Uyumlu Zaman-Kesirli Sharma-Tasso-Olver-Burgers Denkleminin Modifiye Kudryashov Yöntemi ile Yeni Analitik Soliton Molekül Çözümleri

Öz

Bu çalışmada, uyumlu zaman-kesirli Sharma-Tasso-Olver-Burgers (STOB) denklemi için yeni analitik soliton molekül çözümleri, modifiye Kudryashov yöntemi kullanılarak ilk kez elde edilmiştir. STOB denklemi, iyi bilinen Burgers ve Sharma-Tasso-Olver (STO) denklemlerini birleştiren ve zengin soliton moleküler davranış gösteren önemli bir nonlineer diferansiyel denklemdir. Çözümlerimiz, klasik Burgers ve STO denklemlerinin ötesine geçen benzersiz nonlineer dinamiklere sahip, daha önce bilinmeyen soliton molekül yapılarını ortaya koymaktadır. Modifiye Kudryashov yönteminin sistematik uygulaması yoluyla; tek-kink dalga profilleri ve farklı özelliklere sahip soliter dalga çözümleri dahil olmak üzere, farklı özelliklere sahip çoklu tam çözüm aileleri elde edilmektedir. Çalışmamızda kullanılan uyumlu kesirli türev yaklaşımı, fiziksel sistemlerdeki anomal dinamik davranışların ve kesirli ölçeklemenin doğru modellenmesine olanak sağlarken, kesirli türev mertebesindeki değişikliklerin dinamik evrimi nasıl etkilediğini göstermektedir. Kesirli türev mertebe değerine bağlı olarak, subdifüzif dinamikten klasik dinamiklere kadar geniş bir spektrumda farklı dinamik davranışlar keşfetmekteyiz. Elde edilen çözümler; mod-kilitli fiber lazerlerden anomal difüzyon süreçlerine ve optik fiberlerde yavaş ışık yayılımından Bose-Einstein yoğuşuklarına kadar çeşitli fiziksel sistemlerde potansiyel uygulamalara sahiptir. Bu çözümlerin evrimi, kapsamlı grafiksel analiz yoluyla incelenmiştir. Kesirli mertebe α'nın fiziksel etkisini örneklendirmek için 3B yüzeyler, kontur haritaları ve 2B kesit grafikleri kullanılmıştır. Bu görselleştirmeler, dalga profillerinin yayılırken belirgin kesirli dinamikler sergilemesine rağmen kararlılıklarını koruduklarını doğrulamaktadır. Ayrıca, elde edilen sonuçlar kesirli-mertebe sistemlerdeki soliton moleküllerinin anlaşılmasına önemli katkılar sağlamakta; kuantum alan teorisi, plazma fiziği ve nonlineer optik gibi alanlarda yeni araştırma yolları açmaktadır.

Anahtar Kelimeler

• Modifiye Kudryashov yöntemi
• Uyumlu kesirli türev
• Sharma-Tasso-Olver-Burgers denklemi
• Sharma-Tasso-Olver-Burgers denklemi
• Soliton molekülleri

Destekleyen Kurum

bulunmamaktadır

Etik Beyan

bulunmamaktadır

Teşekkür

bulunmamaktadır

New Analytical Soliton Molecule Solutions of the Conformable Time-Fractional Sharma-Tasso-Olver-Burgers Equation Using Modified Kudryashov Method

Öz

In this study, novel analytical soliton molecule solutions for the conformable time-fractional Sharma-Tasso-Olver-Burgers (STOB) equation have been obtained using the modified Kudryashov method for the first time. The STOB equation is an important nonlinear differential equation that combines the well-known Burgers and Sharma-Tasso-Olver (STO) equations, showing rich soliton molecular behavior. Our solutions reveal previously unknown soliton molecule structures with unique nonlinear dynamics that go beyond classical Burgers and STO equations. Through systematic application of the modified Kudryashov method, we obtain multiple families of exact solutions with different characteristics, including single-kink wave profiles and solitary wave solutions with different characteristics. The conformable fractional derivative approach used in our study allows accurate modeling of anomalous dynamic behaviors and fractional scaling in physical systems, while showing how changes in the fractional derivative order affect dynamic evolution. We discover different dynamical behaviors ranging from subdiffusive to classical dynamics across a broad spectrum, depending on the fractional derivative order value. The obtained solutions have potential applications in various physical systems, from mode-locked fiber lasers to anomalous diffusion processes, and from slow light propagation in optical fibers to Bose-Einstein condensates. The evolution of these solutions is examined through comprehensive graphical analysis. 3D surfaces, contour maps, and 2D cross-sectional plots are utilized to illustrate the physical impact of the fractional order α. These visualizations confirm that the wave profiles maintain their stability while exhibiting distinct fractional dynamics as they propagate. Furthermore, the obtained results make significant contributions to the understanding of soliton molecules in fractional-order systems, opening new research avenues in fields such as quantum field theory, plasma physics, and nonlinear optics.

