Skip to main content
SpringerLink
Log in

New perspective on fractional Hamiltonian amplitude equation

  • Published:
Optical and Quantum Electronics Aims and scope Submit manuscript

Abstract

In this paper, we present a pioneering investigation on the fractional Hamiltonian amplitude equation involving the beta fractional derivative for the first time, addressing a research gap in the field of nonlinear fractional dynamics. Our primary objective is to develop effective analytical techniques capable of solving the fractional Hamiltonian amplitude equation and obtaining novel soliton solutions. To achieve this, we introduce two advanced methods: the extended fractional rational \(\sin e_{\delta } - \cos ine_{\delta }\) and the fractional rational \(\sinh_{\delta } - \cosh_{\delta }\) techniques. By employing these cutting-edge approaches, we successfully derive new types of soliton solutions, demonstrating the reliability and efficiency of the proposed methods. Furthermore, the applicability of these techniques extends to various fractional nonlinear evolution models, highlighting their versatility in the realm of fractional dynamics. Finally, we provide a comprehensive presentation of the results, which substantiate the effectiveness of the methods in solving the complex fractional Hamiltonian amplitude equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Data availability

No applicable for this paper.

References

  • Ahmad, H., Khan, T.A., Stanimirovic, P., Ahmad, I.: Modified variational iteration technique for the numerical solution of fifth order KdV type equations. J. Appl. Comput. Mech. 2020, 2197 (2020a)

    Google Scholar 

  • Ahmad, H., Seadawy, A.R., Khan, T.A.: Study on numerical solution of dispersive water wave phenomena by using a reliable modification of variational iteration algorithm. Math. Comput. Simul. 177, 13–23 (2020b)

    Article  MathSciNet  MATH  Google Scholar 

  • Ain, Q.T., Anjum, N., He, C.H.: An analysis of time-fractional heat transfer problem using two-scale approach. GEM. Int. J. Geomathema. 12(1), 1–10 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Akar, M., Ozkan, E.M.: On exact solutions of the (2+1)-dimensional time conformable Maccari system. Int. J. Mod. Phys. B 2023, 2350219 (2023)

    Article  Google Scholar 

  • Demiray, S.T., Bulut, H.: New exact solutions of the new Hamiltonian amplitude equation and Fokas-Lenells equation. Entropy 17(9), 6025–6043 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Duran, S.: An investigation of the physical dynamics of a traveling wave solution called a bright soliton. Phys. Scr. 96(12), 125251 (2021)

    Article  ADS  Google Scholar 

  • Durur, H., Yokuş, A., Duran, S.: Investigation of exact soliton solutions of nematicons in liquid crystals according to nonlinearity conditions. Int. J. Mod. Phys. B 2023, 2450054 (2023)

    Article  Google Scholar 

  • Fendzi-Donfack, E., Nguenang, J.P., Nana, L.: Fractional analysis for nonlinear electrical transmission line and nonlinear Schroedinger equations with incomplete sub-equation. Eur. Phys. J. plus. 133(2), 32 (2018)

    Article  Google Scholar 

  • Fendzi-Donfack, E., Nguenang, J.P., Nana, L.: On the soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission line. Nonlinear Dyn. 104(1), 691–704 (2021)

    Article  Google Scholar 

  • Fendzi-Donfack, E., Temgoua, G.W.K., Djoufack, Z.I., Kenfack-Jiotsa, A., Nguenang, J.P., Nana, L.: Exotical solitons for an intrinsic fractional circuit using the sine-cosine method. Chaos, Solitons Fractals 160, 112253 (2022a)

    Article  MathSciNet  MATH  Google Scholar 

  • Fendzi-Donfack, E., Kumar, D., Tala-Tebue, E., Nana, L., Nguenang, J.P., Kenfack-Jiotsa, A.: Construction of exotical soliton-like for a fractional nonlinear electrical circuit equation using differential-difference Jacobi elliptic functions sub-equation method. Results Phys. 32, 105086 (2022b)

    Article  Google Scholar 

  • Fendzi-Donfack, E., Tala-Tebue, E., Inc, M., Kenfack-Jiotsa, A., Nguenang, J.P., Nana, L.: Dynamical behaviours and fractional alphabetical-exotic solitons in a coupled nonlinear electrical transmission lattice including wave obliqueness. Opt. Quantum Electron. 55(2), 35 (2023)

    Article  Google Scholar 

  • He, J.H., Jiao, M.L., Gepreel, K.A., Khan, Y.: Homotopy perturbation method for strongly nonlinear oscillators. Math. Comput. Simul. 204, 243–258 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  • Irshad, S., Shakeel, M., Bibi, A., Sajjad, M., Nisar, K.S.: A comparative study of nonlinear fractional SchrÖodinger equation in optics, Modern Physics Letters B37(5), 2250219 (2023)

  • Khan, A., Ain, Q.T., Abdeljawad, T., Nisar, K.S.: Exact controllability of Hilfer fractional differential system with non-instantaneous impluleses and state dependent delay. Qual. Theory Dyn. Syst. 22(2), 62 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  • Krishnan, E.V., Peng, Y.Z.: A new solitary wave solution for the new Hamiltonian amplitude equation. J. Phys. Soc. Jpn. 74(3), 896–897 (2005)

