Solitary Wave Solutions of the (3+1)-dimensional Khokhlov–Zabolotskaya–Kuznetsov Equation by Using the (G^'/G,1/G)-Expansion Method

Year 2021, Volume: 11 Issue: 2, 290 - 301, 31.12.2021

Hülya Durur , Serbay Duran , Asıf Yokuş

https://doi.org/10.37094/adyujsci.885861

Abstract

In this study, the (3+1)-dimensional Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, which is a mathematical model of non-absorption and dispersion in the non-linear medium, which sheds light on the sound beam phenomenon, which has a physically important place, is examined. In order to find the exact solution of this equation, an effective and reliable method, (G^'/G,1/G)-expansion method, is used among analytical methods. The purpose of this method is to obtain more than one traveling wave solution classes depending on the conditions of the λ parameter. These classes are categorized into hyperbolic, trigonometric, complex trigonometric and rational forms. The graphics of the solitary waves represented by these successfully obtained solution classes are presented as 2-dimensional, 3-dimensional and contours. This article makes use of ready-made package programs for complex arithmetic operations and graphic drawings.

Keywords

(G'/G,1/G)-Expansion Method, The (3+1)-dimensional Khokhlov–Zabolotskaya–Kuznetsov Equation, Traveling wave solution

(3+1)-Boyutlu Khokhlov–Zabolotskaya–Kuznetsov Denkleminin (G^'/G,1/G)-Açılım Metodu Yardımıyla Solitary Dalga Çözümleri

Abstract

Bu çalışmada, fiziksel olarak önemli bir yere sahip olan ses ışını (sound beam) olayına ışık tutan, özellikle lineer olmayan ortamda dağılım ve soğurma olmayan durumların matematiksel modeli olan (3+1)-boyutlu Khokhlov–Zabolotskaya–Kuznetsov (KZK) denklemi incelendi. Bu denklemin tam çözümünü bulmak için analitik metotlar arasında yer alan etkili ve güvenilir bir yöntem olan (G^'/G,1/G)-açılım metodu kullanıldı. Bu metodun seçilme amacı λ parametresinin durumlarına bağlı olarak birden fazla yürüyen dalga çözüm sınıfları elde edilmesidir. Bu sınıflar hiperbolik, trigonometrik, kompleks trigonometrik ve rasyonel formda kategorize edilir. Başarılı bir şekilde elde edilen bu çözüm sınıflarının temsil ettiği solitary dalgaların grafikleri 2-boyutlu, 3-boyutlu ve kontur olarak sunuldu. Bu makalede karmaşık aritmetik işlemler ve grafik çizimleri için hazır paket programlardan faydalanıldı.

Keywords

(G^, 1/G)-açılım metodu; (3+1)-boyutlu Khokhlov–Zabolotskaya–Kuznetsov Denklemi; Solitary dalga çözümleri.