Anahtar Kelimeler

• Modified Kudryashov method
• Conformable fractional derivative
• Sharma-Tasso-Olver-Burgers equation
• Soliton solutions
• Soliton molecules

Destekleyen Kurum

Not applicable

Etik Beyan

Not applicable

Teşekkür

Not applicable

Kaynakça

1. Bec, J., & Khanin, K. (2007). Burgers turbulence. Physics Reports, 447(1-2), 1–66.
2. Burgers, J. M. (1948). A mathematical model illustrating the theory of turbulence. Advances in Applied Mechanics, 1, 171–199.
3. Cole, J. D. (1951). On a quasi-linear parabolic equation occurring in aerodynamics. Quarterly of Applied Mathematics, 9(3), 225–236.
4. Das, A., Ghosh, N., & Ansari, K. (2018). Bifurcation and exact traveling wave solutions for dual power Zakharov-Kuznetsov-Burgers equation with fractional temporal evolution. Computers & Mathematics with Applications, 75(1), 59–69.
5. Drummond, P. D., Kheruntsyan, K. V., & He, H. (1998). Coherent molecular solitons in Bose-Einstein condensates. Physical Review Letters, 81(15), 3055.
6. Ege, S. M., & Misirli, E. (2014). The modified Kudryashov method for solving some fractional-order nonlinear equations. Advances in Difference Equations, 2014(1), 135.
7. El-Kalaawy, O. H., Moawad, S. M., Tharwat, M. M., & Al-Denari, R. B. (2019). Conservation laws, analytical solutions and stability analysis for the time-fractional Schamel-Zakharov-Kuznetsov-Burgers equation. Advances in Difference Equations, 2019(1), 445.
8. Esen, A., Taşbozan, O., & Yağmurlu, N. M. (2012). Approximate analytical solutions of the fractional Sharma-Tasso-Olver equation using homotopy analysis method and a comparison with other methods. Cankaya University Journal of Science and Engineering, 9(2).

