Abstract
The simplified modified Camassa–Holm (SMCH) model is widely recognized in the fields of plasma physics, bio-mathematics, and optical fibers. In this study, we have explored the SMCH model implementing the improved Bernoulli sub-equation function (IBSEF) method to obtain novel traveling wave solitons. Our investigation has yielded broad-ranging analytical solutions for the SMCH equation, including hyperbolic, trigonometric, and exponential solutions. Furthermore, we have analyzed the impact of wind and friction on water waves by employing the free parameters of the obtained solutions which can play a crucial role in nature. These findings hold significant relevance to the study of natural phenomena. Moreover, we have discussed the parametric effects of the obtained solitons on the wave profile of the system, revealing kink, bright, and dark-type solitons. Our findings suggest that the IBSEF method is reliable and can be used in future studies to determine different and novel soliton explanations of various nonlinear evolution equations encountered in mathematical physics and engineering.
Similar content being viewed by others
Availability of data and materials
No real data is used here.
References
Abdelrahman, M.A., Zahran, E.H., Khater, M.M., et al.: The Exp (-\(\varphi\) (\(\xi\)))-expansion method and its application for solving nonlinear evolution equations. Int. J. Mod. Nonlinear Theory Appl. 4(01), 54539 (2015)
Alam, M.N., Akbar, M.A.: Some new exact traveling wave solutions to the simplified MCH equation and the (1+ 1)-dimensional combined KdV–mKdV equations. J. Assoc. Arab Univ. Basic Appl. Sci. 17, 6–13 (2015)
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 Akbar, M., Ali, N.H.M.: The improved F-expansion method with Riccati equation and its applications in mathematical physics. Cogent Math. 4(1), 1282577 (2017)
Arafat, S.Y., Fatema, K., Islam, M.E., Akbar, M.A.: Promulgation on various genres soliton of Maccari system in nonlinear optics. Opt. Quant. Electron. 54(4), 206 (2022)
Bashar, M.H., Inc, M., Islam, S.R., Mahmoud, K., Akbar, M.A.: Soliton solutions and fractional effects to the time-fractional modified equal width equation. Alex. Eng. J. 61(12), 12539–12547 (2022)
Baskonus, H.M., Bulut, H.: An effective schema for solving some nonlinear partial differential equation arising in nonlinear physics. Open Phys. 13(1), 280–289 (2015)
Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661 (1993)
Chu, Y.M., Fahim, M.R.A., Kundu, P.R., Islam, M.E., Akbar, M.A., Inc, M.: Extension of the sine-Gordon expansion scheme and parametric effect analysis for higher-dimensional nonlinear evolution equations. J. King Saud Univ. Sci. 33(6), 101515 (2021)
Fahim, M.R.A., Kundu, P.R., Islam, M.E., Akbar, M.A., Osman, M.: Wave profile analysis of a couple of (3 + 1)-dimensional nonlinear evolution equations by sine-Gordon expansion approach. J. Ocean Eng. Sci. 7(3), 272–279 (2022)
Irshad, A., Usman, M., Mohyud-Din, S.T.: Exp-function method for simplified modified Camassa–Holm equation. Int. J. Mod. Math. Sci. 4(3), 146–155 (2012)
Islam, M.E., Akbar, M.A.: Stable wave solutions to the Landau–Ginzburg–Higgs equation and the modified equal width wave equation using the IBSEF method. Arab J. Basic Appl. Sci. 27(1), 270–278 (2020)
Islam, M.E., Akbar, M.A.: Study of the parametric effects on soliton propagation in optical fibers through two analytical methods. Opt. Quant. Electron. 53, 1–20 (2021)
Islam, T., Akbar, M.A., Azad, A.K.: Traveling wave solutions to some nonlinear fractional partial differential equations through the rational (G’/G)-expansion method. J. Ocean Eng. Sci. 3(1), 76–81 (2018a)
Islam, S.R., Khan, K., Woadud, K.A.A.: Analytical studies on the Benney–Luke equation in mathematical physics. Waves Random Complex Media 28(2), 300–309 (2018b)
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 (2019)
Islam, M.E., Barman, H.K., Akbar, M.A.: Search for interactions of phenomena described by the coupled Higgs field equation through analytical solutions. Opt. Quant. Electron. 52, 1–19 (2020)
Islam, S.R., Arafat, S.Y., Wang, H.: Abundant closed-form wave solutions to the simplified modified Camassa–Holm equation. J. Ocean Eng. Sci. 8(3), 238–245 (2023)
Khan, K., Akbar, M.A.: Exact and solitary wave solutions for the Tzitzeica–Dodd–Bullough and the modified KdV–Zakharov–Kuznetsov equations using the modified simple equation method. Ain Shams Eng. J. 4(4), 903–909 (2013)
Kundu, P.R., Fahim, M.R.A., Islam, M.E., Akbar, M.A.: The sine-Gordon expansion method for higher-dimensional NLEEs and parametric analysis. Heliyon 7(3), e06459 (2021)
Liu, J.G., Yang, X.J.: Symmetry group analysis of several coupled fractional partial differential equations. Chaos Solitons Fractals 173, 113603 (2023)
Liu, J.G., Yang, X.J., Geng, L.L., Yu, X.J.: On fractional symmetry group scheme to the higher-dimensional space and time fractional dissipative Burgers equation. Int. J. Geom. Methods Mod. Phys. 19(11), 2250173 (2022)
Liu, J.G., Zhang, Y.F., Wang, J.J.: Investigation of the time fractional generalized (2+ 1)-dimensional Zakharov–Kuznetsov equation with single-power law nonlinearity. Fractals (2023). https://doi.org/10.1142/S0218348X23500330
