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Traveling Waves in Two Distinct Equations: The (1+1)-Dimensional cKdV–mKdV Equation and The sinh-Gordon Equation

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Abstract

In this research work, two mathematical models, the (1+1)-dimensional cKdV–mKdV equation and the sinh-Gordon (shG) equation, are studied using an analytical method to obtain solitary wave solutions. The paper presents explicit parameterized traveling wave solutions for these equations, with hyperbolic function solutions resulting in solitary wave solutions when specific parameter values are used. The Hamiltonian function and phase plane are briefly discussed, and the relationship between the phase picture and its corresponding component solutions are depicticted. The phase orbits of the planar dynamical system are also studied to determine all the traveling wave solutions of the analyzed models. In this article we also examines the impact of parameters on wave velocity and profile. According to the research findings, the method employed is efficient and applicable to different mathematical physics nonlinear evolution equations.

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References

  1. Wazwaz, A.-M.: One and two soliton solutions for the sinh-gordon equation in (1+1), (2+1) and (3+1) dimensions. Appl. Math. Lett. 25(12), 2354–2358 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dusunceli, F., Celik, E., Askin, M., Bulut, H.: New exact solutions for the doubly dispersive equation using the improved bernoulli sub-equation function method. Indian J. Phys. 95, 309–314 (2021)

    Article  Google Scholar 

  3. Zhang, R.-F., Li, M.-C., Albishari, M., Zheng, F.-C., Lan, Z.-Z.: Generalized lump solutions, classical lump solutions and rogue waves of the (2+1)-dimensional caudrey-dodd-gibbon-kotera-sawada-like equation. Appl. Math. Comput. 403, 126201 (2021)

    MathSciNet  MATH  Google Scholar 

  4. Khan, K., Salam, M.A., Mondal, M., Akbar, M.A.: Construction of traveling wave solutions of the (2+1)-dimensional modified KdV–KP equation. Math. Methods Appl. Sci. 4, 2042–2054 (2023)

    Article  MathSciNet  Google Scholar 

  5. Ma, W.-X.: Soliton solutions by means of hirota bilinear forms. Partial Differ. Equ. Appl. Math. 5, 100220 (2022)

    Article  Google Scholar 

  6. Ahmed, M.T., Khan, K., Akbar, M.A.: Study of nonlinear evolution equations to construct traveling wave solutions via modified simple equation method. Phys. Rev. Res. Int 3(4), 490–503 (2013)

    Google Scholar 

  7. Barman, H.K., Roy, R., Mahmud, F., Akbar, M.A., Osman, M.: Harmonizing wave solutions to the fokas-lenells model through the generalized kudryashov method. Optik 229, 166294 (2021)

    Article  Google Scholar 

  8. Akbar, M.A., Ali, N.H.M.: Exp-function method for duffing equation and new solutions of (2+1) dimensional dispersive long wave equations. Progress Appl. Math. 1(2), 30–42 (2011)

    Google Scholar 

  9. Khan, K., Liu, S., Schaerf, T.M., Du, Y.: Invasive behaviour under competition via a free boundary model: a numerical approach. J. Math. Biol. 83(3), 1–43 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Khan, K., Schaerf, T.M., Du, Y.: Effects of environmental heterogeneity on species spreading via numerical analysis of some free boundary models. Discr. Contin. Dyn. Syst. B (2022). https://doi.org/10.3934/dcdsb.2022077

    Article  Google Scholar 

  11. Tariq, K.U., Zabihi, A., Rezazadeh, H., Younis, M., Rizvi, S., Ansari, R.: On new closed form solutions: The (2+1)-dimensional bogoyavlenskii system. Modern Phys. Lett. B 35(09), 2150150 (2021)

    Article  MathSciNet  Google Scholar 

  12. Althobaiti, A., Althobaiti, S., El-Rashidy, K., Seadawy, A.R.: Exact solutions for the nonlinear extended kdv equation in a stratified shear flow using modified exponential rational method. Results Phys. 29, 104723 (2021)

