Evolution of complex waves via Davey-Stewartson equation

Yıl 2025, Cilt: 3 Sayı: 1, 42 - 59, 29.05.2025

Yusuf Pandır , Nail Turhan

https://doi.org/10.70500/bjs.1666749

Öz

Davey-Stewartson equations (DSEs) have been discussed to examine the features of wave motion in finite depth water which is affected by gravitational force and surface tension. The new versions of the generalized F-expansion method and the generalized -expansion method are suggested to evaluate the analytical solutions of the DSEs. Thus, single, combined and mixed non-degenerative Jacobi elliptic function solutions (JEFSs) and degenerative solutions of the DSEs are obtained to contribute to the literature. These wave solutions are fresh and unexplored. The visual representations of some solutions are also shown in three and two dimensions.

Anahtar Kelimeler

New version of the generalized F-expansion method , New version of the generalized -expansion method , Davey-Stewartson equations , Single combined and mixed JEFSs

Davey-Stewartson denklemi ile karmaşık dalgaların evrimi

Öz

Davey-Stewartson denklemleri (DSE'ler), yerçekimi kuvveti ve yüzey geriliminden etkilenen sonlu derinlikteki suda dalga hareketinin özelliklerini incelemek için tartışılmıştır. Genelleştirilmiş F-genişleme yönteminin ve genelleştirilmişF'/F-genişleme yönteminin yeni versiyonları, DSE'lerin analitik çözümlerini değerlendirmek için önerilmiştir. Böylece, literatüre katkıda bulunmak için tek, birleşik ve karışık dejeneratif olmayan Jacobi eliptik fonksiyon çözümleri (JEFS'ler) ve DSE'lerin dejeneratif çözümleri elde edilmiştir. Bu dalga çözümleri yeni ve üzerine araştırmalar bulunmamaktadır. Bazı çözümlerin görsel temsilleri de üç ve iki boyutlu olarak gösterilmiştir.

Anahtar Kelimeler

Genelleştirilmiş F-genişleme yönteminin yeni versiyonu , Genelleştirilmiş -genişleme yönteminin yeni versiyonu , Davey-Stewartson denklemleri , Tek birleşik ve karışık JEFS

