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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Computational and Analytical Techniques for Long Dispersive Wave and Construction of Solitary Wave Solutions for Nonlinear Whitham-Broer-Kaup Equation

Journal of Applied Nonlinear Dynamics 14(1) (2025) 211--232 | DOI:10.5890/JAND.2025.03.014

Mujahid Iqbal$^{1}$, Aly R. Seadway$^{2}$, Dianchen Lu$^{1}$, Zhengdi Zhang$^{1}$

$^1$ School of Mathematical Sciences, Jiangsu University, Zhenjiang, Jiangsu 212013, P. R. China

$^2$ Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

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Abstract

The couple system of nonlinear partial differential equation under investigation base on extension of modified rational expansion method. In this article, with the help of symbolic computation Mathematica successfully constructed the various kinds of solitary wave solutions named anti-kink soliton, travelling wave solutions, bright soliton, kink soliton, dark soliton, kink dark solitons, kink bright solitons, anti-kink dark solitons, and anti-kink bright solitons for nonlinear Whitham-Broer-Kaup equation. The calculated results are very interested, different and novel which have not been investigated in past studies. The graphical demonstration of constructed solutions demonstrate by 3-D, 2-D, contour shape by computer software Mathematica. The investigated results will be play keen role in the study of nonlinear physical phenomena in nonlinear sciences and constructed results prove that EMRE approach is very efficient, reliable, powerful for the investigation of other nonlinear partial differential equations.

