PAINLEVÉ- BÄCKLUND DENKLEMİNİN RASYONEL (G'/G) AÇILIM METODU İLE SOLITON ÇÖZÜMLERİ

Yıl 2024, Cilt: 23 Sayı: 45, 1 - 13, 26.06.2024

Sait San , Kübra Kaymak

https://doi.org/10.55071/ticaretfbd.1387780

Öz

Bu çalışmada lineer olmayan oluşum denklemlerinin ilerleyen dalga çözümlerinin bulunmasına yönelik rasyonel (G'/G) açılım yöntemi ele alınmıştır. Bu yöntem sayesinde trigonometrik fonksiyonlar, rasyonel fonksiyonlar ve hiperbolik fonksiyonlara göre düzenlenmiş uygun formdaki çeşitli soliton çözümler elde edilir. Aynı türden başka bir dalgayla çarpıştığında yok olmayan soliton dalgalarını incelemek için lineer olmayan 1+1-boyutlu Painlevé- Bäcklund denklemi üzerinde rasyonel (G'/G) açılım yöntemi uygulanmıştır. Bu yöntem kullanılarak Painlevé- Bäcklund denkleminin keyfi parametreleriyle ilerleyen dalga çözümleri başarıyla elde edilir. Parametrelere özel değerler verildiğinde ise ilerleyen dalgalardan denklemlerin soliter dalga çözümleri bulunarak 3-boyutlu ve kontur grafikleri çizdirilmiştir. Önerilen rasyonel (G'/G) açılım yöntemi doğrudan, basit ve etkilidir. Diğer birçok lineer olmayan ve tam sayı dengelenmeye sahip denklemler için etkili ve güçlü bir matematiksel yöntemdir.

Anahtar Kelimeler

Painlevé - Bäcklund Denklemi, Soliton çözüm, rasyonel açılım metodu

SOLITON SOLUTIONS OF THE PAINLEVÉ- BÄCKLUND EQUATION USING THE RATIONAL (G'/G) EXPANSION METHOD

Öz

In this study, the rational (G'/G) expansion method for finding traveling wave solutions of nonlinear formation equations is discussed. Thanks to this method, various soliton solutions in appropriate form arranged according to trigonometric functions, rational functions and hyperbolic functions are obtained. The rational (G'/G) expansion method was applied on the non-linear 1+1-dimensional Painlevé- Bäcklund equation to examine soliton waves that do not disappear when they collide with another wave of the same type. Using this method, traveling wave solutions with arbitrary parameters of the Painlevé-Bäcklund equation are successfully obtained. When special values were given to the parameters, solitary wave solutions of the equations were found from the traveling waves and 3-dimensional and contour graphs were drawn. The proposed rational (G'/G) expansion method is direct, simple and effective. It is an effective and powerful mathematical method for many other nonlinear and integer balancing equations.

Anahtar Kelimeler

Rational Expansion Method, Painlevé - Bäcklund Equation, Soliton Solution.

Kaynakça

• Akbar, M. A., Abdullah, F. A., Islam, M. T., Al Sharif, M. A., & Osman, M. S. (2023). New solutions of the soliton type of shallow water waves and superconductivity models. Results in Physics, 44, 106180.
• Fan, E., & Zhang, H. (1998). A note on the homogeneous balance method. Physics Letters A, 246(5), 403-406.
• He, J. H., & Wu, X. H. (2006). Exp-function method for nonlinear wave equations. Chaos, Solitons & Fractals, 30(3), 700-708.
• Hirota, R. (2004). The direct method in soliton theory (No. 155). Cambridge University Press.
• Hosseini, K., Sadri, K., Mirzazadeh, M., Chu, Y. M., Ahmadian, A., Pansera, B. A., & Salahshour, S. (2021). A high-order nonlinear Schrödinger equation with the weak non-local nonlinearity and its optical solitons. Results in Physics, 23, 104035.
• Hossen, M. B., Roshid, H. O., & Ali, M. Z. (2017). Modified double sub-equation method for finding complexiton solutions to the (1+ 1) dimensional nonlinear evolution equations. International Journal of Applied and Computational Mathematics, 3(Suppl 1), 679-697.
• Isidore, N. (1996). Exact solutions of a nonlinear dispersive-dissipative equation. Journal of Physics A: Mathematical and General, 29(13), 3679.
• Islam, M. T., Akbar, M. A., & Azad, A. K. (2015). A rational (G’/G)-expansion method and its application to modified KdV-Burgers equation and the (2+ 1)-dimensional Boussineq equation. Nonlinear Stud, 6(4), 1-11.
• Kakutani, T., & Kawahara, T. (1970). Weak ion-acoustic shock waves. Journal of the Physical Society of Japan, 29(4), 1068-1073.
• Malfliet, W. (1992). Solitary wave solutions of nonlinear wave equations. American journal of physics, 60(7), 650-654.
• Matveev, V. B., & Salle, M. A. (1991). Darboux transformations and solitons (Vol. 17). Berlin: Springer.
• Mohyud-Din, S. T., Noor, M. A., & Noor, K. I. (2009). Modified variational iteration method for solving Sine Gordon equations. World Appl. Sci. J, 6(7), 999-1004.
• 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.
• Salahshour, S., Hosseini, K. , Mirzazadeh, M., Ahmadian, A., Baleanu, D., & Khoshrang, A. (2021). The (2+ 1)-dimensional Heisenberg ferromagnetic spin chain equation: its solitons and Jacobi elliptic function solutions. The European Physical Journal Plus, 136(2), 1-9.
• Vakhnenko, V. O., Parkes, E. J., & Morrison, A. J. (2003). A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation. Chaos, Solitons & Fractals, 17(4), 683-692.
• Wang, M., Li, X., & Zhang, J. (2008). The (G′ G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A, 372(4), 417-423.
• Wazwaz, A. M. (2004). A sine-cosine method for handlingnonlinear wave equations. Mathematical and Computer modelling, 40(5-6), 499-508.
• Zhu, Z. N. (1993). Lax pair, Bäcklund transformation, solitary wave solution and finite conservation laws of the general KP equation and MKP equation with variable coefficients. Physics Letters A, 180(6), 409-412.