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A modified approach for a reliable study of new nonlinear equation: two-mode Korteweg–de Vries–Burgers equation

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Abstract

In this paper, we establish a new nonlinear equation which is called the two-mode Korteweg–de Vries–Burgers equation (TMKdV–BE). The new equation describes the propagation of two different wave modes simultaneously. First, we introduce a new Cole–Hopf transformation needed for the modified simplified bilinear method to find necessary conditions on the solution of TMKdV–BE. Also, a finite series in terms of tanh–coth function is presented as an alternative method to study the solution of the equation. Finally, the reliability of the obtained results is discussed in the last section.

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References

  1. Jaradat, H.M., Al-Shara, S., Awawdeh, F., Alquran, M.: Variable coefficient equations of the Kadomtsev–Petviashvili hierarchy: multiple soliton solutions and singular multiple soliton solutions. Phys. Scr. 85(3), 035001 (2012)

    Article  MATH  Google Scholar 

  2. Yan, C.: A simple transformation for nonlinear waves. Phys. Lett. A 224, 77–84 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Yan, Z.Y., Zhang, H.Q.: New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics. Phys. Lett. A 252(6), 291–296 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Yan, Z.Y., Zhang, H.Q.: On a new algorithm of constructing solitary wave solutions for systems of nonlinear evolution equations in mathematical physics. Appl. Math. Mech. 21, 383–388 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alquran, M.: Solitons and periodic solutions to nonlinear partial differential equations by the sine–cosine method. Appl. Math. Inf. Sci. 6(1), 85–88 (2012)

    MathSciNet  MATH  Google Scholar 

  6. Alquran, M., Qawasmeh, A.: Classifications of solutions to some generalized nonlinear evolution equations and systems by the sine–cosine method. Nonlinear Stud. 20(2), 261–270 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Alquran, M., Al-Khaled, K.: The tanh and sine–cosine methods for higher order equations of Korteweg–de Vries type. Phys. Scr. 84(2), 025010 (2011)

    Article  MATH  Google Scholar 

  8. Alquran, M., Ali, M., Al-Khaled, K.: Solitary wave solutions to shallow water waves arising in fluid dynamics. Nonlinear Stud. 19(4), 555–562 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Raslan, K.R.: The application of He’s exp-function method for MKdV and Burgers’ equations with variable coefficients. Int. J. Nonlinear Sci. 7(2), 174–181 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Alquran, M., Katatbeh, Q., Al-Shrida, B.: Applications of first integral method to some complex nonlinear evolution systems. Appl. Math. Inf. Sci. 9(2), 825–831 (2015)

    MathSciNet  Google Scholar 

  11. Qawasmeh, A., Alquran, M.: Reliable study of some new fifth-order nonlinear equations by means of \((G^{\prime }/G)\)-expansion method and rational sine-cosine method. Appl. Math. Sci. 8(120), 5985–5994 (2014)

    Google Scholar 

  12. Qawasmeh, A., Alquran, M.: Soliton and periodic solutions for \( (2+1)\)-dimensional dispersive long water-wave system. Appl. Math. Sci. 8(50), 2455–2463 (2014)

    MathSciNet  Google Scholar 

  13. Korsunsky, S.V.: Soliton solutions for a second-order KdV equation. Phys. Lett. A 185, 174–176 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  14. Xiao, Z.J., Tian, B., Zhen, H.L., Chai, J., Wu, X.Y.: Multi-soliton solutions and Bucklund transformation for a two-mode KdV equation in a fluid. Waves Random Complex Media 31(6), 1–4 (2016)

    MATH  Google Scholar 

  15. Lee, C.T., Liu, J.L.: A Hamiltonian model and soliton phenomenon for a two-mode KdV equation. Rocky Mt. J. Math. 41(4), 1273–1289 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lee, C.C., Lee, C.T., Liu, J.L., Huang, W.Y.: Quasi-solitons of the two-mode Korteweg–de Vries equation. Eur. Phys. J. Appl. Phys. 52, 11–301 (2010)

    Article  Google Scholar 

  17. Lee, C.T., Lee, C.C.: On wave solutions of a weakly nonlinear and weakly dispersive two-mode wave system. Waves Random Complex Media 23, 56–76 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhu, Z., Huang, H.C., Xue, W.M.: Solitary wave solutions having two wave modes of KdV-type and KdV-burgers-type. Chin. J. Phys. 35(6), 633–639 (1997)

    MathSciNet  Google Scholar 

  19. Hong, W.P., Jung, Y.D.: New non-traveling solitary wave solutions for a second-order Korteweg–de Vries equation. Z. Naturforsch. 54a, 375–378 (1999)

    Google Scholar 

  20. Abdou, M.A.: New solitons and periodic wave solutions for nonlinear physical models. Nonlinear Dyn. 52(1–2), 129–136 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Syam, M., Jaradat, H.M., Alquran, M.: A study on the two-mode coupled modified Korteweg–de Vries using the simplified bilinear and the trigonometric-function methods. Nonlinear Dyn. 90(2), 1363–1371 (2017)