    Article  ADS  MATH  Google Scholar 

  • Kumar, S.: A new analytical modelling for fractional telegraph equation via Laplace transform. Appl. Math. Model. 38, 3154–3163 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar, D., Kumar, S.: Some new periodic solitary wave solutions of (3+1)-dimensional generalized shallow water wave equation by Lie symmetry approach. Comput. Math. Appl. 78, 857–877 (2019)

    MathSciNet  MATH  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(1), 41–60 (2012)

    Article  ADS  Google Scholar 

  • Kumar, S., Kumar, R., Agarwal, R.P., Samet, B.: A study of fractional Lotka–Volterra population model using Haar wavelet and Adams-Bashforth–Moulton methods. Math. Methods Appl. Sci. 43(8), 5564–5578 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Liu, J.G., Yang, X.J., Wang, J.J.: A new perspective to discuss Korteweg-de Vries-like equation. Phys. Lett. A 451, 128429 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  • Lu, J.F., Chen, L.: Numerical analysis of a fractal modification of Yao–Cheng oscillator. Results Phys. 38, 105602 (2022)

    Article  Google Scholar 

  • Nadeem, M., He, J.H., He, C.H., Sedighi, H.M., Shirazi, A.H.: A numerical solution of nonlinear fractional Newell-Whitehead-Segel equation using natural transform. TWMS. J. Pure. Appl. Math. 13(2), 168–182 (2022)

    Google Scholar 

  • Ozkan, E.M.: New exact solutions of some important nonlinear fractional partial differential equations with Beta derivative. Fractal Fract. 6(3), 173 (2022)

    Article  Google Scholar 

  • Ozkan, E.M., Ozkan, A.: The soliton solutions for some nonlinear fractional differential equations with Beta-Derivative. Axioms 10(3), 203 (2021)

    Article  Google Scholar 

  • Seadawy, A.R., Cheemaa, N.: Some new families of spiky solitary waves of one-dimensional higher-order K-dV equation with power law nonlinearity in plasma physics. Indian J. Phys. 94(1), 117–126 (2020)

    Article  ADS  Google Scholar 

  • Taghizadeh, N., Mirzazadeh, M.: The first integral method to some complex nonlinear partial differential equations. J. Comput. Appl. Math. 235(16), 4871–4877 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Tarla, S., Ali, K.K., Yilmazer, R., Osman, M.S.: Propagation of solitons for the Hamiltonian amplitude equation via an analytical technique. Mod. Phys. Lett. B. 36(23), 2250120 (2022)

    Article  ADS  Google Scholar 

  • Wang, K.J.: Diverse wave structures to the modified Benjamin-Bona-Mahony equation in the optical illusions field. Mod. Phys. Lett. B. 37(11), 2350012 (2023a)

    Article  ADS  MathSciNet  Google Scholar 

  • Wang, K.L.: New fractal soliton solutions for the coupled fractional Klein–Gordon equation with beta-fractional derivative. Fractals 31(1), 2350003 (2023b). https://doi.org/10.1142/S0218348X23500032

    Article  ADS  MATH  Google Scholar 

  • Wang, K.J., Si, J.: Dynamic properties of the attachment oscillator arising in the nanophysics. Open Phys. 21(1), 20220214 (2023)

    Article  Google Scholar 

  • Wazwaz, A.M.: Solitary wave solutions of the generalized shallow water wave (GSWW) equation by Hirota’s method, tanh-coth method, and Exp-function method. Appl. Math. Comput. 202, 275–286 (2008)

    MathSciNet  MATH  Google Scholar 

  • Yadav, P., Jahan, S., Nisar, K.S.: Fibonacci wavelet collocation method for Fredholm integral equations of second kind. Qual. Theory Dyn. Syst. 22(2), 82 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  • Younas, U., Seadawy, A.R., Younis, M., Rizvi, S.T.: Dispersive of propagation wave structures to the Dullin-gottwald-holm dynamical equation in a shallow water waves. Chin. J. Phys. 68, 348–364 (2020)

    Article  MathSciNet  Google Scholar 

  • Younas, U., Sulaiman, T.A., Ren, J.L.: Propagation of M-truncated optical pulses in nonlinear optics. Opt. Quantum Electron. 55(2), 102 (2023)

    Article  Google Scholar 

Download references

Funding

No funding received for this paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China

    Kang-Le Wang

Authors
  1. Kang-Le Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

KangLe Wang:Conceptualization, Formal analysis, Investigation, Methodology, Project administration, Software, Writingoriginal draft, Writing-review and editing. All authors reviewed the manuscript.

Corresponding author

Correspondence to Kang-Le Wang.

Ethics declarations

Conflict of interest

The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Ethical approval

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, KL. New perspective on fractional Hamiltonian amplitude equation. Opt Quant Electron 55, 1033 (2023). https://doi.org/10.1007/s11082-023-05309-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-023-05309-3

Keywords

Search

Navigation