Skip to main content
SpringerLink
Log in

Approximate Analytical Solutions of Generalized Zakharov–Kuznetsov and Generalized Modified Zakharov–Kuznetsov Equations

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

This paper investigates the generalized Zakharov–Kuznetsov (GZK) equation and generalized modified Zakharov–Kuznetsov equation in the presence of external periodic forcing term together with damping. An approximate analytical solution is obtained by employing the direct assumption technique. The framework staged here reveals number of beautiful wave features such as positive amplitude soliton, rare effective soliton, periodic rational soliton, kink type soliton, etc. Moreover, two new parameters along with a control function is introduced to extend the study of traveling wave solution and to create new types of solitary wave solution that are depicted from a numerical standpoint. It is noticed that the generalized wave solution for GZK in presence of external periodic forcing with a damping, positive potential soliton may transform into a rare effective soliton due to an increase in the nonlinearity of the system.

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
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Data Availability

No data are used in our article.

References

  1. Wazwaz, A.M.: The extended tanh method for Zakharov–Kuznetsov equation, the modified ZK equation and its generalized forms. Commun. Nonlinear Sci. Numer. Simul. 13, 1039 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Biswas, A., Zerrad, E.: Solitary wave solution of the Zakharov–Kuznetsov equation in plasmas with power law nonlinearity. Nonlinear Anal. RWA 11, 3272–3274 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Zhen, H.L., Tian, B., Zhong, H., Jiang, Y.: Dynamic behaviors and soliton solutions of the modified Zakharov–Kuznetsov equation in the electrical transmission line. Comput. Math. Appl. 68, 579–588 (2014)

    Article  MATH  Google Scholar 

  4. Seadawy, A.R.: Stability analysis for two-dimensional ion-acoustic waves in quantum plasmas. Phys. Plasmas 21, 052107 (2014)

    Article  Google Scholar 

  5. Zakharov, V.E., Kuznetsov, E.A.: On three-dimensional solitons. Sov. Phys. 66, 594–597 (1974)

    Google Scholar 

  6. Kaup, D.J.: Finding eigenvalue problems for solving nonlinear evolution equations. Prog. Theor. Phys. 54, 72–78 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  7. Broer, L.J.F.: Approximate equations for long water waves. Appl. Sci. Res. 31, 377–395 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kupershmidt, B.A.: Mathematics of dispersive water waves. Commun. Math. Phys. 99, 51–73 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Martinez, L.: Schrodinger spectral problems with energy-dependent potentials as sources of nonlinear Hamiltonian evolution equations. J. Math. Phys. 21, 2342–9 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sulaiman, T.A., Bulut, H.: The new extended rational SGEEM for construction of optical solitons to the (2+1)-dimensional Kundu–Mukherjee–Naskar model. Appl. Math. Nonlinear Sci. 4(2), 513–522 (2019)

    Article  MathSciNet  Google Scholar 

  11. Yavuz, M., Sulaiman, T.A., Usta, F., Bulut, H.: Analysis and numerical computations of the fractional regularized long-wave equation with damping term. Math. Methods Appl. Sci. (2020)

  12. Sulaiman, T.A.: Three-component coupled nonlinear Schrödinger equation: optical soliton and modulation instability analysis. Phys. Scripta 95(6), 065201 (2020)

    Article  Google Scholar 

  13. Younis, M., Sulaiman, T.A., Bilal, M., Rehman, S.Ur, Younas, U.: Modulation instability analysis, optical and other solutions to the modified nonlinear Schrödinger equation. Commun. Theor. Phys. 72, 065001 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sulaiman, T.A., Bulut, H.: Optical solitons and modulation instability analysis of the \((1+1)\)-dimensional coupled nonlinear Schrödinger equation. Commun. Theor. Phys. 72, 025003 (2020)

    Article  MATH  Google Scholar 

  15. Sulaiman, T.A., Yusuf, A., Atangana, A.: New lump, lump-kink, breather waves and other interaction solutions to the \((3+1)\)-dimensional soliton equation. Commun. Theor. Phys. 72, 085004 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yusuf, A., Sulaiman, T.A., Inc, M., Bayram, M.: Breather wave, lump-periodic solutions and some other interaction phenomena to the Caudrey–Dodd–Gibbon equation. Eur. Phys. J. Plus 135, 563 (2020)

    Article  Google Scholar 

  17. Qureshi, S., Yusuf, A.: A new third order convergent numerical solver for continuous dynamical systems. J. King Saud Univ. Sci. 32(2), 1409–1416 (2020)

