Abstract
In this paper, the exact explicit traveling wave solutions to the simplified modified Camassa–Holm (SMCH) equation. The SMCH equation is a significant nonlinear evolution equation because it can be used to describe certain physical processes in oceanography and fluid dynamics. The SMCH equation has important applications in mathematics, physics, and engineering. Using the modified auxiliary equation method and the extended \(\left( \frac{G'}{G^{2}}\right)\)-expansion method, we achieve precise solutions with traveling wave behavior. The proposed methods are applied for the first time in this work to examine the considered mathematical model. The solutions are extracted in terms of trigonometric, hyperbolic, and rational functions. The obtained results not only confirm the previously reported solutions of SMCH equation but also offer some new results. To clearly illustrate the physical implications of the examined equation, we provide graphical representations using 3D, 2D and contour plots for certain values of the parameters. The illustrations help to clarify the diverse characteristics and behaviors, associated with the solutions, providing useful insights for researchers as well as practitioners across a variety of scientific and technical disciplines.
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References
Abdelrahman, M.A.E., Alharbi, A.: Analytical and numerical investigations of the modified Camassa–Holm equation. Pramana 95, 117 (2021)
Abdullah, F.A., Islam, M.T., Aguilar, J.F.G., Akbar, M.A.: Impressive and innovative soliton shapes for nonlinear Konno–Oono system relating to electromagnetic field. Opt. Quantum Electron. 55, 69 (2022)
Ahmad, H., Alam, M.N., Omri, M.: New computational results for a prototype of an excitable system. Results Phys. 28, 104666 (2021)
Ahmad, H., Alam, M.N., Rahim, M.A., Alotaibi, M.F., Omri, M.: The unified technique for the nonlinear time-fractional model with the beta-derivative. Results Phys. 29, 104785 (2021)
Ahmet, B., Özkan, G.: Exact solutions of nonlinear fractional differential equations by \(\frac{G^{\prime }}{G}\)-expansion method. Chin. Phys. B 22(11), 110202 (2013)
Akbar, M.A., Abdullah, F.A., Islam, M.T., Sharif, M.A.A.: New solutions of the soliton type of shallow water waves and superconductivity models. Results Phys. 44, 106180 (2023)
Akram, G., Sadaf, M., Zainab, I.: The dynamical study of Biswas–Arshed equation via modified auxiliary equation method. Opt.—Int. J. Light Electron Opt. 255, 168614 (2022)
Alam, M.N., Ali, A.M., Tauseef, M.D.S.: A novel \(\left(\frac{G^{\prime }}{G}\right)\)-expansion method and its application to the Boussinesq equation. Chin. Phys. B, 23(2) (2023)
Alam, M.N., Belgacem, F.B.M., Akbar, M.A.: Analytical treatment of the evolutionary \((1 + 1)\)-dimensional combined KdV–mKdV equation via the novel \(\left(\frac{G^{\prime }}{G}\right)\)-expansion method. J. Appl. Math. Phys., 3(12) (2015)
Alam, M.N., Bonyah, E., Asad, M.S., Osman, M.F.A., Abualnaja, K.M.: Stable and functional solutions of the Klein–Fock–Gordon equation with nonlinear physical phenomena. Phys. Scr. 96(5) (2021)
Alam, M.N., Islam, S., Ilhan, O.A., Bulut, H.: Some new results of nonlinear model arising in incompressible visco-elastic Kelvin–Voigt fluid. Math. Methods Appl. Sci. 45(16) (2022)
Alam, M.N., Li, X.: Exact traveling wave solutions to higher order nonlinear equations. J. Ocean Eng. Sci. 4(3) (2019)
Alam, M.N., Li, X.: New soliton solutions to the nonlinear complex fractional Schrödinger equation and the conformable time-fractional Klein-Gordon equation with quadratic and cubic nonlinearity. Phys. Scr. 95, 045224 (2020)
Alam, M.N., Li, X.: Symbolic methods to construct a cusp, breathers, kink, rogue waves and some soliton waves solutions of nonlinear partial differential equations. Comput. Methods Differ. Equ. 8(3) (2020)
Alam, M.N., Osman, M.S.: New structures for closed-form wave solutions for the dynamical equations model related to the ion sound and Langmuir waves. Commun. Theor. Phys. 73, 035001 (2021)
Alam, M.N., Talib, I., Bazighifan, O. Chalishajar,, D.N., Almarri,B.: An analytical technique implemented in the fractional clannish random Walker’s Parabolic equation with nonlinear physical phenomena. Mathematics 9(8) (2021)