Kaynakça

• Ablowitz, M. J., & Clarkson, P. A. (1991). Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press.
• Baojian, H., Dianchen, L., & Fushu, S. (2009). The extended Jacobi elliptic functions expansion method and new exact solutions for the Zakharov equations. World Journal of Modelling Simulation, 3, 216–224.
• Cai, G., Wang, Q., & Huang, J. A. (2006). Modified F-expansion method for solving breaking soliton equation. International Journal of Nonlinear Science, 2, 122–128.
• Chen, H. T., & Hong-Qing, Z. (2006). New double periodic and multiple soliton solutions of the generalized (2 + 1)-dimensional Boussinesq equation. Chaos, Solitons & Fractals, 20, 765–769. https://doi.org/10.1016/j.chaos.2003.08.006
• Chen, Y., & Yan, Z. (2006). The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations. Chaos, Solitons & Fractals, 29, 948–964. https://doi.org/10.1016/j.chaos.2005.08.071
• Chuntao, Y. A. (1996). Simple transformation for nonlinear waves. Physics Letters A, 224, 77–84. https://doi.org/10.1016/S0375-9601(96)00770-0
• Enam, I. (2010). Generalized Jacobi elliptic function method for traveling wave solutions of (2+1)-dimensional breaking soliton equation. Cankaya University Journal of Science and Engineering, 1, 39–50.
• Fan, E. G. (2000). Extended tanh-function method and its applications to nonlinear equations. Physics Letters A, 277, 212–218. https://doi.org/10.1016/S0375-9601(00)00725-8
• Feng, Z. S. (2002). On explicit exact solutions to the compound Burgers-KdV equation. Physics Letters A, 293, 57–66. https://doi.org/10.1016/S0375-9601(01)00825-8
• Fu, Z., Liu, S., Liu, S., & Zhao, Q. (2001). New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Physics Letters A, 290, 72–76. https://doi.org/10.1016/S0375-9601(01)00644-2
• Gurefe, Y., Misirli, E., Ekici, M., & Sonmezoglu, A. (2013). Extended trial equation method to generalized nonlinear partial differential equations. Applied Mathematics and Computation, 219(10), 5253–5260. https://doi.org/10.1016/j.amc.2012.11.046
• Gurefe, Y., Misirli, E., & Sonmezoglu, A. (2011). Application of trial equation method to the nonlinear partial differential equations arising in mathematical physics. Pramana – Journal of Physics, 77(6), 1023–1029. https://doi.org/10.1007/s12043-011-0201-5
• Gurefe, Y., Sonmezoglu, A., & Misirli, E. (2012). Application of an irrational trial equation method to high dimensional nonlinear evolution equations. Journal of Advanced Mathematical Studies, 5(1), 41–47.
• Guo, S., & Zhou, Y. (2010). The extended -expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KdV equations. Applied Mathematics and Computation, 215, 3214–3221. https://doi.org/10.1016/j.amc.2009.10.008
• He, J. H., & Wu, X. H. (2006). Exp-function method for nonlinear wave equations. Chaos, Solitons & Fractals, 30, 700–708. https://doi.org/10.1016/j.chaos.2006.03.020
• Inc, M., & Ergut, M. (2005). Periodic wave solutions for the generalized shallow water wave equation by the improved Jacobi elliptic function. Applied Mathematics E-Notes, 5, 89–96.
• Liu, C. S. (2005). Trial equation method and its applications to nonlinear evolution equations. Acta Physica Sinica, 54(6), 2505–2509. https://doi.org/10.7498/aps.54.2505
• Liu, C. S. (2006). Trial equation method for nonlinear evolution equations with rank inhomogeneous: Mathematical discussions and applications. Communications in Theoretical Physics, 45(2), 219–223. https://doi.org/10.1088/0253-6102/45/2/005
• Liu, C. S. (2010). Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations. Computer Physics Communications, 181(2), 317–324. https://doi.org/10.1016/j.cpc.2009.10.006
• Liu, S., Fu, S., Liu, S., & Zhao, Q. (2001). Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Physics Letters A, 289, 69–74. https://doi.org/10.1016/S0375-9601(01)00580-1
• Lu, H. L., Liu, X. Q., & Niu, L. A. (2010). Generalized -expansion method and its applications to nonlinear evolution equations. Applied Mathematics and Computation, 215(11), 3811–3816. https://doi.org/10.1016/j.amc.2009.11.021
• Malanyuk, T. M. (1991). Finite-gap solutions of the Davey-Stewartson II equations. Communications of the Moscow Mathematical Society, 46(5), 193–194.
• Malanyuk, T. M. (1994). Finite-gap solutions of the Davey-Stewartson I equations. Journal of Nonlinear Science, 4, 1–21.
• Matveev, V. B., & Salle, M. A. (1991). Darboux transformations and solitons. Springer-Verlag.
• Pandir, Y. (2014). New exact solutions of the generalized Zakharov-Kuznetsov modified equal-width equation. Pramana – Journal of Physics, 82(6), 949–964. https://doi.org/10.1007/s12043-014-0748-z
• Pandir, Y., Gurefe, Y., & Misirli, E. (2013). Classification of exact solutions to the generalized Kadomtsev-Petviashvili equation. Physica Scripta, 87(2), 1–12. https://doi.org/10.1088/0031-8949/87/02/025003
• Pandir, Y., Gurefe, Y., & Misirli, E. (2013). The extended trial equation method for some time fractional differential equations. Discrete Dynamics in Nature and Society, 2013, Article ID 491359, 1–8. https://doi.org/10.1155/2013/491359
• Pandir, Y., Gurefe, Y., Kadak, U., & Misirli, E. (2012). Classifications of exact solutions for some nonlinear partial differential equations with generalized evolution. Abstract and Applied Analysis, 2012, Article ID 478531, 1–16. https://doi.org/10.1155/2012/478531
• Qi, W., Yong, C., & Zhang, H. (2005). A new Jacobi elliptic function rational expansion method and its application to (1+1)-dimensional dispersive long wave equation. Chaos, Solitons & Fractals, 23, 477–483. https://doi.org/10.1016/j.chaos.2004.04.029
• Shen, S., & Pan, Z. (2003). A note on the Jacobi elliptic function expansion method. Physics Letters A, 308, 143–148. https://doi.org/10.1016/S0375-9601(02)01802-9
• Tajiri, M., & Arai, T. (2010). Periodic soliton solutions to the Davey-Stewartson equation. Proceedings of the Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine, 30(1), 210–217.
• Wang, M., Li, X., & Zhang, J. (2008). The -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A, 372, 417–424. https://doi.org/10.1016/j.physleta.2007.07.051
• Wang, M., & Li, X. (2005). Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos, Solitons & Fractals, 24, 1257–1268. https://doi.org/10.1016/j.chaos.2004.09.044
• Xia, T., & Zhang, S. (2008). An improved generalized F-expansion method and its application to the (2+1)-dimensional KdV equations. Communications in Nonlinear Science and Numerical Simulation, 13, 1294–1301. https://doi.org/10.1016/j.cnsns.2006.12.008
• Xia, T. A., & Zhang, S. (2007). Generalized F-expansion method with symbolic computation exactly solving Broer–Kaup equations. Applied Mathematics and Computation, 189, 949–955. https://doi.org/10.1016/j.amc.2006.11.143
• Xia, X. Y., Haili, X., & Hongqing, Z. (2012). An extended elliptic equation expansion method and its application in the ZK-MEW equation. International Journal of Nonlinear Science, 3, 316–322.
• Xia, X. Y., Haili, X., & Hongqing, Z. (2012). A new extended Jacobi elliptic function expansion method and its application to the generalized shallow water wave equation. Journal of Applied Mathematics, 2012, Article ID 896748, 1–21. https://doi.org/10.1155/2012/896748
• Xun, C. (2011). Jacobi elliptic function solutions for (2 + 1) dimensional Boussinesq and Kadomtsev-Petviashvili equation. Applied Mathematics, 2, 1313–1316. https://doi.org/10.4236/am.2011.211183
• Yang, K., & Liu, J. (2004). The extended F-expansion method and exact solutions of nonlinear PDEs. Chaos, Solitons & Fractals, 22, 111–121. https://doi.org/10.1016/j.chaos.2003.12.069
• Yong, C., Qi, W., & Biao, L. (2004). Jacobi elliptic function rational expansion method with symbolic computation to construct new doubly-periodic solutions of nonlinear evolution equations. Zeitschrift für Naturforschung A, 59a, 529–536. https://doi.org/10.1515/zna-2004-0901
• Zhang, J. L., Wang, M. L., Wang, Y. M., & Fang, Z. D. (2006). The improved F-expansion method and its applications. Physics Letters A, 350, 103–109. https://doi.org/10.1016/j.physleta.2005.10.099
• Zhang, S., Li, W., Zheng, F., Yu, J., Ji, M., Lu, Z., & Ma, C. A. (2008). Generalized F-expansion method and its application to (2+1)-dimensional breaking soliton equations. International Journal of Nonlinear Science, 5, 25–32.