References

  1. [1]  Li, B.Q. and Ma, Y.L. (2023), Hybrid soliton and breather waves, solution molecules and breather molecules of a (3+1)-dimensional Geng equation in shallow water waves, Physics Letters A, 463, 128672.
  2. [2]  Li, B.Q. and Ma, Y.L. (2023), Optical soliton resonances and soliton molecules for the Lakshmanan–Porsezian–Daniel system in nonlinear optics, Nonlinear Dynamics, 111(7), 6689-6699.
  3. [3] Ma, Y.L. and Li, B.Q. (2023), Dynamics of soliton resonances and soliton moleculesfor the AB system in two-layer fluids, Nonlinear Dynamics, 111, 13327-13341.
  4. [4]  Ma, Y.L. and Li, B.Q. (2023), Interaction behaviors between solitons, breathers and their hybrid forms for a short pulse equation, Qualitative Theory of Dynamical Systems, 22(4), 146.
  5. [5]  Li, B.Q. and Ma, Y.L. (2023), Breather, soliton molecules, soliton fusions and fissions, and lump wave of the Caudrey-Dodd-Gibbon equation, Physica Scripta, 98(9), 095214.
  6. [6]  Eslami, M. and Rezazadeh, H. (2016), The first integral method for Wu–Zhang system with conformable time-fractional derivative, Calcolo, 53, 475-485.
  7. [7]  Ablowitz, M.J. and Clarkson, P.A. (1991), Solitons, Nonlinear Evolution Equations and Inverse Scattering, 149, Cambridge university press.
  8. [8]  Chen, Y., Yan, Z., and Zhang, H. (2003), New explicit solitary wave solutions for (2+ 1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation, Physics Letters A, 307(2-3), 107-113.
  9. [9]  Zhang, S. (2007), Exp-function method for solving Maccari's system, Physics Letters A, 371(1-2), 65-71.
  10. [10]  Xue-Qin, Z. and Hong-Yan, Z. (2008), An improved F-expansion method and its application to coupled Drinfel'd–Sokolov–Wilson equation, Communications in Theoretical Physics, 50(2), 309.
  11. [11]  Wen, X. (2010), Construction of new exact rational form non-travelling wave solutions to the (2+1)-dimensional generalized Broer–Kaup system, Applied Mathematics and Computation, 217(4), 1367-1375.
  12. [12]  Seadawy, A.R. and El-Rashidy, K. (2013), Traveling wave solutions for some coupled nonlinear evolution equations, Mathematical and Computer Modelling, 57(5-6), 1371-1379.
  13. [13]  Lin, Y., Liu, Y., and Li, Z. (2013), Exact solutions for pattern formation in a reaction-diffusion system, International Journal of Nonlinear Sciences and Numerical Simulation, 14(5), 307-315.
  14. [14]  Kochanov, M.B., Kudryashov, N.A., and Sinel'Shchikov, D.I. (2013), Non-linear waves on shallow water under an ice cover. Higher order expansions, Journal of Applied Mathematics and Mechanics, 77(1), 25-32.
  15. [15]  Haq, S. and Ishaq, M. (2014), Solution of coupled Whitham–Broer–Kaup equations using optimal homotopy asymptotic method, Ocean Engineering, 84, 81-88.
  16. [16]  Lü, X. and Lin, F. (2016), Soliton excitations and shape-changing collisions in alpha helical proteins with interspine coupling at higher order, Communications in Nonlinear Science and Numerical Simulation, 32, 241-261.
  17. [17]  Iqbal, M., Seadawy, A.R., Lu, D., and Zhang, Z. (2023), Structure of analytical and symbolic computational approach of multiple solitary wave solutions for nonlinear Zakharov‐Kuznetsov modified equal width equation, Numerical Methods for Partial Differential Equations, 39(5), 3987-4006.
  18. [18]  Iqbal, M., Seadawy, A.R., Khalil, O.H., and Lu, D. (2020), Propagation of long internal waves in density stratified ocean for the (2+1)-dimensional nonlinear Nizhnik-Novikov-Vesselov dynamical equation, Results in Physics, 16, 102838.
  19. [19]  Iqbal, M., Seadawy, A.R., Lu, D., and Zhang, Z. (2023), Computational approach and dynamical analysis of multiple solitary wave solutions for nonlinear coupled Drinfeld–Sokolov–Wilson equation, Results in Physics, 54, 107099.
  20. [20]  Seadawy, A.R. and Iqbal, M. (2023), Dispersive propagation of optical solitions and solitary wave solutions of Kundu-Eckhaus dynamical equation via modified mathematical method, Applied Mathematics-A Journal of Chinese Universities, 38(1), 16-26.
  21. [21]  Wazwaz, A.M. (2010), Partial Differential Equations and Solitary Waves Theory, Springer Science \& Business Media.
  22. [22]  Gandzha, I.S. and Sedletsky, Y.V. (2017), Bright and dark solitons on the surface of finite-depth fluid below the modulation instability threshold, Physics Letters A, 381(21), 1784-1790.
  23. [23]  Liu, H. and Yan, F. (2011), The bifurcation and exact travelling wave solutions for (2+1)-dimensional nonlinear models generated by the Jaulent-Miodek hierarchy, International Journal of Nonlinear Science, 11(2), 200-205.
  24. [24]  McLean, W. (1986), A spectral Galerkin method for a boundary integral equation, Mathematics of Computation, 47(176), 597-607.
