Rasyonel sine-Gordon metodu ile (2+1) boyutlu Zoomeron denkleminin analitik çözümü

Yıl 2024, Cilt: 26 Sayı: 2, 507 - 517, 15.07.2024

Beyhan Kemaloğlu , Gülnur Yel , Hasan Bulut

https://doi.org/10.25092/baunfbed.1395997

Öz

Mevcut çalışma matematiğin fiziğin önemli modellerinden biri olan Zoomeron denkleminin çözümü ile ilgilidir. Bu çalışmada denklemin çözümü için rasyonel Sine-Gordon açılım metodu kullanılıp analitik çözümler elde edildi. Diğer metotlara göre bu metot oldukça etkili olup istenilen sonuçlar elde edildi. Bununla birlikte Matematika programı ile çözümlerin iki ve üç boyutlu uzayda geometrik yorumu yapılmıştır.

Anahtar Kelimeler

Rasyonel Sine-Gordon açılım metodu, Zoomeron denklemi, Analitik metot

Analytical solution of the (2+1)-dimensional Zoomeron equation by rational sine-Gordon Method

Öz

The current study is about the solution of the Zoomeron equation, one of the important models of mathematics and physics. In this study, the rational Sine-Gordon expansion method (RSGEM) is used to obtain various analytical solutions of the model. Compared to other methods, this method is quite effective and the desired results were obtained. Although there are many analytical solutions to the model used in the literature, we present rational type solutions for the first time with this method. We obtained rational hyperbolic function solutions, and also classified all soliton solutions (kink-like, kink, singular kink, anti-kink, dark, bright). In addition, geometric representations of the solutions in two-, and three-dimensional space and contour shape are made with the Mathematica software program.

