İkinci mertebeden Benjamin-Ono denkleminin korunum kanunları yardımıyla çift indirgemesi ve tam çözümleri

Yıl 2021, Cilt: 23 Sayı: 1, 210 - 223, 29.01.2021

Yeşim Sağlam Özkan

https://doi.org/10.25092/baunfbed.848234

Öz

Bu çalışmada, ilk kez tabakalı sıvılardaki iç dalgaları tanımlamak için sunulan Benjamin-Ono denklemini ele alınmıştır. Lie nokta simetrileri ve yerel korunum vektörleri arasındaki ilişkiyi kullanarak hem değişken sayısında hem de denklemin mertebesinde bir indirgeme elde edilmiştir. İndirgenen denkleme yardımcı denklem metodu başarılı bir şekilde uygulanmş ve farklı tipte çözümler elde edilmiştir. Ayrıca çözümlerdeki parametrelerin özel değerleri için bazı grafik temsilleri verilmiştir.

Anahtar Kelimeler

Çift indirgeme yöntemi, korunum vektörleri, Benjamin-Ono denklemi

Double reduction of second order Benjamin-Ono equation via conservation laws and the exact solutions

Öz

In this study, the Benjamin-Ono equation which was first introduced to describe internal waves in stratified fluids are considered. Using the association between Lie point symmetries and local conserved vectors, a reduction in both the number of variables and the order of the equation is achieved. The auxiliary equation method successfully applied to the reduced equation and different types of solutions are obtained. Moreover, some graphical representations for special values of the parameters in solutions are presented.