9. Faraz, N., Sadaf, M., Akram, G., Zainab, I., & Khan, Y. (2021). Effects of fractional order time derivative on the solitary wave dynamics of the generalized ZK-Burgers equation. Results in Physics, 25, 104217.
10. Forte, S. (1992). Quantum mechanics and field theory with fractional spin and statistics. Reviews of Modern Physics, 64(1), 193.
11. Grelu, P., & Akhmediev, N. (2012). Dissipative solitons for mode-locked lasers. Nature photonics, 6(2), 84-92.
12. He, J. H., & Wu, X. H. (2006). Exp-function method for nonlinear wave equations. Chaos, Solitons & Fractals, 30(3), 700–708.
13. Hu, X., Miao, Z., & Lin, S. (2021). Solitons molecules, lump and interaction solutions to a (2+ 1)-dimensional Sharma-Tasso-Olver-Burgers equation. Chinese Journal of Physics, 74, 175–183.
14. Hussain, E., Li, Z., Shah, S. A. A., Az-Zo'bi, E. A., & Hussien, M. (2023). Dynamics study of stability analysis, sensitivity insights and precise soliton solutions of the nonlinear (STO)-Burger equation. Optical and Quantum Electronics, 55(14), 1274.
15. Jadaun, V., & Singh, N. R. (2020). Soliton solutions of generalized (3+ 1)-dimensional Yu-Toda-Sasa- Fukuyama equation using Lie symmetry analysis. Analysis and Mathematical Physics, 10(4), 42.
16. Khalil, R., Al Horani, M., Yousef, A., & Sababheh, M. (2014). A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264, 65–70.
17. Kudryashov, N. A. (1988). Exact soliton solutions of the generalized evolution equation of wave dynamics. Journal of Applied Mathematics and Mechanics, 52(3), 361–365.
18. Kudryashov, N. A. (2005). Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos, Solitons & Fractals, 24(5), 1217–1231.
19. Kudryashov, N. A. (2012). One method for finding exact solutions of nonlinear differential equations. Communications in Nonlinear Science and Numerical Simulation, 17(6), 2248–2253.
20. Kumar, D., Singh, J., Tanwar, K., & Baleanu, D. (2019). A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler laws. International Journal of Heat and Mass Transfer, 138, 1222–1227.
21. Liu, H., Sun, Y., Yao, Y., Liu, Y., Luo, Y., & Sun, Q. (2024, November). Encoding of the Soliton Molecules in a Passively Mode-Locked Fiber Laser. In 2024 Asia Communications and Photonics Conference (ACP) (pp. 1–4). IEEE.
22. Lonngren, K. E. (1983). Soliton experiments in plasmas. Plasma Physics, 25(9), 943.
23. Lou, S. Y., & Huang, F. (2017). Alice-Bob physics: coherent solutions of nonlocal KdV systems. Scientific Reports, 7(1), 869.
24. Mohyud-Din, S. T., & Bibi, S. (2017). Exact solutions for nonlinear fractional differential equations using exponential rational function method. Optical and Quantum Electronics, 49(2), 64.
25. Naderifard, A. Z. A. D. E. H., Hejazi, S. R., & Dastranj, E. L. H. A. M. (2020). Conservation laws of the time-fractional Zakharov-Kuznetsov-Burgers equation. Kragujevac Journal of Mathematics, 44(1), 75-88.
26. Parkes, E. J., & Duffy, B. R. (1996). An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Computer Physics Communications, 98(3), 288–300.
27. Parkes, E. J., Duffy, B. R., & Abbott, P. C. (2002). The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations. Physics Letters A, 295(5-6), 280–286.
28. Ray, S. S., & Sahoo, S. (2016). Two efficient reliable methods for solving fractional fifth order modified Sawada-Kotera equation appearing in mathematical physics. Journal of Ocean Engineering and Science, 1(3), 219–225.
29. Sheikh, M. A. N., Taher, M. A., Hossain, M. M., & Akter, S. (2023). Variable coefficient exact solution of Sharma-Tasso-Olver model by enhanced modified simple equation method. Partial Differential Equations in Applied Mathematics, 7, 100527.
30. Selvaraj, R., Venkatraman, S., Ashok, D. D., & Krishnaraja, K. (2020). Exact solutions of time-fractional generalised Burgers–Fisher equation using generalised Kudryashov method. Pramana, 94(1), 137.
31. Stratmann, M., Pagel, T., & Mitschke, F. (2005). Experimental observation of temporal soliton molecules. Physical review letters, 95(14), 143902.
32. Thabet, H., & Kendre, S. (2023). A fractional mathematical model with nonlinear partial differential equations for transmission dynamics of severe acute respiratory syndrome coronavirus 2 infection. Healthcare Analytics, 4, 100209.
33. Ullah, N., Rehman, H. U., Asjad, M. I., Ashraf, H., & Taskeen, A. (2024). Dynamic study of Clannish Random Walker's parabolic equation via extended direct algebraic method. Optical and Quantum Electronics, 56(2), 183.
34. Vitanov, N. K. (2010). Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity. Communications in Nonlinear Science and Numerical Simulation, 15(8), 2050–2060.
35. Wright, L. G., Christodoulides, D. N., & Wise, F. W. (2017). Spatiotemporal mode-locking in multimode fiber lasers. Science, 358(6359), 94–97.
36. Xu, S., Viry, A., & Sander, M. Y. (2024). Dissimilar soliton molecule formed by dissipative pulses in a single-mode mode-locked fiber laser. Optics Express, 32(21), 37073–37084.
37. Yan, Z., & Lou, S. (2020). Soliton molecules in sharma-tasso-olver-burgers equation. Applied Mathematics Letters, 104, 106271.
38. Yousif, A. A., AbdulKhaleq, F. A., Mohsin, A. K., Mohammed, O. H., & Malik, A. M. (2023). A developed technique of homotopy analysis method for solving nonlinear systems of Volterra integro-differential equations of fractional order. Partial Differential Equations in Applied Mathematics, 8, 100548.
39. Zhao, H. Q. (2017). On a new semi-discrete integrable combination of Burgers and Sharma-Tasso-Olver equation. Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(2).
40. Zhou, Y., & Zhuang, J. (2022). Dynamics and exact traveling wave solutions of the Sharma-Tasso-Olver-Burgers equation. Symmetry, 14(7), 1468.
41. Zhou, T. Y., Tian, B., & Chen, Y. Q. (2023). Elastic two-kink, breather, multiple periodic, hybrid and half-/local-periodic kink solutions of a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves. Qualitative Theory of Dynamical Systems, 22(1), 34.
42. Zhou, Y., Yang, K., Tsia, K. K., Zeng, H., & Wong, K. K. (2025). Intelligent soliton molecules control in an ultrafast thulium fiber laser. Advanced Photonics Nexus, 4(1), 016012–016012.