    Article  Google Scholar 

  13. Wazwaz, A.-M.: The tanh method: solitons and periodic solutions for the dodd–bullough–mikhailov and the tzitzeica–dodd–bullough equations. Chaos Solitons Fractals 25(1), 55–63 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ali, A.T.: New generalized jacobi elliptic function rational expansion method. J. Comput. Appl. Math. 235(14), 4117–4127 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zheng, B.: A new bernoulli sub-ode method for constructing traveling wave solutions for two nonlinear equations with any order. UPB Sci. Bull. Ser. A 73(3), 85–94 (2011)

    MathSciNet  MATH  Google Scholar 

  16. Khan, K., Akbar, M.A.: Study of explicit traveling wave solutions of nonlinear evolution equations. Partial Differ. Equ. Appl. Math. 7, 100475 (2023)

    Article  Google Scholar 

  17. Arafat, S.M.Y., Fatema, K., Islam, S.M.R., Islam, M.E., Akbar, M.A., Osman, M.: The mathematical and wave profile analysis of the maccari system in nonlinear physical phenomena. Opt. Quantum Electron. 55(2), 136 (2023)

    Article  Google Scholar 

  18. Islam, S.M.R., Kumar, D., Fendzi-Donfack, E., et al.: Impacts of nonlinearity and wave dispersion parameters on the soliton pulses of the (2+1)-dimensional kundu-mukherjee-naskar equation. Revista Mexicana de Física 68(6), 061301 (2022)

    Google Scholar 

  19. Akbulut, A., Islam, S.M.R., Arafat, S.M.Y., Tascan, F.: A novel scheme for smch equation with two different approaches. Comput. Methods Differ. Equ. (2022)

  20. Raut, S., Roy, S., Kairi, R.R., Chatterjee, P.: Approximate analytical solutions of generalized zakharov–kuznetsov and generalized modified zakharov–kuznetsov equations. Int. J. Appl. Comput. Math. 7, 1–25 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Duran, S., Durur, H., Yokuş, A.: Traveling wave and general form solutions for the coupled Higgs system. Math. Methods Appl. Sci. (2023). https://doi.org/10.1002/mma.9024

    Article  MathSciNet  Google Scholar 

  22. Weiss, J., Tabor, M., Carnevale, G.: The painlevé property for partial differential equations. J. Math. Phys. 24(3), 522–526 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  23. Roy, S., Raut, S., Kairi, R. R., and Chatterjee, P. Bilinear bäcklund, lax pairs, breather waves, lump waves and soliton interaction of (2+1)-dimensional non-autonomous kadomtsev–petviashvili equation. Nonlinear Dyn. 1–21 (2022)

  24. Roy, S., Raut, S., Kairi, R.R., Chatterjee, P.: Integrability and the multi-soliton interactions of non-autonomous zakharov–kuznetsov equation. Eur. Phys. J. Plus 137(5), 1–14 (2022)

    Article  Google Scholar 

  25. Weiss, J.: The painlevé property for partial differential equations. ii: Bäcklund transformation, lax pairs, and the schwarzian derivative. J. Math. Phys. 24(6), 1405–1413 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  26. Grauel, A.: shG equation, painlevé property and bäcklund transformation. Phys. A Stat. Mech. Appl. 132(2–3), 557–568 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  27. Chern, S.-S.: Geometrical interpretation of the shG equation. Ann. Polonici Math. 1, 63–69 (1981)

    Article  MATH  Google Scholar 

  28. Boiti, M., Leon, J.-P., Pempinelli, F.: Integrable two-dimensional generalisation of the sine-and shG equations. Inverse Probl. 3(1), 37 (1987)

    Article  MATH  Google Scholar 

  29. Jost, J., Wang, G., Ye, D., Zhou, C.: The blow up analysis of solutions of the elliptic shG equation. Calc. Var. Partial Differ. Equ. 31, 2 (2008)

    Article  Google Scholar 

  30. Eslami, M., Neirameh, A.: New solitary and double periodic wave solutions for a generalized shG equation. Eur. Phys. J. Plus 129, 1–6 (2014)

    Article  Google Scholar 

  31. Neirameh, A.: Soliton solutions of the generalized shG equation by the binary \((g^{\prime }/g g^{\prime }/g)\)-expansion method. Pramana 85, 739–745 (2015)