Anahtar Kelimeler

Double reduction method, conservation vectors, Benjamin-Ono equation

Kaynakça

• Ablowitz, M.J., Kaup, D.J., Newell, A.C. ve Segur, H., Method for solving the sine-Gordon equation, Physical Review Letters, 30, 25, 1262, (1973).
• Cahn, J.W., ve Hilliard, J.E., Free energy of a nonuniform system. I. Interfacial free energy, The Journal of chemical physics, 28,2, 258-267, (1958).
• Temam, R., Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, (1979).
• Lee, S.T. ve Brockenbrough, J.R., A new approximate analytic solution for finite-conductivity vertical fractures, SPE Formation Evaluation, 1,01, 75–88, (1986).
• Durur, H., Different types analytic solutions of the (1+ 1)-dimensional resonant nonlinear Schrödinger’s equation using (G′/G)-expansion method, Modern Physics Letters B, 34, 03, 2050036, (2020).
• Chen, C.J. ve Chen, H.C., Finite analytic numerical method for unsteady two-dimensional Navier-Stokes equations, Journal of Computational Physics, 53, 209–226, (1984).
• Ahmad, H., Khan, T.A., Durur, H., Ismail, G.M., ve Yokus, A., Analytic approximate solutions of diffusion equations arising in oil pollution, Journal of Ocean Engineering and Science, (2020).
• Yokus, A., Durur, H., Ahmad, H., ve Yao, S.W., Construction of Different Types Analytic Solutions for the Zhiber-Shabat Equation, Mathematics, 8, 6, 908, (2020).
• Yokus, A., Durur, H., ve Ahmad, H., Hyperbolic type solutions for the couple Boiti-Leon-Pempinelli system, Facta Universitatis, Series: Mathematics and Informatics, 35, 2, 523-531, (2020).
• Morales-Delgado, V.F., Gómez-Aguilar, J.F., Yépez-Martínez, H., Baleanu, D., Escobar-Jimenez, R.F. ve Olivares-Peregrino, V.H., Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Advances in Difference Equations, 2016, 164, (2016).
• Gao, W., Senel, M., Yel, G., Baskonus, H.M., ve Senel, B., New complex wave patterns to the electrical transmission line model arising in network system, AIMS Math, 5, 3, 1881-1892, (2020).
• Yokuş, A., Durur, H., Abro, K.A., ve Kaya, D., Role of Gilson–Pickering equation for the different types of soliton solutions: a nonlinear analysis, The European Physical Journal Plus, 135, 8, 1-19, (2020).
• Younis, M., Optical solitons in (n+1) dimensions with Kerr and power law nonlinearities, Modern Physics Letters B, 31,15, 1750186, (2017).
• Yokus, A., On the exact and numerical solutions to the FitzHugh–Nagumo equation, International Journal of Modern Physics B, 2050149, (2020).
• Durur, H., Ilhan, E., ve Bulut, H., Novel Complex Wave Solutions of the (2+ 1)-Dimensional Hyperbolic Nonlinear Schrödinger Equation, Fractal and Fractional, 4, 3, 41, (2020).
• Osman, M.S., Rezazadeh, H., Eslami, M., Neirameh, A. ve Mirzazadeh, M., Analytical study of solitons to Benjamin-Bona-Mahony-Peregrine equation with power law nonlinearity by using three methods, University Politehnica of Bucharest Scientific Bulletin-Series A-Applied Mathematics and Physics, 80, 4, 267-278, (2018).
• Durur, H., Tasbozan, O., ve Kurt, A., New Analytical Solutions of Conformable Time Fractional Bad and Good Modified Boussinesq Equations, Applied Mathematics and Nonlinear Sciences, 5, 1, 447-454, (2020).
• Biswas, A., Yıldırım, Y., Yaşar, E., Zhou, Q., Moshokoa, S.P. ve Belic, M., Optical soliton solutions to Fokas-lenells equation using some different methods, Optik, 173, 21-31, (2018).
• Durur, H., ve Yokuş, A., Analytical solutions of Kolmogorov–Petrovskii–Piskunov equation, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 22, 2, 628-636, (2020).
• Durur, H., Kurt, A., ve Tasbozan, O., New Travelling Wave Solutions for KdV6 Equation Using Sub Equation Method, Applied Mathematics and Nonlinear Sciences, 5, 1, 455-460, (2020).
• Hosseini, K., Seadawy, A.R., Mirzazadeh, M., Eslami, M., Radmehr, S., ve Baleanu, D., Multiwave, multicomplexiton, and positive multicomplexiton solutions to a (3+ 1)-dimensional generalized breaking soliton equation, Alexandria Engineering Journal, (2020).
• Durur, H., ve Yokuş, A., Vakhnenko-Parkes Denkleminin Hiperbolik Tipte Yürüyen Dalga Çözümü, Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 13, 2, 550-556, (2020).
• Saglam Özkan, Y., Yaşar, E., ve Seadawy, A.R., A third-order nonlinear Schrödinger equation: the exact solutions, group-invariant solutions and conservation laws, Journal of Taibah University for Science, 14, 1, 585-597, (2020).
• Hereman, W., Banerjee, P.P., Korpel, A., Assanto, G., Van Immerzeele, A. ve Meerpoel, A., Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method, Journal of Physics A: Mathematical and General, 19, 5, 607, (1986).
• Korpel, A. ve Banerjee, P.P., A heuristic guide to nonlinear dispersive wave equations and soliton-type solutions, Proceedings of the IEEE, 72, 9, 1109-1130, (1984).
• Lai, H. ve Ma, C., The lattice Boltzmann model for the second-order Benjamin- Ono equations, Journal of Statistical Mechanics: Theory and Experiment, 2010, 04, P04011, (2010).
• Yan, Z.Y., New families of solitons with compact support for Boussinesq-like B(m, n) equations with fully nonlinear dispersion, Chaos Solitons Fractals, 14, 1151-1158, (2002).
• Fu, Z.T., Liu, S.K., Liu, S.D. ve Zhao, Q., The JEFE method and periodic solutions of two kinds of nonlinear wave equations, Communications in Nonlinear Science and Numerical Simulation, 8, 67-75, (2003).
• Xu, Z.H., Xian, D.Q. ve Chen, H.L., New periodic solitary-wave solutions for the Benjamin Ono equation, Applied Mathematics and Computation, 215, 4439-4442 (2010).
• Taghizadeh, N., Mirzazadeh, M. ve Farahrooz, F., Exact soliton solutions for second-order Benjamin-Ono equation, Applications and Applied Mathematics, 6, 384-395 (2011).
• Zhen, W., De-Sheng, L., Hui-Fang, L. ve Hong-Qing, Z., A method for constructing exact solutions and application to Benjamin Ono equation, Chinese Physics, 14, 11, 2158, (2005).
• Bessel-Hagen, E., Über die erhaltungssätze der elektrodynamik, Mathematische Annalen, 84, 3-4, 258-276, (1921).
• Ibragimov, N.H., CRC handbook of Lie group analysis of differential equations (Vol. 3), CRC press, (1995).
• Kara, A.H. ve Mahomed, F.M., Action of Lie–Bäcklund symmetries on conservation laws, Modern Group Analysis, 7, (1997).
• Kara, A.H. Ve Mahomed, F.M., Relationship between Symmetries and Conservation Laws, International Journal of Theoretical Physics, 39, 1, 23-40, (2000).
• Steeb, W.H. ve Strampp, W., Diffusion equations and Lie and Lie-Bäcklund transformation groups, Physica A: Statistical Mechanics and its Applications, 114, 1-3, 95-99, (1982).
• Bokhari, A.H., Al-Dweik, A.Y., Zaman, F.D., Kara, A.H. ve Mahomed, F.M., Generalization of the double reduction theory. Nonlinear Analysis: Real World Applications, 11, 5, 3763-3769, (2010).
• Sabi’u, J., Rezazadeh, H., Tariq, H. ve Bekir, A., Optical solitons for the two forms of Biswas–Arshed equation, Modern Physics Letters B, 33, 25, 1950308, (2019).
• Rocha Filho, T.M. ve Figueiredo, A., [SADE] a Maple package for the symmetry analysis of differential equations, Computer Physics Communications, 182, 2, 467-476, (2011).
• Kaplan, M., San, S. ve Bekir, A., On the exact solutions and conservation laws to the Benjamin-Ono equation, Journal of Applied Analysis and Compututation, 8, 1, 1-9, (2018).
• Naz, R., Ali, Z., ve Naeem, I., Reductions and new exact solutions of ZK, Gardner KP, and modified KP equations via generalized double reduction theorem. In Abstract and Applied Analysis, 2013, Hindawi, (2013).