Lu, D., Seadawy, A.R., Iqbal, M.: Construction of new solitary wave solutions of generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony and simplified modified form of Camassa–Holm equations. Open Phys. 16(1), 896–909 (2018)
Mirza, A., ul Hassan, M.: Bilinearization and soliton solutions of N = 1 supersymmetric coupled dispersionless integrable system. J. Nonlinear Math. Phys. 24(1), 107–115 (2017)
Mirzazadeh, M., Eslami, M., Zerrad, E., Mahmood, M.F., Biswas, A., Belic, M.: Optical solitons in nonlinear directional couplers by sine-cosine function method and Bernoulli’s equation approach. Nonlinear Dyn. 81, 1933–1949 (2015)
Naher, H., Begum, F.A.: Application of linear ODE as auxiliary equation to the nonlinear evolution equation. Am. J. Appl. Math. Stat. 3(1), 23–28 (2015)
Najafi, M., Arbabi, S., Najafi, M.: He’s semi-inverse method for Camassa–Holm equation and simplified modified Camassa–Holm equation. Int. J. Phys. Res. 1, 1–6 (2013)
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). https://doi.org/10.1016/j.joes.2022.06.012
Qian, T., Tang, M.: Peakons and periodic cusp waves in a generalized Camassa–Holm equation. Chaos Solitons Fractals 12(7), 1347–1360 (2001)
Shen, J., Xu, W.: Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa–Holm equation. Chaos Solitons Fractals 26(4), 1149–1162 (2005)
Singh, S., Kaur, L., Sakthivel, R., Murugesan, K.: Computing solitary wave solutions of coupled nonlinear Hirota and Helmholtz equations. Physica A 560, 125114 (2020)
Wazwaz, A.M.: New compact and noncompact solutions for two variants of a modified Camassa–Holm equation. Appl. Math. Comput. 163(3), 1165–1179 (2005)
Wazwaz, A.M., Kaur, L.: Optical solitons for nonlinear Schrödinger (NLS) equation in normal dispersive regimes. Optik 184, 428–435 (2019)
Yao, S.W., Ullah, N., Rehman, H.U., Hashemi, M.S., Mirzazadeh, M., Inc, M.: Dynamics on novel wave structures of non-linear Schrödinger equation via extended hyperbolic function method. Results Phys. 48, 106448 (2023)
Yaşar, E., Yıldırım, Y., Zhou, Q., et al.: Perturbed dark and singular optical solitons in polarization preserving fibers by modified simple equation method. Superlattices Microstruct. 111, 487–498 (2017)
Younas, U., Ren, J.: On the study of optical soliton molecules of Manakov model and stability analysis. Int. J. Mod. Phys. B 36(26), 2250180 (2022a)
Younas, U., Ren, J.: Construction of optical pulses and other solutions to optical fibers in absence of self-phase modulation. Int. J. Mod. Phys. B 36(32), 2250239 (2022b)
Younas, U., Ren, J., Sulaıman, T.A., Bilal, M., Yusuf, A.: On the lump solutions, breather waves, two-wave solutions of (2 + 1)-dimensional Pavlov equation and stability analysis. Mod. Phys. Lett. B 36(14), 2250084 (2022)
Younas, U., Baber, M., Yasin, M., Sulaiman, T., Ren, J.: The generalized higher-order nonlinear Schrödinger equation: optical solitons and other solutions in fiber optics. Int. J. Mod. Phys. B 37(18), 2350174 (2023a)
Younas, U., Sulaiman, T., Ren, J.: Dynamics of optical pulses in fiber optics with stimulated Raman scattering effect. Int. J. Mod. Phys. B 37(08), 2350080 (2023b)
Zdravković, S., Kavitha, L., Satarić, M.V., Zeković, S., Petrović, J.: Modified extended tanh-function method and nonlinear dynamics of microtubules. Chaos Solitons Fractals 45(11), 1378–1386 (2012)
Zhang, S., Li, J., Zhang, L.: A direct algorithm of exp-function method for non-linear evolution equations in fluids. Therm. Sci. 20(3), 881–884 (2016)
Zhong, W.P., Belić, M., Assanto, G., Malomed, B.A., Huang, T.: Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient. Phys. Rev. A 84(4), 043801 (2011a)
Zhong, W.P., Belić, M.R., Assanto, G., Malomed, B.A., Huang, T.: Self-trapping of scalar and vector dipole solitary waves in Kerr media. Phys. Rev. A 83(4), 043833 (2011b)
Acknowledgements
The authors thank to Ministry of NST, Bangladesh for supporting project ID SRG-226676 under the letter 39.00.0000.009.99.024.22-193 (2022-23)
Funding
The authors have no funding information.
Author information
Authors and Affiliations
Contributions
MAM: Conceptualization, Software, Visualization, Methodology, Formal analysis; RCB: Software, Formal analysis, Writing-original draft; MTK: Conceptualization, Software; MEI: Investigation, Formal analysis; USB: Data curation, Writing-review editing, Resources; MAA: Validation, Project administration, Supervision.
Corresponding author
Ethics declarations
Ethical approval
All authors listed on the title page have read the manuscript, attest to the validity and legitimacy of the data and the contents of this manuscript will not be copyrighted, submitted, or published elsewhere.
Conflict of interest
The authors declare that they have no financial or personal interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Mannaf, M.A., Bhowmik, R.C., Khatun, M.T. et al. Optical solitons of SMCH model in mathematical physics: impact of wind and friction on wave. Opt Quant Electron 56, 71 (2024). https://doi.org/10.1007/s11082-023-05641-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-023-05641-8