    Article  Google Scholar 

  32. Wang, G., Yang, K., Gu, H., Guan, F., Kara, A.: A (2+1)-dimensional sine-Gordon and shG equations with symmetries and kink wave solutions. Nuclear Phys. B 953, 114956 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  33. Alzaleq, L., Al-zaleq, D., Alkhushayni, S.: Traveling waves for the generalized shG equation with variable coefficients. Mathematics 10(5), 822 (2022)

    Article  Google Scholar 

  34. Zhao, Q., Liu, S.-K., Fu, Z.-T.: New soliton-like solutions for combined kdv and mkdv equation. Commun. Theor. Phys 43, 615–616 (2005)

    Article  MathSciNet  Google Scholar 

  35. Kaya, D., Gülbahar, S., Yokuş, A., Gülbahar, M.: Solutions of the fractional combined kdv–mkdv equation with collocation method using radial basis function and their geometrical obstructions. Adv. Differ. Equ. 2018(1), 1–16 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lu, D., Shi, Q.: New Jacobi elliptic functions solutions for the cKdV–mKdV equation. Int. J. Nonlinear Sci 10(3), 320–325 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Lu, D., Shi, Q.: New solitary wave solutions for the cKdV–mKdV equation. J. Inf. Comput. Sci 8(7), 1733–1737 (2010)

    Google Scholar 

  38. Naher, H., Abdullah, F.: Some new solutions of the cKdV–mKdV equation by using the improved g/g-expansion method. World Appl. Sci. J. 16(11), 1559–1570 (2012)

    Google Scholar 

  39. Hu, H., Tan, M., Hu, X.: New interaction solutions to the combined kdv-mkdv equation from cte method. J. Assoc. Arab Univ. Basic Appl. Sci. 21, 64–67 (2016)

    Google Scholar 

  40. Huang, Y., Wu, Y., Meng, F., Yuan, W.: All exact traveling wave solutions of the cKdV–mKdV equation. Adv. Differ. Equ. 2014, 1–11 (2014)

    Article  Google Scholar 

  41. Chen, C., Jiang, Y.-L.: Lie group analysis, exact solutions and new conservation laws for combined kdv-mkdv equation. Differ. Equ. Dyn. Syst. 28, 827–840 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  42. Malik, S., Kumar, S., and Das, A. A (2+1)-dimensional combined kdv–mkdv equation: integrability, stability analysis and soliton solutions. Nonlinear Dyn. 1–13 (2022)

  43. Roy, S., Raut, S., Kairi, R.R.: Nonlinear analysis of the ion-acoustic solitary and shock wave solutions for non-extensive dusty plasma in the framework of modified Korteweg-de Vries-Burgers equation. Pramana 96(2), 67 (2022)

    Article  Google Scholar 

  44. Sarkar, T., Roy, S., Raut, S., Mali, P.: Studies on the dust acoustic shock, solitary, and periodic waves in an unmagnetized viscous dusty plasma with two-temperature ions. Braz. J. Phys. 53(1), 12 (2023)

    Article  Google Scholar 

  45. Sarkar, T., Raut, S., Mali, P.: The Classification of the Exact Single traveling Wave Solutions to the Constant Coefficient KP–mKP Equation Employing Complete Discrimination System for Polynomial Method. Comput. Math. Methods (2022)

  46. Shabat, A., Zakharov, V.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media Sov. Phys. JETP 34(1), 62 (1972)

    Google Scholar 

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Authors and Affiliations

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

    Kamruzzaman Khan & S. M. Rayhanul Islam

  2. School of Science and Technology, University of New England, Armidale, NSW, 2351, Australia

    Kamruzzaman Khan

  3. School of Mathematical and Computing Science, Fiji National University, Suva, Fiji

    Rajnesh K. Mudaliar

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  1. Kamruzzaman Khan
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Khan, K., Mudaliar, R.K. & Islam, S.M.R. Traveling Waves in Two Distinct Equations: The (1+1)-Dimensional cKdV–mKdV Equation and The sinh-Gordon Equation. Int. J. Appl. Comput. Math 9, 21 (2023). https://doi.org/10.1007/s40819-023-01503-9

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