    Article  Google Scholar 

  18. Sulaiman, T.A., Bulut, H.: Boussinesq equations: M-fractional solitary wave solutions and convergence analysis. J. Ocean Eng. Sci. 4, 1–6 (2019)

    Article  Google Scholar 

  19. Yusuf, A., Inc, M., Aliyu, A.I., Baleanu, D.: Efficiency of the new fractional derivative with nonsingular Mittag–Leffler kernel to some nonlinear partial differential equations. Chaos Solitons Fractals 116, 220–226 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Inc, M., Yusuf, A., Aliyu, A.Isa, Baleanu, D.: Investigation of the logarithmic-KdV equation involving Mittag–Leffler type kernel with Atangana–Baleanu derivative. Physica A 506, 520–531 (2018)

    Article  MathSciNet  Google Scholar 

  21. Tchier, F., Inc, M., Yusuf, A.: Symmetry analysis, exact solutions and numerical approximations for the space-time Carleman equation in nonlinear dynamical systems. Eur. Phys. J. Plus 134, 250 (2019)

    Article  Google Scholar 

  22. Qureshi, S., Yusuf, A., Shaikh, A.A., Inc, M.: Transmission dynamics of varicella zoster virus modeled by classical and novel fractional operators using real statistical data. Physica A 534, 122149 (2019)

    Article  MathSciNet  Google Scholar 

  23. Ghosh, U., Raut, S., Sarkar, S., Das, S.: Solution of space time fractional generalized KdV equation, KdV Burger equation and Bona–Mahonay–Burgers equation with dual power-law nonlinearity using complex fractional transformation. J. Math. Comput. Sci. 81, 114–129 (2018)

    Google Scholar 

  24. Ablowitz, M.J., Clarkson, P.A.: Nonlinear Evolution Equations and Inverse Scattering. Cambridge University, Cambridge (1991)

    Book  MATH  Google Scholar 

  25. Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math. 53(4), 249–315 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  26. Dai, C.Q., Zhang, J.F.: Jacobian elliptic function method for nonlinear differential–difference equations. Chaos Solitons Fractals 27(4), 1042–7 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yu, Y.X., Wang, Q., Zhang, H.Q.: The extended Jacobi elliptic function method to solve a generalized Hirota–Satsuma coupled KdV equation. Chaos Solitons Fractals 26(5), 1415–21 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hirota, R.: Exact envelope soliton solutions of a nonlinear wave equation. J. Math. Phys. 14, 805–810 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hirota, R., Satsuma, J.: Soliton solutions of a coupled KDV equation. Phys. Lett. A 85, 404–408 (1981)

    Article  Google Scholar 

  30. Wazwaz, A.M.: The tanh method: solitons and periodic solutions for the Dodd–Bullough–mikhailov and theb Tzitzeica–Dodd–Bullough equations. Chaos Solitons Fractals 25(1), 55–63 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wazwaz, A.M.: The tanh method for generalized forms of nonlinear heat conduction and Burgers–Fisher equations. Appl. Math. Comput. 169, 321–338 (2005)

    MathSciNet  MATH  Google Scholar 

  32. Fan, E.G.: Extended tanh-method and its applications to nonlinear equations. Phys. Lett. A 277, 212–218 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wazwaz, A.M.: The extended tanh method for the Zakharov–Kuznetsov (ZK) equation, the modified ZK equation and its generalized forms. Commun. Nonlinear Sci. Numer. Simul. 13, 1039–1047 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhu, Y.G., Chang, Q.S., Wu, S.C.: Construction of exact solitary solutions for Boussinesq-like B(m, n) equations with fully nonlinear dispersion by the decomposition method. Chaos Solitons Fractals 26(3), 897–903 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  35. El Sayed, S.M., Kaya, D., Zarea, S.: The decomposition method applied to solve high-order linear Volterra–Fredholm integro-differential equations. Int. J. Nonlinear Sci. Numer. Simul. 5(2), 105–112 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kaw, P., Dawson, J.: Relativistic nonlinear propagation of laser beams in cold overdense plasmas. Phys. Fluids 13, 472 (1970)

    Article  Google Scholar 

  37. Bhattacharyya, B.: Dominance of ion motion over electron motion in some intensity-induced wave processes in a magnetized plasma. Phys. Rev. A 27, 568 (1983)

    Article  Google Scholar 

  38. He, J.H.: Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlinear Sci. Numer. Simul. 6(2), 207–208 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rashidi, M.M., Domairry, G., Dinarvand, S.: Commun. Nonlinear Sci. Numer. Simul. 14(3), 708 (2009)

