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A study on the two-mode coupled modified Korteweg–de Vries using the simplified bilinear and the trigonometric-function methods

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Abstract

In this paper, we study the system of the two-mode coupled mKdV using the simplified bilinear method. We find the necessary conditions that make the solutions exists. In addition, we investigate the multiple soliton and multiple singular soliton solutions of this system. We find the necessary conditions to have N-soliton solutions. To verify the efficiency of our approach, we apply the trigonometric-function methods. The trigonometric-function methods produce 27 different solutions to this system. These solutions are the same solutions that are produced by the simplified bilinear method. Up to our knowledge, this study is new and we can apply the same idea to the other coupled systems.

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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  MATH  MathSciNet  Google Scholar 

  2. Wazwaz, A.M.: Multiple soliton solutions and other exact solutions for a two-mode KdV equation. Math. Methods Appl. Sci. (2007). doi:10.1002/mma.4138

    Google Scholar 

  3. Wazwaz, A.M.: A two-mode burgers equation of weak shock waves in a fluid: multiple kink solutions and other exact solutions. Int. J. Appl. Comput. Math. (2016). doi:10.1007/s40819-016-0302-4

    Google Scholar 

  4. Wazwaz, A.M., Xu, G.Q.: Negative-order mKdV equations:multiple soliton and multiple singular soliton solutions. Math. Methods Appl. Sci. 39, 661–667 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  5. Khater, A.H., Temsah, R.S., Hassan, M.M.: A Chebyshev spectral collocation method for solving Burgers-type equations. J. Comput. Appl. Math. 222(2), 333–350 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Rashid, A., Ismail, A.I.B.: A Fourier pseudospectral method for solving coupled viscous Burgers equations. Comput. Methods Appl. Math. 9(4), 412–420 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Kaya, D.: An explicit solution of coupled viscous Burgers’ equation by the decomposition method. IJMMS 27(11), 675–680 (2001)

    MATH  MathSciNet  Google Scholar 

  8. Dehghan, M., Hamidi, A., Shakourifar, M.: The solution of coupled Burgers’ equations using Adomian–Pade technique. Appl. Math. Comput. 189, 1034–1047 (2007)

    MATH  MathSciNet  Google Scholar 

  9. Abdou, M.A., Soliman, A.: A Variational iteration method for solving Burgers’ and coupled Burgers’ equations. J. Comput. Appl. Math. 181, 245–251 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Soliman, A.A.: The modified extended tanh-function method for solving Burgers-type equations. Physica A 361, 394–404 (2006)

    Article  MathSciNet  Google Scholar 

  11. Wazwaz, A.M.: Two-mode fifth-order KdV equations: necessary conditions for multiple-soliton solutions to exist. Nonlinear Dyn. (2016). doi:10.1007/s11071-016-3144-z

    Google Scholar 

  12. 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 

  13. Hirota, R.: Exact solution of the modified Korteweg–de Vries equation for multiple collisions of solitons. J. Phys. Soc. Jpn. 33, 1456–1458 (1972)

    Article  Google Scholar 

  14. 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  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. 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), 035001 (2012)

    Article  MATH  Google Scholar 

  19. 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 

  20. 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 

  21. 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 

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

    MATH  MathSciNet  Google Scholar 

  23. 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)

    MATH  MathSciNet  Google Scholar 

  24. 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)

    MATH  MathSciNet  Google Scholar 

  25. 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)

    MATH  MathSciNet  Google Scholar 

  26. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  28. 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 

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

    MATH  MathSciNet  Google Scholar 

  30. Alquran, M., Al-Khaled, K.: Sinc and solitary wave solutions to the generalized Benjamin–Bona–Mahony–Burgers equations. Phys. Scr. 83, 065010 (2011)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  32. Alquran, Marwan, Qawasmeh, Aminah: Classifications of solutions to some generalized nonlinear evolution equations and systems by the sine–cosine method. Nonlinear Stud. 20(2), 263–272 (2013)

    MATH  MathSciNet  Google Scholar 

  33. Chaudhary, N.I., Raja, M.A.Z.: Identification of Hammerstein nonlinear ARMAX systems using nonlinear adaptive algorithms. Nonlinear Dyn. 2(79), 1385–1397 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  34. Shah, S.M., Samar, R., Khan, N.M., Raja, M.A.Z.: Design of fractional-order variants of complex LMS and NLMS algorithms for adaptive channel equalization. Nonlinear dyn. 88(2), 839–858 (2017)

    Article  Google Scholar 

  35. Chaudhary, N.I., Raja, M.A.Z., Khan, A.U.R.: Design of modified fractional adaptive strategies for Hammerstein nonlinear control autoregressive systems. Nonlinear Dyn. 4(82), 1811–1830 (2015)

    Article  Google Scholar 

  36. Aslam, M.S., Chaudhary, N.I., Raja, M.A.Z.: A sliding-window approximation-based fractional adaptive strategy for Hammerstein nonlinear ARMAX systems. Nonlinear Dyn. 1(87), 519–533 (2016)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, United Arab Emirates University, Al Ain, United Arab Emirates

    Muhammed Syam

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

    H. M. Jaradat

  3. Department of Mathematics and Applied Sciences, Dhofar University, Salalah, Oman

    H. M. Jaradat

  4. Department of Mathematics and Statistics, Jordan University of Science and Technology, P.O.Box(3030), Irbid, 22110, Jordan

    Marwan Alquran

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

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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, 1363–1371 (2017). https://doi.org/10.1007/s11071-017-3732-6

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  • DOI: https://doi.org/10.1007/s11071-017-3732-6

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