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Influence of the Free Parameters and Obtained Wave Solutions from CBS Equation

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Abstract

This manuscript investigates the exact travelling wave solutions of the (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. By employing the unified method, a large amount of new soliton solutions are derived. All soliton solution are trigonometric and hyperbolic function solutions. Based on the obtained solutions, we have discussed the values of wave variables and the effect of the free parameters, which was not in the previous literature. The outcomes are normally helpful in learning the interaction of waves in new specialized structures with high-dimensional systems.

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References

  1. Hirota, R.: The direct method in soliton theory. Cambridge University Press (2004). https://doi.org/10.1017/CBO9780511543043

    Article  MATH  Google Scholar 

  2. Wang, G.: A novel (3+ 1)-dimensional sine-Gorden and a sinh-Gorden equation: derivation, symmetries and conservation laws. Appl. Math. Lett. 113, 106768 (2021). https://doi.org/10.1016/j.aml.2020.106768

    Article  MathSciNet  MATH  Google Scholar 

  3. Al-Qarni A A, Alshaery A A and Bakodah H O 2020 Optical solitons for the Lakshmanan-Porsezian-Daniel model by collective variable method Results Optics 1 100017 https://doi.org/10.1016/j.rio.2020.100017

  4. He, J.H., Yusry, O.E.D.: The reducing rank method to solve third-order duffing equation with the homotopy perturbation numer. Methods Partial Differential Eq. 37, 1800–1808 (2021). https://doi.org/10.1002/num.22609

    Article  Google Scholar 

  5. Hossain, A.K.M.K.S., Akbar, M.A.: Solitary wave solutions of few nonlinear evolution equations. AIMS Math. 5, 1199–1215 (2020). https://doi.org/10.3934/math.2020083

    Article  MathSciNet  MATH  Google Scholar 

  6. Seadawy AR, El-Rashidy K (2018) Dispersive solitary wave solutions of Kadomtsev-Petviashvili and modified Kadomtsev-Petviashvili dynamical equations in unmagnetized dust plasma, Results Phys. 8: 1216-1222. Doi: https://doi.org/10.1016/j.rinp.2018.01.053

  7. Islam SMR, Khan K, Akbar MA (2015) Exact solutions of unsteady Korteweg-de Vries and time regularized long wave equations, Springer Plus 4: 124 Doi: https://doi.org/10.1186/s40064-015-0893-y

  8. Islam S M R 2015 Application of the -expansion method to find exact traveling wave solutions of the Benney-Luke equation in mathematical physics Am. J. Appl. Math. 3 100–105 https://doi.org/10.11648/j.ajam.20150303.14

  9. Hosseini, K., Aligoli, M., Mirzazadeh, M., Eslami, M., Gomez-Aguilar, F.: Dynamics of rational solutions in a new generalized Kadomtsev–Petviashvili equation Mod. Phys. Lett. B. 33, 1950437 (2019). https://doi.org/10.1142/S0217984919504372

    Article  MathSciNet  Google Scholar 

  10. Wazwaz A M and El-Tantawy S A 2016 A new integrable (3+1)-dimensional KdV-like model with its multiple-soliton solutions Nonlinear Dyn. 83: 1529–1534 https://doi.org/10.1007/s11071-015-2427-0

  11. Khater, M.M.A., Lu, D.C., Attia, R.A.M., Inç, M.: Analytical and approximate solutions for complex nonlinear Schrödinger equation via generalized auxiliary equation and numerical schemes Commun. Theor. Phys. 71, 1267 (2019). https://doi.org/10.1088/0253-6102/71/11/1267

    Article  MathSciNet  MATH  Google Scholar 

  12. Khan K, Akbar M A and Islam SMR 2014 Exact solutions for (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations Springer Plus 3 724 https://doi.org/10.1186/2193-1801-3-724

  13. Islam M E, Kundu P R, Akbar M A, Kheled A G and Hammad A 2021 Study of the parametric effect of self-control waves of the Nizhnik-Novikov-Veselov equation by the analytical solutions, Results Phys. 22 103887 DOI: https://doi.org/10.1016/j.rinp.2021.103887

  14. Devi, M., Yadav, S., Arora, R.: Optimal system, invariance analysis of fourth-Order nonlinear ablowitz–Kaup–Newell–Segur water wave dynamical equation using lie symmetry approach Appl. Math. Comput. 404, 126230 (2021). https://doi.org/10.1016/j.amc.2021.126230