    Article  Google Scholar 

  40. Wazwaz, A.M.: The sine-cosine method for obtaining solutions with compact and noncompact structures. Appl. Math. Comput. 159(2), 559–76 (2004)

    MathSciNet  MATH  Google Scholar 

  41. Wazwaz, A.M.: A sine-cosine method for handling nonlinear wave equations. Math. Comput. Model. 40(56), 499–508 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. Saadatmandia, A., Dehghan, M.: He’s variational iteration method for solving a partial differential equation arising in modelling of the water waves. Z. Naturforsch. 64a, 783–787 (2009)

    Article  Google Scholar 

  43. Tatari, M., Dehghan, M.: On the convergence of He’s variational iteration method. J. Comput. Appl. Math. 207, 121–128 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  44. Gardner, L.R.T., Gardner, G.A.: Solitary waves of the regularized long-wave equation. J. Comput. Phys. 91(2), 441–459 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  45. Gardner, L.R.T., Gardner, G.A.: Solitary waves of the equal width wave equation. J. Comput. Phys. 101(1), 218–223 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  46. Morrison, P.J., Meiss, J.D., Cary, J.R.: Scattering of regularized-long-wave solitary waves. Physica D 11(3), 324–336 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  47. Gardner, L.R.T., Gardner, G.A., Geyikli, T.: The boundary forced MKdV equation. J. Comput. Phys. 113(1), 5–12 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  48. Abdulloev, KhO, Bogolubsky, I.L., Makhankov, V.G.: One more example of inelastic soliton interaction. Phys. Lett. A 56(6), 427–428 (1976)

    Article  MathSciNet  Google Scholar 

  49. Ali, R., Saha, A., Chatterjee, P.: Dynamics of the positron acoustic waves in electron-positron-ion magnetosplasmas. Indian J. Phys. Phys. Plasmas 24, 122106 (2017)

    Article  Google Scholar 

  50. Ali, R., Saha, A., Chatterjee, P.: Analytical solitary wave solution of the dust ion acoustic waves for the damped forced Korteweg–de Vries equation in super thermal plasmas. Z. Naturforsch. 73(2), 151–159 (2018)

    Article  Google Scholar 

  51. Choudhuri, S., Mandi, L., Chatterjee, P.: Effect of externally applied periodic force on ion acoustic waves in superthermal plasmas. Phys. Plasmas 25, 042112 (2018)

    Article  Google Scholar 

  52. Mandi, L., Saha, A., Chatterjee, P.: Dynamics of ion-acoustic waves in Thomas–Fermi plasmas with source term. Adv. Space Res. 64, 427–435 (2019)

    Article  Google Scholar 

  53. Pal, N., Mondal, K.K., Chatterjee, P.: Effect of dust ion collision on dust ion acoustic solitary waves for nonextensive plasmas in the framework of damped Korteweg–de Vries–Burgers equation. Z. Naturforsch. 74(10), 861–867 (2019)

    Article  Google Scholar 

  54. Mandi, L., Mondal, K.K., Chatterjee, P.: Analytical solitary wave solution of the dust ion acoustic waves for the damped forced modified Korteweg–de Vries equation in q-nonextensive plasmas. Eur. Phys. J. Spec. Top. 228, 2753–2768 (2019)

    Article  Google Scholar 

  55. Mondal, K.K., Roy, A., Chatterjee, P., Raut, S.: Propagation of ion-acoustic solitary waves for damped forced Kuznetsov equation in a realistic rotating magnetized electron-positron-ion plasma. Int. J. Appl. Comput. Math. 66(3), 1–17 (2020)

    MATH  Google Scholar 

  56. Sen, A., Tiwari, S., Mishra, S., Kaw, P.: Nonlinear wave excitations by orbiting charged space debris objects. Adv. Space Res. 56(3), 429 (2015)

    Article  Google Scholar 

  57. Aslanov, V.S., Yudintsev, V.V.: Dynamics, analytical solutions and choice of parameters for towed space debris with flexible appendages. Adv. Space Res. 55, 660–667 (2015)

    Article  Google Scholar 

  58. Schamel, H.: A modified Korteweg–de-Vries equation for ion acoustic waves due to resonant electrons. J. Plasma Phys. 9(3), 377–387 (1973)

    Article  MathSciNet  Google Scholar 

  59. Munro, S., Parkes, E.J.: The derivation of a modified Zakharov–Kuznetsov equation and the stability of its solutions. J. Plasma Phys. 62(3), 305–317 (1999)

