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A Reliable Study of New Nonlinear Equation: Two-Mode Kuramoto–Sivashinsky

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Abstract

In this letter, we use the sense made by Korsunsky (Phys Lett A 185:174–176, 1994 ) to establish a new nonlinear equation called the two-mode Kuramoto–Sivashinsky (TMKS). A finite series in terms of the Tanh function is proposed to be a candidate solution to this new model. Also, we study possible solutions of TMKS by means of the modified simplified bilinear method where a new Cole-Hopf transformation is considered. The new model describes the propagation of two different wave modes simultaneously.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. 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 

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

    MathSciNet  MATH  Google Scholar 

  4. 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 

  5. 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 

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

    MathSciNet  Google Scholar 

  7. 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 

  8. Kuramoto, Y.: Diffusion-induced chaos in reaction systems. Suppl. Prog. Theor. Phys. 64, 346–367 (1978)

    Article  Google Scholar 

  9. Sirashinsky, G.I.: On flame propagation under conditions of stoichiometry. SIAM J. Appl. Math. 39, 67–82 (1980)

    Article  MathSciNet  Google Scholar 

  10. Wazwaz, A.M.: A variety of distinct kinds of multiple soliton solutions for a (3+1)-dimensional nonlinear evolution equation. Math. Meth. Appl. Sci. 36(3), 349–357 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Maliet, W., Hereman, W.: The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys. Scr. 54, 563–568 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Alquran, M., Al-khaled, K.: Mathematical methods for a reliable treatment of the (2+1)-dimensional Zoomeron equation. Math. Sci. 6(12), 11 (2012)

    Article  MATH  Google Scholar 

  13. Alquran, M., Al-Khaled, K.: The tanh and sine-cosine methods for higher order equations of Korteweg-de Vries type. Phys. Scr. 84(025010), 4 (2011)

    MATH  Google Scholar 

  14. Alquran, M., Al-Khaled, K.: Sinc and solitary wave solutions to the generalized BenjaminBona-Mahony-Burgers equations. Phys. Scr. 83(065010), 6 (2011)

    Google Scholar 

  15. 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 

  16. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. 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, 1 (2012)

    Article  MATH  Google Scholar 

  21. 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, Romanian. J. Phys. 60(3–4), 324–343 (2015)

    Google Scholar 

  22. 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 

  23. 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 

  24. Wazwaz, A.M.: Multiple-soliton solutions for the Boussinesq equation. Appl. Math. Comput. 192, 479–486 (2007)

    MathSciNet  MATH  Google Scholar 

  25. Jaradat, H.M.: New solitary wave and multiple soliton solutions for the time-space fractional boussinesq equation. Ital. J. Pure Appl. Math. 36, 367–376 (2016)

    MathSciNet  MATH  Google Scholar 

  26. Alsayyed, O., Jaradat, H.M., Jaradat, M.M.M., Mustafa, Zead, Shatat, Feras: 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 

  27. Jaradat, H.M.: Dynamic behavior of traveling wave solutions for a class for the time-space coupled fractional kdV system with time-dependent coefficients. Ital. J. Pure Appl. Math. 36, 945–958 (2016)

    MathSciNet  MATH  Google Scholar 

  28. 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 

  29. Jaradat, H.M., Al-Shara’, S., Alquran, M., Momani, S., Jaradat, M.M.M., Mustafa, Zead, Alsayyed, O., Alquran, Marwan, Abohassan, Khedr M., Momani, S.: New solitary wave and multiple soliton solutions for the time-space coupled fractional mKdV system with time-dependent coefficients. J. Comput. Theor. Nanosci. 13(12), 9082–9089 (2016)

    Article  Google Scholar 

  30. Jaradat, H.M.: Dynamic behavior of traveling wave solutions for new couplings of the burgers equations with time-dependent variable coefficients. Adv. Differ. Equ. 2017, 167 (2017). https://doi.org/10.1186/s13662-017-1223-1

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Authors and Affiliations

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

    H. M. Jaradat

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

    Marwan Alquran

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

    Muhammed I. Syam

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

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Jaradat, H.M., Alquran, M. & Syam, M.I. A Reliable Study of New Nonlinear Equation: Two-Mode Kuramoto–Sivashinsky. Int. J. Appl. Comput. Math 4, 64 (2018). https://doi.org/10.1007/s40819-018-0497-7

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  • DOI: https://doi.org/10.1007/s40819-018-0497-7

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