    Article  MATH  Google Scholar 

  15. Lu J, Duan X, Li C and Hong X 2021 Explicit solutions for the coupled nonlinear Drinfeld–Sokolov–Satsuma–Hirota system, Results Phys. 24 104128 https://doi.org/10.1016/j.rinp.2021.104128

  16. Feng Y, Bilige S 2021 Multiple rough wave solutions of (2+1)-dimensional YTSF equation via Hirota bilinear method, Waves Random Complex Media https://doi.org/10.1080/17455030.2021.1900625

  17. Ma, W.X., Yong, X., Zhang, H.Q.: Diversity of interaction solutions to the (2+1)-dimensional Ito equation. Comput Math. Appl. 75, 289–295 (2018). https://doi.org/10.1016/j.camwa.2017.09.013

    Article  MathSciNet  MATH  Google Scholar 

  18. Demirkuş, D.: Nonlinear dark solitary SH waves in a heterogeneous layer TWMS. J. Appl. Eng. Math. 11, 386–394 (2020)

    Google Scholar 

  19. Bashar, M.H., Islam, S.M.R.: Exact solutions to the (2 + 1)-Dimensional Heisenberg ferromagnetic spin chain equation by using modified simple equation and improve F-expansion methods. Phys Open. 5, 100027 (2020). https://doi.org/10.1016/j.physo.2020.100027

    Article  Google Scholar 

  20. Zhanh, R.F., Li, M.C., Albishari, M., Zheng, F.C., Lan, Z.Z.: Generalized lump solutions, classical lump solutions and rough waves of the (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada-like equation. Appl. Math. Comput. 403, 126201 (2021). https://doi.org/10.1016/j.amc.2021.126201

    Article  MATH  Google Scholar 

  21. Ma, W.X., Zhu, Z.: Solving the (3+1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl Math. Comput. 218, 11871–11879 (2012). https://doi.org/10.1016/j.amc.2012.05.049

    Article  MathSciNet  MATH  Google Scholar 

  22. Ma, W.X., Batwa, S.: A binary Darboux transformation for multicomponent NLS equations and their reductions. Anal. Math. Phys. 11, 44 (2021). https://doi.org/10.1007/s13324-021-00477-5

    Article  MathSciNet  MATH  Google Scholar 

  23. Tariq, K.U., Zabihi, A., Rezazadeh, H., Younis, M., Rizvi, S.T.R., Ansari, R.: On new closed form solutions: The (2+1)-dimensional Bogoyavlenskii system Mod. Phys. Lett. B. 35, 2150150 (2021). https://doi.org/10.1142/S0217984921501505

    Article  Google Scholar 

  24. Kumar A, Ilhan E, Ciancio A, Yel G and Baskonus H M 2021 Extractions of some new travelling wave solutions to the conformable Date-Jimbo-Kashiwara-Miwa equation, AIMS Mathematics 6 4238-4264 https://doi.org/10.3934/math.2021251

  25. Rehman SU, Bilal M and Ahmad J 2021 New exact solitary wave solutions for the 3D-FWBBM model in arising shallow water waves by two analytical methods, Results Phys 25: 104230 https://doi.org/10.1016/j.rinp.2021.104230

  26. Cinar, M., Onder, I., Secer, A., Yusuf, A., Sulaiman, T.A., Bayram, M., Aydin, H.: The analytical solutions of Zoomeron equation via extended rational sin-cos and sinh-cosh methods. Phys. Scr. 96, 094002 (2021). https://doi.org/10.1088/1402-4896/ac0374

    Article  MATH  Google Scholar 

  27. Majeed, A., Kamran, M., Asghar, N., Baleanu, D.: Numerical approximation of inhomogeneous time fractional Burgers-Huxley equation with B-spline functions and Caputo derivative Eng. Comput (2021). https://doi.org/10.1007/s00366-020-01261-y

    Article  Google Scholar 

  28. Shakeel, M., Mohyud-Din, S.T.: Improved (G′/G)-expansion and extended tanh methods for (2+ 1)-dimensional Calogero–Bogoyavlenskii–Schiff equation. Alexandria Eng. J. 54, 27–33 (2015). https://doi.org/10.1016/j.aej.2014.11.003