Anahtar Kelimeler

RSGEM, Zoomeron Equation, Analytical Method

Kaynakça

• Abazari, R.: The solitary wave solutions of Zoomeron equation. Appl. Math. Sci. 5(59), 2943–2949 (2011)
• Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform. Cambridge University Press, Cambridge (1991)
• Arshad, M., Seadawy, A.R., Lu, D.: Bright-dark solitary wave solutions of generalized higher-order nonlinear Schrödinger equation and its applications in optics. J. Electromagn. Waves Appl. 31, 1711–1721 (2017)
• Ata E. Kıymaz O. New generalized Mellin transform and applications to partial and fractional differential equations, International Journal of Mathematics and Computer in Engineering, 1(2023)
• Ata, E., & Kıymaz, İ. O. Generalized Gamma, Beta and Hypergeometric Functions Defined by Wright Function and Applications to Fractional Differential Equations. Cumhuriyet Science Journal, 43(4), 684-695, (2022)
• Baskonus, H.M., Sulaiman, T.A., Bulut, H.: On the new wave behavior to the Klein–Gordon–Zakharov equations in plasma physics. Indian J. Phys. 93(3), 393–399 (2019)
• Baskonus, H. M., Bulut, H., Atangana, A. On the complex and hyperbolic structures of the longitudinal wave equation in a magneto-electro-elastic circular rod. Smart Materials and Structures, 25(3), 035022 (2016)
• Baskonus, H.M., Bulut, H., Sulaiman, T.A.: New complex hyperbolic structures to the Lonngren-wave equation by using sine-Gordon expansion method. Appl. Math. Nonlinear Sci. 4(1), 141–150 (2019)
• Bulut, H., Sulaiman, T.A., Baskonus, H.M.: Dark, bright and other soliton solutions to the Heisenberg ferromagnetic spin chain equation. Superlattices Microstruct. 123, 12–19 (2018)
• Degasperis, A., Rogers, C., Schief, W.K.: Isothermic surfaces generated via Bäcklund and Moutard Transformations: Boomeron and Zoomeron connections. Stud. Appl. Math. 109, 39–65 (2002)
• Dokuyucu M. A., Çelik E., Bulut H., Baskonus H. M., Cancer treatment model with the Caputo-Fabrizio fractional derivative, The European Physical Journal Plus, 133, 1-6 (2018)
• Durur, H., Ilhan, E., Bulut, H. Novel Complex Wave Solutions of the (2+1)-Dimensional Hyperbolic Nonlinear Schrödinger Equation. Fractal and Fractional, 4(3), 41. (2020).
• Gao W., Rezazadeh, H. Pinar Z., Baskonus H. M., Sarwar S. and Yel G., (2020) Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique, Optical and Quantum Electronics 52:52 https://doi.org/10.1007/s11082-019-2162-8
• Ilhan OA., Sulaiman TA., Bulut H. and Baskonus HM, On the new wave solutions to a nonlinear model arising in plasma physics, Eur. Phys. J. Plus 133: 27 (2018)
• Ismael, H. F. Bulut, H., Baskonus, H. M. Optical soliton solutions to the Fokas–Lenells equation via sine-Gordon expansion method and (m+(G'/G))-expansion method. Pramana, 94(1), 35 (2020).
• Khalique, C.M., Mhlanga, I.E. Travelling waves and conservation laws of a (2+1)-dimensional coupling system with Korteweg-de Vries equation. Appl. Math. Nonlinear Sci. 3(1), 241–254 (2018)
• Khalique, C.M., Adeyemo, O.D., Simbanefayi, I.: On optimal system, exact solutions and conservation laws of the modifed equal-width equation. Appl. Math. Nonlinear Sci. 3(2), 409–418 (2018)
• Khan, K., Akbar, A.M.: Traveling wave solutions of the (2+1)-dimensional Zoomeron equation and the Burgers equations via the MSE method and the exp-function method. Ain Shams Eng. J. 5, 247–256 (2014)
• Kundu,P.R. Fahim Md. R. Islam Md. E. and Akbar, M.A. The sine-Gordon expansion method for higher-dimensional NLEEs and parametric analysis, Heliyon, 7(3), e06459 (2021)
• Ma, W.X., Huang, T., Zhang, Y.: A multiple exp-function method for nonlinear diferential equations and its application. Phys. Scr. 82(065003), 1–10 (2010)
• Morris, M.R., Leach, P.G.L.: Symmetry reductions and solutions to the Zoomeron equation. Phys. Scr. 90(015202), 1–5 (2014)
• Pandey, P.K.: A new computational algorithm for the solution of second order initial value problems in ordinary diferential equations. Appl. Math. Nonlinear Sci. 3(1), 167–174 (2018)
• Peng, Y. Z., & Shen, M. On exact solutions of the Bogoyavlenskii equation. Pramana, 67(3), 449-456. (2006)
• Raza, N., Javid, A.: Optical dark and singular solitons to the Biswas–Milovic equation in nonlinear optics with spatio-temporal dispersion. Optik 158, 1049–1057 (2018)
• Seadawy, A.R.: Exact solutions of a two dimensional nonlinear Schrödinger equation. Appl. Math. Lett. 25, 687–691 (2017)
• Seadawy, A.R., Lu, D.: Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Schrödinger equation and its stability. Results Phys. 7, 43–48 (2017)
• Sulaiman, T.A., Bulut, H., Yel, G., Atas, S.S.: Optical solitons to the fractional perturbed Radhakrishnan– Kundu–Lakshmanan model. Opt. Quant. Electron. 50(372), 372–378 (2018b)
• Sulaiman, T.A., Bulut, H., Baskonus, H.M.: Optical solitons to the fractional perturbed NLSE in nano-fibers. Discrete Contin. Dyn. Syst. S 13(3), 925–936 (2020)
• Veeresha, P., Prakasha, DG, Baskonus,HM. New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives, Chaos: An Interdisciplinary Journal of Nonlinear Science 29 (1) (2019)
• Wazwaz, A.M.: Partial Diferential Equations: Methods and Applications. Balkema, Leiden (2002)
• Wazwaz, A.M.: Partial Diferential Equations and Solitary Wave Theory. Higher Education Press, Beijing and Springer-Verlag, Berlin Heidelberg (2009)
• Yamgou, S.B., Deffo G.R. ,and Pelap, F. C., A new rational sine-Gordon expansion method and its application to nonlinear wave equations arising in mathematical physics, Eur. Phys. J. Plus 134: 380 (2019)
• Yel, G., Baskonus, H.M., Bulut, H.: Novel archetypes of new coupled Konno-Oono equation by using sine– Gordon expansion method. Opt. Quant. Electron. 49(285), 1–10 (2017)
• Yel, G., Baskonus, H.M., Bulut, H.: Regarding some novel exponential travelling wave solutions to the Wu– Zhang system arising in nonlinear water wave model. Indian J. Phys. 93(8), 1031–1039 (2019)
• Yel, G., New wave patterns to the doubly dispersive equation in nonlinear dynamic elasticity, Pramana – J. Phys. 94(1):79 (2020)
• Zhao, Z., Han, B.: Lump solutions of a (3+1)-dimensional B-type KP equation and its dimensionally reduced equations. Anal. Math. Phys. 9(1), 119–130 (2019)