Alam, M.N., Akbar, M.A.: The new approach of the generalized \(\frac{G^{\prime }}{G}\)-expansion method for nonlinear evolution equations. Ain Shams Eng. J. 5(2), 595–603 (2014)
Alam, M.N., Ilhan, O.A., Uddin, M.S., Rahim, M.A.: Regarding on the results for the fractional clannish random Walker’s Parabolic equation and the nonlinear fractional Cahn-Allen equation. Adv. Math. Phys. 5635514, 2022 (2022)
Alhami, R., Alquran, M.: Extracted different types of optical lumps and breathers to the new generalized stochastic potential-KdV equation via using the Cole-Hopf transformation and Hirota bilinear method. Opt. Quantum Electron. 54, 553 (2022)
Ali, M., Alquran, M., Salman, O.B.: A variety of new periodic solutions to the damped \((2+1)\)-dimensional Schrödinger equation via the novel modified rational sine cosine functions and the extended tanh coth expansion methods. Results Phys. 37, 105462 (2022)
Ali, A., Iqbal, M.A., Mohyud-Din, S.T.: Traveling wave solutions of generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony and simplified modified form of Camassa-Holm equation \(\exp (-\phi (\eta ))\)-Expansion method. Egypt. J. Basic Appl. Sci. 3(2), 134–140 (2016)
Ali, A., Iqbal, M.A., Din, S.T.M.: Traveling wave solutions of generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony and simplified modified form of Camassa-Holm equation \(exp(\phi (\eta ))\)-Expansion method. Egypt. J. Basic Appl. Sci. 3(2), 134–140 (2016)
Alquran, M., Alhami, R.: Analysis of lumps, single-stripe, breather-wave, and two-wave solutions to the generalized perturbed-KdV equation by means of Hirota’s bilinear method. Nonlinear Dyn. 109(9) (2022)
Alquran, M.: Classification of single-wave and bi-wave motion through fourth-order equations generated from the Ito model. Phys. Scr. 98(8) (2023)
Alquran, M.: Necessary conditions for convex-periodic, elliptic-periodic, inclined-periodic, and rogue wave-solutions to exist for the multi-dispersions Schrödinger equation. Phys. Scr. 99, 025248 (2024)
Alquran, M.: New interesting optical solutions to the quadratic cubic Schrödinger equation by using the Kudryashov-expansion method and the updated rational sine-cosine functions. Opt. Quantum Electron. 54, 666 (2022)
Alquran, M.: Optical bidirectional wave-solutions to new two-mode extension of the coupled KdV–Schrödinger equations. Opt. Quantum Electron. 53, 588 (2021)
Alquran, M.: Physical properties for bidirectional wave solutions to a generalized fifth-order equation with third-order time-dispersion term. Results Phys. 28, 104577 (2021)
Arafat, S.M.Y., Fatema, K., Islam, M.E., Akbar, M.A.: Promulgation on various genres soliton of Maccari system in nonlinear optics. Opt. Quantum Electron. 54, 206 (2020)
Arafat, S.M.Y., Fatema, K., Islam, S.M.R., Islam, M.E., Akbar, M.A., Osman, M.S.: The mathematical and wave profile analysis of the Maccari system in nonlinear physical phenomena. Opt. Quantum Electron. 55, 136 (2022)
Arshed, S., Sadaf, M., Akram, G., Yasin, M.M.: Analysis of Sasa Satsuma equation with beta fractional derivative using extended \(\left(\frac{G^{\prime }}{G^{2}}\right)\)-expansion technique and \(\exp (-\phi (\xi ))\)-expansion technique. Opt.—Int. J. Light Electron Opt. 271, 170087 (2022)
Bekir, A., Güner, Ö.: The \(\frac{G^{\prime }}{G}\)-expansion method using modified Riemann–Liouville derivative for some space-time fractional differential equations. Ain Shams Eng. J. 5(3), 959–965 (2014)
Ege, S.M., Misirli, E.: Solutions of space-time fractional foam drainage equation and the fractional Klein–Gordon equation by use of modified Kudryashov method. Int. J. Res. Advent Technol.: IJRAT 2, 384–388 (2014)
Faraz, N., Khan, Y., Yildirim, A.: Analytical approach to two-dimensional viscous flow with a shrinking sheet via variational iteration algorithm-II. J. King Saud Univer.-Sci. 23, 77–81 (2011)
Fatema, K., Islam, M.E., Arafat, S.M.Y., Akbar, M.A.: Solitons behavior of waves by the effect of linearity and velocity of the results of a model in magnetized plasma field. J. Ocean Eng. Sci. (2022)
Gupta, S., Kumar, D., Singh, J.: Application of homotopy perturbation transform method for solving time-dependent functional differential equations. Int. J. Appl. Nonlinear Sci. 16, 37–49 (2013)