    Article  Google Scholar 

  29. Aminakbari N, Gu Y and Yuan W 2020 Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation De Gruyter 18 1342–1351 https://doi.org/10.1515/math-2020-0099

  30. Wang KL 2022 Exact solitary wave solution for fractal shallow water wave model by He's variational method, Mod Phys Lett B 2150602 https://doi.org/10.1142/S0217984921506028

  31. Wang KL 2022 Solitary wave solution of nonlinear Bogoyavlenskii system by variational analysis method, Int J Mod Phys B 2250015 https://doi.org/10.1142/S0217979222500151

  32. Wang, K.L.: New variational theory for coupled nonlinear fractal Schrodinger system. Int J Nume Meths Heat Fluid Flow 32, 589–597 (2022). https://doi.org/10.1108/HFF-02-2021-0136

    Article  Google Scholar 

  33. Islam, S.M.R., Arafat, S.M.Y., Wang, H.F.: Abundunt closed-form wave solutions to the simplified modified Camassa-Holm equation. J Ocean Engi Sci. (2022). https://doi.org/10.1016/j.joes.2022.01.012

    Article  Google Scholar 

  34. Kumar D, Park C, Tamanna N, Paul G C and Osman M S 2020 Dynamics of two-mode Sawada-Kotera equation: Mathematical and graphical analysis of its dual-wave solutions, Results Phys. 19: 103581 Doi: https://doi.org/10.1016/j.rinp.2020.103581

  35. Islam, S.M.R., Bashar, M.H., Muhammad, N.: Immeasurable soliton solutions and enhanced (G’G)-expansion method. Phys Open. 9, 100086 (2021). https://doi.org/10.1016/j.physo.2021.100086

    Article  Google Scholar 

  36. Islam, S.M.R.: Application of an enhanced to find exact solutions of nonlinear PDEs in particle physics. Am J Appl Sci 12, 836–846 (2015). https://doi.org/10.3844/ajassp.2015.836.846

    Article  Google Scholar 

  37. Akbulut A, Islam SMR, Rezazadeh H, Tascan F 2022 Obtaining exact solutions of nonlinear partial differential equations via two different methods Int J mod Phys B 2250041 https://doi.org/10.1142/S0217979222500412

  38. Bogoyaylenskii O I 1990 Overturning solitons in new two-dimensional integrable equations (Russian) Uspekhi Math. Nauk. 4(274) 17–77 192; translation in Russian Math. Surveys 45 4 1–86

  39. Schiff J 1992 Integrability of Chern–Simons–Higgs vortex equations and a reduction of the self-dual Yang–Mills equations to three dimensions. Workshop proceedings at the NATO Advanced research Workshop Painleve Trascendents, Their Asymptotics and Physical Applications Plenum New York 393

  40. Wazwaz, A.M.: Multiple-soliton solutions for the Calogero–Bogoyavlenskii–Schiff, Jimbo–Miwa and YTSF equations. Appl. Math. Comput. 203, 592–597 (2008). https://doi.org/10.1016/j.amc.2007.06.002

    Article  MathSciNet  MATH  Google Scholar 

  41. Al-Amr, M.O.: Exact solutions of the generalized (2 + 1)-dimensional nonlinear evolution equations via the modified simple equation method. Comput. Math. Appl. 69, 390–397 (2015). https://doi.org/10.1016/j.camwa.2014.12.011

    Article  MathSciNet  MATH  Google Scholar 

  42. Kaplan M, Bekir A, Akbulut A (2016) A generalized Kudryashov method to some nonlinear evolution equations in mathematical physics Nonlinear Dyn. 85: 2843-2850 Doi: https://doi.org/10.1007/s11071-016-2867-1

  43. Chen, S.T., Ma, W.X.: Lump solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation. Comput. Math. Appl. 76, 1680–1685 (2018). https://doi.org/10.1016/j.camwa.2018.07.019

    Article  MathSciNet  MATH  Google Scholar 

  44. Gözükızıl OM, Akçağıl S, Aydemir T (2016) Unification of all hyperbolic tangent function methods, Open Phys. 14: 524–541 Doi: https://doi.org/10.1515/phys-2016-0051

  45. Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60, 650–654 (1992). https://doi.org/10.1119/1.17120

