractional Benjamin–Ono equation in fluid dynamics. Modern Phys. Lett. B, p. 2150363 (2021)

  • Sun, J.: Auxiliary equation method for solving nonlinear partial differential equations. Phys. Lett. A 309(5–6), 387–396 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Taghizadeh, N., Mirzazadeh, M.: The first integral method to some complex nonlinear partial differential equations. J. Comput. Appl. Math. 235, 4871–4877 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Taghizadeh, N., Najand, M.: Exact solutions of the new Hamiltonian amplitude equation by the (G’ / G) -expansion method. Int. J. Appl. Math. Comput. 4, 390–395 (2012)

    MATH  Google Scholar 

  • Tariq, H., Ahmed, H., Rezazadeh, H., Javeed, S., Alimgeer, K.S., Nonlaopon, K., Khedher, K.M.: New travelling wave analytic and residual power series solutions of conformable Caudrey-Dodd-Gibbon-Sawada-Kotera equation. Results Phys., p. 104591 (2021)

  • Tarla, S., Ali, K.K., Yilmazer, R., Osman, M.S.: The dynamic behaviors of the Radhakrishnan-Kundu-Lakshmanan equation by Jacobi elliptic function expansion technique. Opt. Quant. Electron. 54(5), 1–12 (2022)

    Article  Google Scholar 

  • Wadati, M., Segur, H., Ablowitz, M.J.: A new hamiltonian amplitude equation governing modulated wave instabilities. J. Phys. Soc. Jpn. 61, 1187–1193 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Wan, R., Manafian, J., Ismael, H.R., Mohammed, S.A.: Investigating one-, two-, and triple wave solutions via multiple exp-function method arising in engineering sciences. Adv. Math. Phys. 2020, 1–18 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Wei, G., Veeresha, P., Baskonus, H.M., Prakasha, D.G., Kumar, P.: A new study of unreported cases of 2019-nCOV epidemic outbreaks. Chaos Solitons Fract. 138, 109929 (2020)

    Article  MathSciNet  Google Scholar 

  • Yokus, A., Münevver, T.: An application of a new version of (G’/G)-expansion method. AIP Conference Proceedings. Vol. 1798. No. 1. AIP Publishing LLC, (2017)

  • Yokuş, A.: Comparison of Caputo and conformable derivatives for time-fractional Korteweg-de Vries equation via the finite difference method. Int. J. Modern Phys. B 32(29), 1850365 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Yokuş, A., Hülya, D., Abro, K.A.: Symbolic computation of Caudrey-Dodd-Gibbon equation subject to periodic trigonometric and hyperbolic symmetries. Eur. Phys. J. Plus 136(4), 1–16 (2021)

    Article  Google Scholar 

  • Yokuş, A., Durur, H., Duran, S., Islam, M.: Ample felicitous wave structures for fractional foam drainage equation modeling for fluid-flow mechanism. Comput. Appl. Math. 41(4), 1–13 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  • Yomba, E.: The general projective riccati equations method and exact solutions for a class of nonlinear partial differential equations. Chin. J. Phys. 43, 991–1003 (2005)

    MathSciNet  Google Scholar 

  • Yuming, C., et al.: Application of modified extended tanh technique for solving complex ginzburg-landau equation considering kerr law nonlinearity. CMC-Comput. Mater. Continua 66(2), 1369–1378 (2021)

    Article  Google Scholar 

  • Zafar, A., et al.: On optical soliton solutions of new Hamiltonian amplitude equation via Jacobi elliptic functions. Eur. Phys. J. Plus 135(8), 1–17 (2020)

    Article  ADS  Google Scholar 

  • Zayed-Elsayed, M.E., Al-Nowehy, A.-G.: Many new exact solutions to the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms using three different techniques. Optik 143, 84–103 (2017)

    Article  ADS  Google Scholar 

  • Zayed-Elsayed, M.E., Al-Nowehy, A.-G., Elshater, M.E.M.: New \(\varphi ^{6}\)-model expansion method and its applications to the resonant nonlinear Schrödinger equation with parabolic law nonlinearity. Eur. Phys. J. Plus 133(10), 417 (2018)

    Google Scholar 

  • Zhou, Q., et al.: Optical solitons with nonlinear dispersion in polynomial law medium. J. Optoelectron. Adv. Mater. 17, 82–86 (2015)

    Google Scholar 

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  1. Department of Mathematics, Faculty of Science, Firat University, Elazig, Turkey

    Asif Yokus & Muhammad Abubakar Isah

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  1. Asif Yokus
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  2. Muhammad Abubakar Isah
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Correspondence to Asif Yokus.

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Yokus, A., Isah, M.A. Investigation of internal dynamics of soliton with the help of traveling wave soliton solution of Hamilton amplitude equation. Opt Quant Electron 54, 528 (2022). https://doi.org/10.1007/s11082-022-03944-w

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