    Article  Google Scholar 

  60. Munro, S., Parkes, E.J.: Stability of solitary-wave solutions to a modified Zakharov–Kuznetsov equation. J. Plasma Phys. 64(3), 411–426 (2000)

    Article  Google Scholar 

  61. Dehgan, M., Manafian, J., Saadatmandi, A.: Analytical treatment of some partilal differential equations arising in mathematical physics by using the exp-function method. Int. J. Mod. Phys. B 25(22), 2965–2981 (2011)

    Article  MATH  Google Scholar 

  62. Wang, D.S., Zhang, H.Q.: Further improved F-expansion method and new exact solutions of Konopelchenko–Dubrovsky equation. Chaos Solitons Fractals 25(3), 601–610 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  63. Ren, Y.J., Zhang, H.Q.: A generalized F-expansion methoa to find abundant families of Jacobi elliptic function solutions of the \((2+1)\)-dimensional Nizhnik–Novikov–Veselov equation. Chaos Solitons Fractals 27(4), 959–979 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  64. He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos Soliton Fractal 30, 700–708 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  65. Korteweg, D.J., de Vries, G.: On the change of form of longwaves advancing in a rectangular channel, and on a newtype of long stabilizing wave. Philos. Mag. 39, 422–443 (1895)

    Article  MathSciNet  MATH  Google Scholar 

  66. Zabusky, N.J., Kruskal, M.D.: Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)

    Article  MATH  Google Scholar 

  67. Ismail, M.S., Biswas, A.: 1-Soliton solution of the generalized KdV equation with generalized evolution. Appl. Math. Comput. 216, 1673–1679 (2010)

    MathSciNet  MATH  Google Scholar 

  68. Gepreel, K.A., Shehata, A.R.: Exact complexiton soliton solutions for nonlinear partial differential equations in mathematical physics. Sci. Res. Essays 7(2), 149–157 (2012)

    Google Scholar 

  69. Remoissenet, M.: Waves Called Solitons: Concepts and Experiments. Springer, New York (2013)

    MATH  Google Scholar 

  70. Lomdahl, P.S., Soerensen, O.H., Christiansen, P.L.: Soliton excitations in Josephson tunnel junctions. Phys. Rev. B 25(9), 5737–5748 (1982)

    Article  Google Scholar 

  71. Wazwaz, A.M.: Exact solutions with solitons and periodic structures for the Zakharov–Kuznetsov (ZK) equation and its modified form. Commun. Nonlinear Sci. Numer. Simul. 10, 597–606 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  72. Raut, S., Mondal, K.K., Chatterjee, P., Roy, A.: Propagation of dust-ion-acoustic solitary waves for damped modified Kadomtsev-Petviashvili-Burgers equation in dusty plasma with a q-nonextensive nonthermal electron velocity distribution. SeMA 78(1), 1–23 (2021)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are thankful to the reviewers for their useful comments and suggestions which helped us to improve the quality of the paper. One of the authors Mr. Subrata Roy (JRF) gratefully acknowledges University Grants Commission (UGC) for the financial support vide sanctioned No. 1106/2018 for pursuing this research work.

Funding

Subrata Roy is thankful to the University Grants Commission (UGC), India, for financial support in pursuing this research work.

Author information

Authors and Affiliations

  1. Department of Mathematics, Mathabhanga College, Cooch Behar, West Bengal, 736146, India

    Santanu Raut

  2. Department of Mathematics, Cooch Behar Panchanan Barma University, Cooch Behar, West Bengal, 736101, India

    Subrata Roy & Rishi Raj Kairi

  3. Department of Mathematics, Siksha Bhavana, Visva-Bharati, Santiniketan, 731235, India

    Prasanta Chatterjee

Authors
  1. Santanu Raut
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Subrata Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rishi Raj Kairi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Prasanta Chatterjee
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All the authors equally contribute their knowledge and skill to complete the tusk.

Corresponding author

Correspondence to Prasanta Chatterjee.

Ethics declarations

Conflict of interest

We declare that there is no conflict of interest between the authors.

Code Availability

Mat-lab codes for drawing the graphs are attached in supplementary materials.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Raut, S., Roy, S., Kairi, R.R. et al. Approximate Analytical Solutions of Generalized Zakharov–Kuznetsov and Generalized Modified Zakharov–Kuznetsov Equations. Int. J. Appl. Comput. Math 7, 157 (2021). https://doi.org/10.1007/s40819-021-01034-1

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-021-01034-1

Keywords

Search

Navigation