    Article  MathSciNet  MATH  Google Scholar 

  46. Wazwaz, A.M.: 2007 The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations. Appl. Math. Comput. 184, 1002–1014 (2007). https://doi.org/10.1016/j.amc.2006.07.002

    Article  MathSciNet  MATH  Google Scholar 

  47. El-Wakil, S.A., El-Labany, S.K., Zahran, M.A., Sabry, R.: Modified extended tanh-function method and its applications to nonlinear equations. Appl. Math. Comput. 161, 403–412 (2005). https://doi.org/10.1016/j.amc.2003.12.035

    Article  MathSciNet  MATH  Google Scholar 

  48. Khuri, S.A.: A complex tanh-function method applied to nonlinear equations of Schrödinger type. Chaos Solitons Fractals 20, 1037–1040 (2004). https://doi.org/10.1016/j.chaos.2003.09.042

    Article  MathSciNet  MATH  Google Scholar 

  49. Akcagil S, Aydemir T (2018) A new application of the unified method new trends Math. Sci. 6 185–199 https://doi.org/10.20852/ntmsci.2018.261

  50. He JH, Wu XH (2006) Exp-function method for nonlinear wave equations, Chaos Solitons Fract; 30: 700–8

  51. Yusufoglu, E.: New solitary solutions for the MBBM equations using Exp-function method. Phys Lett A 372, 442–446 (2008)

    Article  MathSciNet  Google Scholar 

  52. Ma, W.X., Fuchssteiner, B.: Explicit and exact solutions to a Kolmogorov–Petrovskii–Piskunov equation. Int J nonlinear Mech. 31, 329–338 (1996). https://doi.org/10.1016/0020-7462(95)00064-X

    Article  MathSciNet  MATH  Google Scholar 

  53. Ma WX, Lee JH A (2009) Transformed Rational Function Method and Exact Solutions to the (3+1) Dimensional Jimbo-Miwa Equation Chaos Solitons Fractals 42 (3) 2009 1356–1363 https://doi.org/10.1016/j.chaos.2009.03.043

  54. Ma, W.X.: Binary Darboux transformation for general matrix mKdV equations and reduced counterparts. Chaos Solitons Fractals 146, 110824 (2021). https://doi.org/10.1016/j.chaos.2021.110824

    Article  MathSciNet  MATH  Google Scholar 

  55. Ma WX (2020) N-soliton solutions and the Hirota conditions in (2+1)-dimensions. Opt Quantum Elects; 52(12). https://doi.org/10.1007/s11082-020-02628-7

  56. Ma, W.X.: N-soliton solutions and the Hirota conditions in (1 + 1)-dimensions. Int J Nonlinear Sci Numer Simula. 23, 123–133 (2021). https://doi.org/10.1515/ijnsns-2020-0214

    Article  MathSciNet  MATH  Google Scholar 

  57. Ma, W.X.: N-soliton solution of a combined pKP–BKP equation. J Geo Phys. 165, 104191 (2021). https://doi.org/10.1016/j.geomphys.2021.104191

    Article  MathSciNet  MATH  Google Scholar 

  58. Ma, W.X., Yong, X., Lü, X.: Soliton solutions to the B-type Kadomtsev–Petviashvili equation under general dispersion relations. Wave Motion 103, 102719 (2021). https://doi.org/10.1016/j.wavemoti.2021.102719

    Article  MathSciNet  MATH  Google Scholar 

  59. Ma WX (2021) N-soliton solution and the Hirota condition of a (2+1)-dimensional combined equation. Math Comput Simula; 190(C): 270–279. 2022. https://doi.org/10.1016/j.matcom.2021.05.020

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Acknowledgements

We would express our sincere thanks to referee for his enthusiastic help and valuable suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Pabna University of Science and Technology, Pabna, 6600, Bangladesh

    S. M. Yiasir Arafat & S. M. Rayhanul Islam

  2. Department of Mathematics, European University of Bangladesh, Gabtali, Dhaka, 1216, Bangladesh

    Md Habibul Bashar

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Correspondence to S. M. Rayhanul Islam.

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Arafat, S.M.Y., Islam, S.M.R. & Bashar, M.H. Influence of the Free Parameters and Obtained Wave Solutions from CBS Equation. Int. J. Appl. Comput. Math 8, 99 (2022). https://doi.org/10.1007/s40819-022-01295-4

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