  25. [25]  Iqbal, M., Seadawy, A.R., Lu, D., and Zhang, Z. (2023), Physical structure and multiple solitary wave solutions for the nonlinear Jaulent–Miodek hierarchy equation, Modern Physics Letters B, 38(16), p.2341016.
  26. [26]  Matinfar, M., Eslami, M., and Roshandel, S. (2015), The first integral method to study the (2+1)-dimensional Jaulent–Miodek equations, Pramana, 85, 593-603.
  27. [27]  Bilige, S., Chaolu, T., and Wang, X. (2013), Application of the extended simplest equation method to the coupled Schr\"{o}dinger-Boussinesq equation, Applied Mathematics and Computation, 224, 517-523.
  28. [28]  Wazwaz, A.M. (2008), The Hirota's direct method for multiple-soliton solutions for three model equations of shallow water waves, Applied Mathematics and Computation, 201(1-2), 489-503.
  29. [29]  Kabir, M.M., Khajeh, A., Abdi Aghdam, E., and Yousefi Koma, A. (2011), Modified Kudryashov method for finding exact solitary wave solutions of higher‐order nonlinear equations, Mathematical methods in the Applied Sciences, 34(2), 213-219.
  30. [30]  Chen, Y. and Yan, Z. (2005), New exact solutions of (2+1)-dimensional Gardner equation via the new sine-Gordon equation expansion method, Chaos, Solitons $\&$ Fractals, 26(2), 399-406.
  31. [31]  Seadawy, A.R., Lu, D., and Iqbal, M. (2019), Application of mathematical methods on the system of dynamical equations for the ion sound and Langmuir waves, Pramana, 93, 1-12.
  32. [32]  Iqbal, M., Seadawy, A.R., and Lu, D. (2019), Applications of nonlinear longitudinal wave equation in a magneto-electro-elastic circular rod and new solitary wave solutions, Modern Physics Letters B, 33(18), 1950210.
  33. [33]  Iqbal, M., Seadawy, A.R., and Lu, D. (2018), Construction of solitary wave solutions to the nonlinear modified Kortewege-de Vries dynamical equation in unmagnetized plasma via mathematical methods, Modern Physics Letters A, 33(32), 1850183.
  34. [34]  Zhao, Y.M. (2013), F-expansion method and its application for finding new exact solutions to the Kudryashov-Sinelshchikov equation, Journal of Applied Mathematics, (1), 895760.
  35. [35]  Li, B.Q. and Ma, Y.L. (2010), New application of the $(G'/G)$-expansion method to excite soliton structures for nonlinear equation, Zeitschrift für Naturforschung A, 65(6-7), 518-524.
  36. [36]  Javidi, M. and Golbabai, A. (2007), Numerical studies on nonlinear Schr\"{o}dinger equations by spectral collocation method with preconditioning, Journal of Mathematical Analysis and Applications, 333(2), 1119-1127.
  37. [37]  Zhu, Z.N. (1993), Painleve property, Backlund transformation, Lax pair and soliton-like solution for a variable coefficient KP equation, Physics Letters A, 182(2-3), 277-281.
  38. [38]  Mohamad Jawad, A.J.A., Mirzazadeh M., and Biswas A. (2014), Solitary wave solutions to nonlinear evolution equations in mathematical physics, Pramana, 83(4), 457-471.
  39. [39]  Iqbal, M., Seadawy, A.R., and Lu, D. (2018), Dispersive solitary wave solutions of nonlinear further modified Korteweg–de Vries dynamical equation in an unmagnetized dusty plasma, Modern Physics Letters A, 33(37), 1850217.
  40. [40]  Seadawy, A.R. and Iqbal, M. (2021), Propagation of the nonlinear damped Korteweg‐de Vries equation in an unmagnetized collisional dusty plasma via analytical mathematical methods, Mathematical Methods in the Applied Sciences, 44(1), 737-748.
  41. [41]  Seadawy, A.R., Iqbal, M., and Lu, D. (2020), Propagation of kink and anti-kink wave solitons for the nonlinear damped modified Korteweg–de Vries equation arising in ion-acoustic wave in an unmagnetized collisional dusty plasma, Physica A: Statistical Mechanics and its Applications, 544, 123560.
  42. [42]  Alruwaili, A.D., Seadawy, A.R., Iqbal, M., and Beinane, S.A.O. (2022), Dust-acoustic solitary wave solutions for mixed nonlinearity modified Korteweg-de Vries dynamical equation via analytical mathematical methods, Journal of Geometry and Physics, 176, 104504.
  43. [43]  Fan, E. (2011), The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials, Physics Letters A, 375(3), 493-497.
  44. [44]  El-Ganaini, S. (2016), Solutions of some class of nonlinear PDEs in mathematical physics, Journal of the Egyptian Mathematical Society, 24(2), 214-219.
  45. [45]  Iqbal, M., Seadawy, A.R., Lu, D., and Xianwei, X. (2020), Construction of a weakly nonlinear dispersion solitary wave solution for the Zakharov–Kuznetsov-modified equal width dynamical equation, Indian Journal of Physics, 94, 1465-1474.
  46. [46]  Lu, D., Seadawy, A.R., and Iqbal, M. (2018), Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations, Open Physics, 16(1), 896-909.
