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A two-mode coupled Korteweg–de Vries: multiple-soliton solutions and other exact solutions

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Abstract

In this paper, we introduce the new nonlinear two-mode coupled Korteweg–de Vries. We find the necessary conditions of dispersion parameter and the nonlinearity parameter that make this newly coupled give multiple-soliton solutions and multiple singular soliton solutions by using the simplified form of Hirota’s direct method. We determine more exact solutions to this new coupled by using other methods such as the sine/cosine method and the sech-expansion method to conduct this study. Finally, all obtained solutions in this paper are new and this coupled is not solved in any other paper.

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References

  1. Wazwaz, A.M.: Multiple soliton solutions and other exact solutions for a two-mode KdV equation. Math. Methods Appl. Sci. (2016). doi:10.1002/mma.4138

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. 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 (2016). doi:10.1080/17455030.2016.1185193

    Google Scholar 

  4. 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 

  5. 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, 11301 (2010)

    Article  Google Scholar 

  6. 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(1), 56–76 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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 

  8. 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 

  9. 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 

  10. Zhang, J.L., Wang, M.L., Feng, Z.D.: The improved F-expansion method and its applications. Phys. Lett. A 350, 103–109 (2006)

    Article  Google Scholar 

  11. Hirota, R., Satsuma, J.: Solition solutions of a coupled Korteweg–de Vries equation. Phys. Lett. A 85, 407–408 (1981)

    Article  MathSciNet  Google Scholar 

  12. Ganji, D.D., Rafei, M.: Solitary wave solutions for a generalized Hirota–Satsuma coupled-KdV equation by homotopy perturbation method. Phys. Lett. A 356, 131–137 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Attili, B., Furati, K., Syam, M.: An efficient implicit Runge–Kutta method for second order systems. Appl. Math. Comput. 178(2), 229–238 (2016)

    MathSciNet  MATH  Google Scholar 

  14. El-sayed, M., Syam, M.: Electrohydrodynamic instability of a dielectric compressible liquid sheet streaming into an ambient stationary compressible gas. Arch. Appl. Mech. 77(9), 613–626 (2007)

    Article  MATH  Google Scholar 

  15. Syam, M., Attili, B.: Numerical solution of singularly perturbed fifth order two point boundary value problem. Appl. Math. Comput. 170(2), 1085–1094 (2005)

    MathSciNet  MATH  Google Scholar 

  16. Gokdogan, A., Yildirim, A., Merdan, M.: Solving coupled-KdV equations by differential transformation method. World Appl. Sci. J. 19(12), 1823–1828 (2012)

    MATH  Google Scholar 

  17. Caom, D.B., Yan, J.R., Zang, Y.: Exact solutions for a new coupled MKdV equations and a coupled KdV equations. Phys. Lett. A 297, 68–74 (2002)

    Article  MathSciNet  Google Scholar 

  18. Zayed, E.M.E., Zedan, H.A., Gepreel, K.A.: On the solitary wave solutions for non-linear Hirota–Satsuma coupled-KdV of equations. Chaos Solitons Fractals 22, 285–303 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wazwaz, A.M.: Two-mode fifth-order KdV equations: necessary conditions for multiple-soliton solutions to exist. Nonlinear Dyn. (2017). doi:10.1007/s11071-016-3144-z

  20. 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 

  21. 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 

  22. Wazwaz, A.M.: A study on the (2+1)-dimensional and the (2+1)-dimensional higher-order Burgers equations. Appl. Math. Lett. 25, 1495–1499 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wazwaz, A.M.: Combined equations of the Burgers hierarchy: multiple kink solutions and multiple singular kink solutions. Phys. Scr. 82, 025001 (2010)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. 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 

  26. Wazwaz, A.M.: Multiple kink solutions for two coupled integrable (2+1)-dimensional systems. Appl. Math. Lett. 58, 1–6 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. 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 

  29. 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 

  30. 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 

  31. 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 

  32. Wazwaz, A.M.: Multiple soliton solutions for the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov equation. Nonlinear Anal. 72, 1314–1318 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  34. 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)

  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)

    MathSciNet  MATH  Google Scholar 

  36. 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 

  37. 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 

  38. Jaradat, H.M., Alquran, M., Jaradat, M.M.M., Mustafa, Z.: Mathematical analysis and physical interpretation on new multiple solitonic solutions of n-coupled modified KdV system. J. Math. Anal. 7(6), 118–129 (2016)

    MathSciNet  MATH  Google Scholar 

  39. Jaradat, H.M., Al-Shara, S., Jaradat, M.M., Mustafa, Z., Alsayyed, O., Alquran, M., Abohassan, K.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), 1–8 (2016)

    Article  Google Scholar 

  40. 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 

  41. 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 

  42. 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 

  43. 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 

  44. 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 

  45. 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  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  47. 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 

  48. Alquran, M.: Bright and dark soliton solutions to the Ostrovsky–Benjamin–Bona–Mahony (OSBBM) equation. J. Math. Comput. Sci. 2(1), 15–22 (2012)

    MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to express their appreciation for the valuable comments of the reviewers.

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Authors and Affiliations

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

    H. M. Jaradat

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

    Muhammed Syam

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

    Marwan Alquran

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  1. H. M. Jaradat
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  2. Muhammed Syam
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Correspondence to Muhammed Syam.

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Jaradat, H.M., Syam, M. & Alquran, M. A two-mode coupled Korteweg–de Vries: multiple-soliton solutions and other exact solutions. Nonlinear Dyn 90, 371–377 (2017). https://doi.org/10.1007/s11071-017-3668-x

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