    Article  MathSciNet  Google Scholar 

  22. Jaradat, H.M., Syam, M., Alquran, M.: A two-mode coupled Korteweg–de Vries: multiple-soliton solutions and other exact solutions. Nonlinear Dyn. 90(1), 371–377 (2017)

    Article  MathSciNet  Google Scholar 

  23. Wazwaz, A.M.: Two-mode Sharma-Tasso-Olver equation and two-mode fourth-order Burgers equation: multiple kink solutions. Eng. J. Alex (2017). https://doi.org/10.1016/j.aej.2017.04.003

  24. Wazwaz, A.M.: Multiple soliton solutions and other exact solutions for a two-mode KdV equation. Math. Methods Appl. Sci. 40(6), 2277–2283 (2017)

    MathSciNet  MATH  Google Scholar 

  25. Alquran, M., Jarrah, A.: Jacobi elliptic function solutions for a two-mode KdV equation. J. King Saud Univ.-Sci. (2017). https://doi.org/10.1016/j.jksus.2017.06.010

    Google Scholar 

  26. Jaradat, H.M.: Two-mode coupled Burgers equation: multiple-kink solutions and other exact solutions. Alex. Eng. J. (2017). https://doi.org/10.1016/j.aej.2017.06.014

  27. Hirota, R.: Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)

    Article  MATH  Google Scholar 

  28. Wazwaz, A.M.: Multiple kink solutions and multiple singular kink solutions for two systems of coupled Burgers’ type equations. Commun. Nonlinear Sci. Numer. Simul. 14, 2962–2970 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wazwaz, A.M.: Kinks and travelling wave solutions for Burgers-like equations. Appl. Math. Lett. 38, 174–179 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn. 83, 591–596 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hirota, R.: Exact N-soliton solutions of a nonlinear wave equation. J. Math. Phys. 14, 805–809 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  32. Jaradat, H.M., Awawdeh, F., Al-Shara’, S., Alquran, M., Momani, S.: Controllable dynamical behaviors and the analysis of fractal burgers hierarchy with the full effects of inhomogeneities of media. Rom. J. Phys. 60(3–4), 324–343 (2015)

    Google Scholar 

  33. Awawdeh, F., Jaradat, H.M., Al-Shara’, S.: Applications of a simplified bilinear method to ion-acoustic solitary waves in plasma. Eur. Phys. J. D 66, 1–8 (2012)

    Article  Google Scholar 

  34. Awawdeh, F., Al-Shara’, S., Jaradat, H.M., Alomari, A.K., Alshorman, R.: Symbolic computation on soliton solutions for variable coefficient quantum Zakharov–Kuznetsov equation in magnetized dense plasmas. Int. J. Nonlinear Sci. Numer. Simul. 15(1), 35–45 (2014)

    Article  MathSciNet  Google Scholar 

  35. Alsayyed, O., Jaradat, H.M., Jaradat, M.M.M., Mustafa, Z., Shatat, F.: Multi-soliton solutions of the BBM equation arisen in shallow water. J. Nonlinear Sci. Appl. 9(4), 1807–1814 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Alquran, M., Jaradat, H.M., Al-Shara’, S., Awawdeh, F.: A new simplified bilinear method for the n-soliton solutions for a generalized FmKdV equation with time-dependent variable coefficients. Int. J. Nonlinear Sci. Numer. Simul. 16, 259–269 (2015)

    MathSciNet  Google Scholar 

  37. Jaradat, H.M.: Dynamic behavior of traveling wave solutions for new couplings of the burgers equations with time-dependent variable coefficients. Differ. Equ. Adv (2017). https://doi.org/10.1186/s13662-017-1223-1

    Google Scholar 

  38. Wazwaz, A.M.: A variety of distinct kinds of multiple soliton solutions for a \((3+1)\)-dimensional nonlinear evolution equation. Math. Methods Appl. Sci. 7(19), 349–357 (2012)

    MathSciNet  MATH  Google Scholar 

  39. Feudel, F., Steudel, H.: Nonexistence of prolongation structure for the Korteweg–de Vries Burgers equation. Phys. Lett. A 107, 5–8 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  40. Dodd, R., Fordy, A.: The prolongation structures of quasipolynomial flow. Proc. R. Soc. Lond. A 385, 389–429 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the reviewers for the valuable comments.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Jordan University of Science and Technology, P.O. Box: 3030, Irbid, 22110, Jordan

    Marwan Alquran

  2. Department of Mathematics, Al al-Bayt University, Mafraq, Jordan

    H. M. Jaradat

  3. Department of Mathematical Sciences, UAE University, Al-Ain, United Arab Emirates

    Muhammed I. Syam

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  1. Marwan Alquran
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  2. H. M. Jaradat
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  3. Muhammed I. Syam
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Correspondence to Marwan Alquran.

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Alquran, M., Jaradat, H.M. & Syam, M.I. A modified approach for a reliable study of new nonlinear equation: two-mode Korteweg–de Vries–Burgers equation. Nonlinear Dyn 91, 1619–1626 (2018). https://doi.org/10.1007/s11071-017-3968-1

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  • DOI: https://doi.org/10.1007/s11071-017-3968-1

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