  47. [47]  Seadawy, A.R., Iqbal, M., and Lu, D. (2020), The nonlinear diffusion reaction dynamical system with quadratic and cubic nonlinearities with analytical investigations, International Journal of Modern Physics B, 34(09), 2050085.
  48. [48]  Iqbal, M., Seadawy, A.R., Lu, D., and Xia, X. (2019), Construction of bright–dark solitons and ion-acoustic solitary wave solutions of dynamical system of nonlinear wave propagation, Modern Physics Letters A, 34(37), 1950309.
  49. [49]  Seadawy, A.R., Iqbal, M., and Lu, D. (2019), Applications of propagation of long-wave with dissipation and dispersion in nonlinear media via solitary wave solutions of generalized Kadomtsev–Petviashvili modified equal width dynamical equation, Computers $\&$ Mathematics with Applications, 78(11), 3620-3632.
  50. [50]  Seadawy, A.R., Iqbal, M., and Lu, D. (2020), Propagation of long-wave with dissipation and dispersion in nonlinear media via generalized Kadomtsive–Petviashvili modified equal width-Burgers equation, Indian Journal of Physics, 94, 675-687.
  51. [51]  Zahed, H., Seadawy, A.R., and Iqbal, M. (2022), Structure of analytical ion-acoustic solitary wave solutions for the dynamical system of nonlinear wave propagation, Open Physics, 20(1), 313-333.
  52. [52]  Zhang, S. and Xia, T. (2006), Further improved extended Fan sub-equation method and new exact solutions of the (2+1)-dimensional Broer–Kaup–Kupershmidt equations, Applied Mathematics and Computation, 182(2), 1651-1660.
  53. [53]  Engui, F. and Hongqing, Z. (1998), Backlund transformation and exact solutions for Whitham-Broer-Kaup equations in shallow water, Applied Mathematics and Mechanics, 19(8), 713-716.
  54. [54]  Xu, G. and Li, Z. (2005), Exact travelling wave solutions of the Whitham–Broer–Kaup and Broer–Kaup–Kupershmidt equations, Chaos, Solitons $\&$ Fractals, 24(2), 549-556.
  55. [55] Jiao, X. and Zhang, H. Q. (2006), An extended method and its application to Whitham–Broer–Kaup equation and two-dimensional perturbed KdV equation, Applied Mathematics and Computation, 172(1), 664-677.
  56. [56]  Chen, X. and Zhou, X. (2012), Solving solitary wave solutions of Whitham-Broer-Kaup equation with homotopy analysis method, In Information Engineering and Applications: International Conference on Information Engineering and Applications (IEA 2011) (pp. 877-883), Springer London.
  57. [57]  Khan, M.D., Naeem, I., and Imran, M. (2014), Analytical solutions of time–space fractional, advection–dispersion and Whitham–Broer–Kaup equations, Pramana, 83, 885-906.
  58. [58]  Arshad, M., Seadawy, A.R., Lu, D., and Wang, J. (2017), Travelling wave solutions of Drinfel'd–Sokolov–Wilson, Whitham–Broer–Kaup and (2+1)-dimensional Broer–Kaup–Kupershmit equations and their applications, Chinese Journal of Physics, 55(3), 780-797.
  59. [59]  Lu, D., Seadawy, A.R., and Iqbal, M. (2018), Mathematical methods via construction of traveling and solitary wave solutions of three coupled system of nonlinear partial differential equations and their applications, Results in Physics, 11, 1161-1171.
  60. [60]  Seadawy, A.R., Zahed, H., and Iqbal, M. (2022), Solitary Wave solutions for the higher dimensional Jimo-Miwa dynamical equation via new mathematical techniques, Mathematics, 10(7), 1011.
  61. [61]  Seadawy, A.R., Iqbal, M., Althobaiti, S., and Sayed, S. (2021), Wave propagation for the nonlinear modified Kortewege–de Vries Zakharov–Kuznetsov and extended Zakharov–Kuznetsov dynamical equations arising in nonlinear wave media, Optical and Quantum Electronics, 53, 1-20.
  62. [62]  Iqbal, M., Seadawy, A.R., and Althobaiti, S. (2022), Mixed soliton solutions for the (2+1)-dimensional generalized breaking soliton system via new analytical mathematical method, Results in Physics, 32, 105030.
  63. [63]  Zahran, E.H., Arqub, O.A., Bekir, A., and Abukhaled, M. (2023), New diverse types of soliton solutions to the Radhakrishnan-Kundu-Lakshmanan equation, AIMS Math, 8, 8985-9008.
  64. [64]  Momani, S., Abu Arqub, O., and Maayah, B. (2020), Piecewise optimal fractional reproducing kernel solution and convergence analysis for the Atangana–Baleanu–Caputo model of the Lienard's equation, Fractals, 28(08), 2040007.
  65. [65]  Momani, S., Maayah, B., and Arqub, O.A. (2020), The reproducing kernel algorithm for numerical solution of Van der Pol damping model in view of the Atangana–Baleanu fractional approach, Fractals, 28(08), 2040010.
  66. [66]  Abu Arqub, O., Alsulami, H., and Alhodaly, M. (2022), Numerical Hilbert space solution of fractional Sobolev equation in 1+1-dimensional space, Mathematical Sciences, 1-12.

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