Islam, M.T., Akbar, M.A., Ahmad, H., Ilhan, O.A.: Diverse and novel soliton structures of coupled nonlinear Schrödinger type equations through two competent techniques. Mod. Phys. Lett. B 36, 2250004 (2022)
Islam, M.E., Akbar, M.A.: Study of the parametric effects on soliton propagation in optical fibers through two analytical methods. Opt. Quantum Electron. 53, 585 (2021)
Islam, M.T., Akter, M.A., Aguilar, J.F.G., Akbar, M.A., Careta, E.P.: Innovative and diverse soliton solutions of the dual core optical fiber nonlinear models via two competent techniques. J. Nonlinear Opt. Phys. Mater. 32(4) (2023)
Islam, M.T., Akter, M.A., Aguilar, J.F.G., Akbar, M.A.: Novel optical solitons and other wave structures of solutions to the fractional order nonlinear Schrödinger equations. Opt. Quantum Electron. 54, 520 (2022)
Islam, M.T., Akter, M.A., Ryehan, S., Aguilar, J.F.G., Akbar, M.A.: A variety of solitons on the oceans exposed by the Kadomtsev Petviashvili-modified equal width equation adopting different techniques. J. Ocean Eng. Sci. (2022)
Islam, M.N., Asaduzzaman, M., Ali, M.S.: Exact wave solutions to the simplified modified Camassa–Holm equation in mathematical physics. AIMS Math. 5(1) (2020)
Islam, M.E., Barman, M.A., Akbar, H.K.: Search for interactions of phenomena described by the coupled Higgs field equation through analytical solutions. Opt. Quantum Electron. 52, 468 (2020)
Islam, M.T., Ryehan, S., Abdullah, F.A., Aguilar, J.F.G.: The effect of brownian motion and noise strength on solutions of stochastic Bogoyavlenskii model alongside conformable fractional derivative. Optik 287, 171140 (2023)
Islam, M.T., Sarkar, T.R., Abdullah, F.A., Aguilar, J.F.G.: Characteristics of dynamic waves in incompressible. Phys. Scr. 98, 085230 (2023)
Islam, M.N., Akbar, M.A.: New exact wave solutions to the space-time fractional-coupled Burgers equations and the space-time fractional foam drainage equation. Cogent Phys. 5(1), 1422957 (2018)
Islam, M.N., Asaduzzaman, M., Ali, M.S.: Exact wave solutions to the simplified modified Camassa–Holm equation in mathematical physics. AIMS Math. 5(1), 26–41 (2020)
Islam, M.E., Hossainb, M.M., Helal, K.M., Basak, U.S.: Solitary wave analysis of the Kadomtsev–Petviashvili model in mathematical physics. Arab J. Basic Appl. Sci. 30(1), 329–340 (2023)
Javeed, S., Abbasi, M.A., Imran, T., Fayyaz, R., Ahmad, H., Botmart, T.: New soliton solutions of simplified modified Camassa–Holm equation, Klein Gordon–Zakharov equation using first integral method and exponential function method. Results Phys. 38, 105506 (2022)
Khan, Y., Faraz, N.: Application of modified Laplace decomposition method for solving boundary layer equation. J. King Saud Univer.-Sci. 23, 115–119 (2011)
Onder, I., Cinar, M., Secer, A., Bayram, M.: Analytical solutions of simplified modified Camassa-Holm equation with conformable and M-truncated derivatives: a comparative study. J. Ocean Eng. Sci. (2022)
Wang, H., Alam, M.N., Ilhan, O.A., Singh, G., Manafian, J.: New complex wave structures to the complex Ginzburg–Landau model. AIMS Math. 6(8) (2021)
Wang, G.W., Xu, T.Z.: The modified fractional sub-equation method and its applications to nonlinear fractional partial differential equations. Rom. J. Phys. 59, 636–645 (2014)
Wazwaz, A.M.: The modified decomposition method and Pade approximants for a boundary layer equation in unbounded domain. Appl. Math. Comput. 177, 737–744 (2006)
Wazwaz, A.M.: Solitary wave solutions for modified forms of Degasperis–Procesi and Camassa–Holm equations. Phys. Lett. A 352, 500–504 (2006)
Yokus, A., Durur, H., Duran, S., Islam, M.T.: Ample felicitous wave structures for fractional foam drainage equation modeling for fluid-flow mechanism. Comput. Appl. Math. 41(4) (2022)
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G.A. and M.S. wrote the main manuscript text and S.A. and M.ABI prepared figures. All authors reviewed the manuscript.
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Akram, G., Sadaf, M., Arshed, S. et al. Simulations of exact explicit solutions of simplified modified form of Camassa–Holm equation. Opt Quant Electron 56, 1037 (2024). https://doi.org/10.1007/s11082-024-06940-4
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DOI: https://doi.org/10.1007/s11082-024-06940-4