References

• [1] Bulut, H., Baskonus, H.M., Pandir, Y., The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation, Abstract and Applied Analysis, vol. 2013, Article ID 636802, 2013.
• [2] Arnous, A.H., Seadawy, A.R., Alqahtani, R.T., Biswas, A., Optical solitons with complex Ginzburg–Landau equation by modified simple equation method, Optik, 144, 475-480, 2017.
• [3] Xiong, M., Chen, L., Li, C., Wang, J., Exact Solutions for (2+ 1)–Dimensional Nonlinear Schrödinger Schrodinger Equation Based on Modified Extended tanh Method, In the International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery, 224-231, 2019.
• [4] Zayed, E.M., Shohib, R.M., Optical solitons and other solutions to Biswas–Arshed equation using the extended simplest equation method, Optik, 185, 626-635, 2019.
• [5] Gürefe, Y., Aktürk, T., Investigation of Analytical Solutions of the Nonlinear Mathematical Model Representing Gas Overflowing, Adıyaman University Journal of Science, 11(1), 182-190, 2021.
• [6] Duran, S., Karabulut, B., Nematicons in liquid crystals with Kerr Law by sub-equation method, Alexandria Engineering Journal, 61(2), 1695-1700, 2022.
• [7] Durur, H., Taşbozan, O., Kurt, A., Şenol, M., New Wave Solutions of Time Fractional Kadomtsev-Petviashvili Equation Arising In the Evolution of Nonlinear Long Waves of Small Amplitude, Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 12(2), 807-815, 2019.
• [8] Sulaiman, T.A., Yavuz, M., Bulut, H., Baskonus, H.M., Investigation of the fractional coupled viscous Burgers’ equation involving Mittag-Leffler kernel, Physica A: Statistical Mechanics and its Applications, 527, 121126, 2019.
• [9] Tozar, A., Tasbozan, O., Kurt, A., Analytical solutions of Cahn-Hillard phase-field model for spinodal decomposition of a binary system, Europhysics Letters, 130(2), 24001, 2020.
• [10] Yavuz, M., Ozdemir, N., Baskonus, H.M., Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, The European Physical Journal Plus, 133(6), 1-11, 2018.
• [11] Aktürk, T., Kubal, Ç., Analysis of wave solutions of (2+1)-dimensional Nizhnik-Novikov-Veselov equation, Ordu Üniversitesi Bilim ve Teknoloji Dergisi, 11(1), 13-24, 2021.
• [12] 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.
• [13] Yokus, A., Durur, H., Ahmad, H., Hyperbolic type solutions for the couple Boiti-Leon-Pempinelli system, Facta Universitatis, Series: Mathematics and Informatics, 35(2), 523-531, 2020.
• [14] Duran, S., Yokuş, A., Durur, H., Kaya, D., Refraction simulation of internal solitary waves for the fractional Benjamin–Ono equation in fluid Dynamics, Modern Physics Letters B, 2150363, 2021.
• [15] Li, L., Li, E., Wang, M., The (G′/G, 1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations, Applied Mathematics-A Journal of Chinese Universities, 25, 454–462, 2010.
• [16] Duran, S., Solitary Wave Solutions of the Coupled Konno-Oono Equation by using the Functional Variable Method and the Two Variables (G'/G, 1/G)-Expansion Method, Adıyaman Üniversitesi Fen Bilimleri Dergisi, 10(2), 585-594, 2020.
• [17] Yokus, A., Durur, H., Ahmad, H., Yao, S.W., Construction of Different Types Analytic Solutions for the Zhiber-Shabat Equation, Mathematics, 8(6), 908, 2020.
• [18] Yokuş, A., Durur, H. Duran, S. Simulation and refraction event of complex hyperbolic type solitary wave in plasma and optical fiber for the perturbed Chen-Lee-Liu equation, Optical and Quantum Electronics, 53, 402, 1-17, 2021.
• [19] Yavuz, M., Yokus, A., Analytical and numerical approaches to nerve impulse model of fractional‐order, Numerical Methods for Partial Differential Equations, 36(6), 1348-1368, 2020.
• [20] Duran, S., Exact Solutions for Time-Fractional Ramani and Jimbo-Miwa Equations by Direct Algebraic Method, Advanced Science, Engineering and Medicine, 12(7), 982-988, 2020.
• [21] Kaya, D., Yokuş, A., Demiroğlu, U., Comparison of exact and numerical solutions for the Sharma–Tasso–Olver equation, In Numerical Solutions of Realistic Nonlinear Phenomena, 53-65, 2020.
• [22] Yokuş, A., Durur, H., Abro, K. A., 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.
• [23] Tozar, A., Tasbozan, O., Kurt, A., Optical soliton solutions for the (1+1)-dimensional resonant nonlinear Schröndinger’s equation arising in optical fibers, Optical and Quantum Electronics, 53(6), 1-8, 2021.
• [24] Rezazadeh, H., Kurt, A., Tozar, A., Tasbozan, O., Mirhosseini-Alizamini, S. M., Wave behaviors of Kundu–Mukherjee–Naskar model arising in optical fiber communication systems with complex structure, Optical and Quantum Electronics, 53(6), 1-11, 2021.
• [25] Yokus, A., Durur, H., Ahmad, H., Thounthong, P., Zhang, Y. F., Construction of exact traveling wave solutions of the Bogoyavlenskii equation by (G′/G, 1/G)-expansion and (1/G′)-expansion techniques, Results in Physics, 103409, 2020.
• [26] Yavuz, M., Sene, N., Approximate solutions of the model describing fluid flow using generalized ρ-laplace transform method and heat balance integral method, Axioms, 9(4), 123, 2020.
• [27] Duran, S., Breaking theory of solitary waves for the Riemann wave equation in fluid dynamics. International Journal of Modern Physics B, 35(9), 2150130, 2021.
• [28] Saleem, S., Hussain, M.Z., Aziz, I., A reliable algorithm to compute the approximate solution of KdV-type partial differential equations of order seven, Plos One, 16(1), e0244027, 2021.
• [29] Duran, S., Kaya, D., Applications of a new expansion method for finding wave solutions of nonlinear differential equations, World Applied Sciences Journal, 18(11), 1582-1592, 2012.
• [30] Kumar, M., Kumar, R., Kumar, A., On similarity solutions of Zabolotskaya–Khokhlov equation, Computers & Mathematics with Applications, 68(4), 454-463, 2014.
• [31] Zabolotskaya, E.A., Khokhlov, R.V., Quasi-plane waves, in the nonlinear acoustics of confined beams, Soviet Physics Acoustics, 15, 35-40, 1969.
• [32] Akçağıl, Ş., Aydemir, T., New exact solutions for the Khokhlov-Zabolotskaya-Kuznetsov, the Newell-Whitehead-Segel and the Rabinovich wave equations by using a new modification of the tanh-coth method, Cogent Mathematics, 3(1), 1193104, 2016.
• [33] Chirkunov, Y.A., Belmetsev, N.F., Invariant submodels and exact solutions of Khokhlov–Zabolotskaya–Kuznetsov model of nonlinear hydroacoustics with dissipation, International Journal of Non-Linear Mechanics, 95, 216-223, 2017.
• [34] Kuznetsov, V.P., Equations of nonlinear acoustics, Soviet Physics Acoustics, 16, 467-470, 1971.
• [35] Ray, S.S., New analytical exact solutions of time fractional KdV–KZK equation by Kudryashov methods, Chinese Physics B, 25(4), 040204, 2016.
• [36] Zhang, L., Ji, J., Jiang, J., Zhang, C., The new exact analytical solutions and numerical simulation of (3+ 1)-dimensional time fractional KZK equation, International Journal of Computing Science and Mathematics, 10(2), 174-192, 2019.
• [37] Demiray, Ş.T., Kastal, S., MEFM For Exact Solutions Of The (3+1) Dimensional KZK Equation and (3+1) Dimensional JM Equation, Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, 21(1), 97-105, 2021.
• [38] Duran, S., Extractions of travelling wave solutions of (2+ 1)-dimensional Boiti–Leon–Pempinelli system via (Gʹ/G, 1/G)-expansion method, Optical and Quantum Electronics, 53(6), 1-12, 2021.