Skip to main content
SpringerLink
Log in

Propagation of traveling wave solutions to the Vakhnenko-Parkes dynamical equation via modified mathematical methods

  • Published:
Applied Mathematics-A Journal of Chinese Universities Aims and scope Submit manuscript

Abstract

In this paper, we investigate some new traveling wave solutions to Vakhnenko-Parkes equation via three modified mathematical methods. The derived solutions have been obtained including periodic and solitons solutions in the form of trigonometric, hyperbolic, and rational function solutions. The graphical representations of some solutions by assigning particular values to the parameters under prescribed conditions in each solutions and comparing of solutions with those gained by other authors indicate that these employed techniques are more effective, efficient and applicable mathematical tools for solving nonlinear problems in applied science.

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

Similar content being viewed by others

References

  1. Abdullah Sonmezoglu, Min Yao, Mehmet Ekici, Mohammad Mirzazadeh, Qin Zhou. Explicit solitons in the parabolic law nonlinear negative-index materials, Nonlinear Dyn, 2017, 88: 595–607.

    Article  Google Scholar 

  2. Yuanyuan Yan, Wenjun Liu, Qin Zhou, Anjan Biswas. Dromion-like structures and periodic wave solutions for variable-coefficients complex cubic-quintic Ginzburg-Landau equation influenced by higher-order effects and nonlinear gain, Nonlinear Dynamics, 2020, 99: 1313–1319.

    Article  Google Scholar 

  3. Suzhi Liu, Qin Zhou, Anjan Biswas, Wenjun Liu. Phase-shift controlling of three solitons in dispersion-decreasing fibers, Nonlinear Dynamics, 2019, 98: 395–401.

    Article  Google Scholar 

  4. Weitian Yu, Wenjun Liu, Houria Triki, Qin Zhou, Anjan Biswas. Phase shift oscillation and collision of the anti-dark solitons for the (3+1)-dimensional coupled nonlinear Schrödinger equation in an optical fiber communication system, Nonlinear Dynamics, 2019, 97: 1253–1262.

    Article  Google Scholar 

  5. Xiaoyu Fan, Tianqi Qu, Shucheng Huang, Xiaoxiao Chen, Menghua Cao, Qin Zhou, Wenjun Liu. Analytic study on the influences of higher-order effects on optical solitons in fiber laser, Optik, 2019, 186: 326–331.

    Article  Google Scholar 

  6. Chenjian Wang, Zizhuo Nie, Weijie Xie, Jingyi Gao, Qin Zhou, Wenjun Liu. Dark soliton control based on dispersion and nonlinearity for third-order nonlinear Schrodinger equation, Optik, 2019, 187: 360–367.

    Google Scholar 

  7. Seadawy A R, Dianchen Lu, Chen Yue. Travelling wave solutions of the generalized nonlinear fifth-order KdV water wave equations and its stability, Journal of Taibah University for Science, 2017, 11: 623–633.

    Article  Google Scholar 

  8. Mujahid Iqbal, Aly R Seadawy, Omer H Khalil, Dianchen Lu. Propagation of long internal waves in density stratified ocean for the (2+1)-dimensional nonlinear Nizhnik-Novikov-Vesselov dynamical equation, Results in Physics, 2020, 16: 102838.

    Article  Google Scholar 

  9. Aly R Seadawy, Mujahid Iqbal, Dianchen Lu. Nonlinear wave solutions of the Kudryashov-Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity, Journal of Taibah University for Science, 2019, 13(1): 1060–1072.

    Article  Google Scholar 

  10. Yesim Glam Ozkan, Emrullah Yasar, Aly Seadawy. A third-order nonlinear Schrodinger equation: the exact solutions, group-invariant solutions and conservation laws, Journal of Taibah University for Science, 2020, 14(1): 585–597.

    Article  Google Scholar 

  11. Yan Z. Abundant families of Jacobi elliptic function solutions of the (2+1)dimensional integrable Davey-Stewartson-type equation via a new method, Chaos, Solitons and Fractals, 2003, 18(2): 299–309.

    Article  MathSciNet  Google Scholar 

  12. A Ali, Seadawy A R, D Lu. Soliton solutions of the nonlinear Schrödinger equation with the dual power law nonlinearity and resonant nonlinear Schrödinger equation and their modulation instability analysis, Optik, 2017, 145: 79–88.

    Article  Google Scholar 

  13. Seadawy Aly. Modulation instability analysis for the generalized derivative higher order nonlinear Schrödinger equation and its the bright and dark soliton solutions, J Electromagn Waves Appl, 2017, 31(14): 1353–1362.

    Article  Google Scholar 

  14. D Lu, Seadawy A R, Arshad M. Applications of extended simple equation method on unstable nonlinear Schrodinger equations, Optik, 2017, 140: 136–144.

    Article  Google Scholar 

  15. S Zhang, J L Tong, Wang W. A generalized (G/G’)-expansion method for the mKdvequation with variable coefficients, Phys Lett A, 2008, 372: 2254–2257.

    Article  Google Scholar 

  16. Z J Zhou, J Z Fu, Z B Li. Maple packages for computing Hirotas bilinear equation and multisoliton solutions of nonlinear evolution equations, Appl Math Comput, 2010, 217(1): 92–104.

    MathSciNet  MATH  Google Scholar 

  17. Bekir A, Aksoy E. Exact solutions of nonlinear evolution equations with variable coefficients using exp-function method, Appl Math Comput, 2010, 217(1): 430–436.

    MathSciNet  MATH  Google Scholar 

  18. Abdullah, Aly Seadawy, Jun Wang. Mathematical methods and solitary wave solutions of three-dimensional Zakharov-Kuznetsov-Burgers equation in dusty plasma and its applications, Results in Physics, 2017, 7: 4269–4277.

    Article  Google Scholar 

  19. Aly Seadawy, K El-Rashidy. Dispersive Solitary wave solutions of Kadomtsev-Petviashivili and modified Kadomtsev-Petviashivili dynamical equations in unmagnetized dust plasma, Results in Physics, 2018, 8: 1216–1222.

    Article  Google Scholar 

  20. Dianchen Lu, Aly R Seadawy, Asghar Ali. Applications of Exact Traveling wave solutions of Modified Liouville and the Symmetric Regularized Long Wave equations via two new techniques, Results in Physics, 2018, 9: 1403–1410.

    Article  Google Scholar 

  21. Mujahid Iqbal, Aly R Seadawy, Dianchen Lu. Construction of solitary wave solutions to the nonlinear modified Kortewege-de Vries dynamical equation in unmagnetized plasma via mathematical methods, Modern Physics Letters A, 2018, 33: 1850183.

    Article  MathSciNet  Google Scholar 

  22. Arshad M, Seadawy A R, D Lu, J Wang. Travelling wave solutions of Drinfeld-Sokolov-Wilson, Whitham-Broer-Kaup and (2+1)-dimensional Broer-Kaup- Kupershmit equations and their applications, Chin J Phys, 2017, 55: 780–797.

    Article  Google Scholar 

  23. D Lu, Seadawy A R, Arshad M, W Jun. New solitary wave solutions of (3 + 1)-dimensional nonlinear extended, Zakharov-Kuznetsov and modified KdV Zakharov-Kuznetsov equations and their applications, Results Phys, 2017, 7: 899–909.

    Article  Google Scholar 

  24. K Tariq, Seadawy A R. Bistable Bright-Dark solitary wave solutions of the (3+1)-dimensional Breaking soliton, Boussinesq equation with dual dispersion and modified Korteweg-de Vries-Kadomtsev-Petviashvili equations and their applications, Results Phys, 2017, 7: 11439.

    Article  Google Scholar 

  25. Helal M A, Seadawy A R. Exact soliton solutions of a d-dimensional nonlinear Schrodinger equation with damping and diffusive terms, Z Angew Math Phys(ZAMP), 2011, 62: 839–847.

    Article  MathSciNet  Google Scholar 

  26. Seadawy A R. New exact solutions for the kdv equation with higher order nonlinearity by using the variational method, Computers and Mathematics with Applications, 2011, 62: 3741–3755.

    Article  MathSciNet  Google Scholar 

  27. Helal M A, Seadawy A R. Benjamin-Feir-instability in nonlinear dispersive waves, Computers and Mathematics with Applications, 2012, 64: 3557–3568.

    Article  MathSciNet  Google Scholar 

  28. Seadawy A R. Travelling wave solutions of a weakly nonlinear two-dimensional higher order Kadomtsev-Petviashvili dynamical equation for dispersive shallow water waves, Eur Phys J Plus, 2017, 132: 29.

    Article  Google Scholar 

  29. Seadawy A R. Traveling wave solutions of the Boussinesq and generalized fifth-order KdV equations by using the direct algebraic method, Appl Math Sci, 2012, 6(82): 4081–4090.

    MathSciNet  MATH  Google Scholar 

  30. Seadawy A R. Stability analysis solutions for nonlinear three-dimensional modified Korteweg-de Vries Zakharov Kuznetsov equation in a magnetized electron-positron plasma, Physica A: Statistical Mechanics and its Applications Physica A, 2016, 455: 44–51.

    Article  MathSciNet  Google Scholar 

  31. A Aasaraai. The Application of Modified F-expansion Method Solving the Maccaris System, BJMCS, 2015, 11(5): 1–14.

    Article  Google Scholar 

  32. H Jafari, N Kadkhodaa, C M Khalique. Travelling wave solutions of nonlinear evolution equations using the simplest equation method, Computers and Mathematics with Applications, 2012, 64: 2084–2088

    Article  MathSciNet  Google Scholar 

  33. H Rashid, M Uddin, M Hossain, M Rahman. Exact traveling wave solutions to Vakhnenko-Parkes Equation, iJournals: International Journal of Software & Hardware Research in Engineering, 2014, 2(5): 178–182.

    Google Scholar 

  34. F Majida, T Houria, H Tasawar, M Omar, A Biswas. Nonlinear Analysis, Modelling and Control, 2013, 18(1): 60–66.

    Google Scholar 

  35. V O Vakhnenko, E J Parkes. Approach in Theory of Nonlinear Evolution Equations:The Vakhnenko-Parkes Equation, Advances in Mathematical Physics, 2016, 34: 2916582.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics department, Faculty of science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

    Aly R. Seadawy

  2. Mathematics Department, University of Education, Lahore, Multan Campus, Pakistan

    Asghar Ali

  3. Mathematics Department, Faculty of science, King Abdulaziz University, Jeddah, Saudi Arabia

    Wafaa A. Albarakati

  4. Department of Mathematics, Cankaya University, Ankara, Turkey

    Dumitru Baleanu

Authors
  1. Aly R. Seadawy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wafaa A. Albarakati
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Asghar Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dumitru Baleanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aly R. Seadawy.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Seadawy, A.R., Albarakati, W.A., Ali, A. et al. Propagation of traveling wave solutions to the Vakhnenko-Parkes dynamical equation via modified mathematical methods. Appl. Math. J. Chin. Univ. 37, 21–34 (2022). https://doi.org/10.1007/s11766-022-4056-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11766-022-4056-y

MR Subject Classification

Keywords

Search

Navigation