Auto-B𝐚̈cklund Transformation for Travelling Wave Solutions of Some Nonlinear Partial Differential Equations

Öz

In this paper, we implemented Auto-Bcklund transformation for finding the travelling wave solutions of the complexly coupled KdV equations and the sixth order equation of the Burgers hierarchy. These solutions are hyperbolic function solutions and exponential function solutions. The Auto- Bcklund transformation used in this article is a powerful method for finding traveling wave solutions of nonlinear partial differential equations.

Anahtar Kelimeler

• Auto-Ba ̈cklund transformation,Complexly coupled KdV equations

Kaynakça

1. [1]. Shang, Y. 2007. Backlund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation. Applied Mathematics and Computation;, 187: 1286-1297.
2. [2]. Bock, TL, Kruskal, MD. 1979. A two-parameter Miura transformation of the Benjamin-Ono equation. Physics Letters A; 74: 173-176.
3. [3]. Abourabia, A, El Horbaty MM. 2006. On solitary wave solutions for the two-dimensional nonlinear modified Kortwegde Vries-Burger equation. Chaos Solitons Fractals; 29: 354-364.
4. [4]. Malfliet, W. 1992. Solitary wave solutions of nonlinear wave equations. American Journal of Physics; 60: 650-654.
5. [5]. Chuntao, Y. 1996. A simple transformation for nonlinear waves. Physics Letters A; 224: 77-84.
6. [6]. Cariello, F, Tabor, M. 1989. Painleve expansions for nonintegrable evolution equations. Physica D; 39: 77-94.
7. [7]. Fan, E. 2000. Two new application of the homogeneous balance method. Physics Letters A; 265: 353-357.
8. [8]. Clarkson, PA. 1989. New similarity solutions for the modified boussinesq equation, Journal of Physics A: Mathematical and General; 22: 2355-2367.

9. [9]. Fan, E. 2000. Extended tanh-function method and its applications to nonlinear equations, Physics Letters A; 277: 212-218.
10. [10]. Elwakil, S A, El-labany, SK, Zahran, MA,d Sabry R. 2002. Modified extended tanh-function method for solving nonlinear partial differential equations. Physics Letters A; 299: 179-188.
11. [11]. Chen, H, Zhang, H. 2004. New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation. Chaos, Solitons and Fractals; 19: 71-76.
12. [12]. Wazwaz, AM. 2008. Analytic study on the one and two spatial dimensional potential KdV equations. Chaos Solitons and Fractals; 36: 175–181.
13. [13]. Kudryashov, N.A. 2019. The Painlevé approach for finding solitary wave solutions of nonlinear nonintegrable differential equations. Optik; 183: 642–649.
14. [14]. Chen, H T, Hong-Qing, Z. 2004. New double periodic and multiple soliton solutions of the generalized (2+1)-dimensional Boussinesq equation. Chaos, Solitons and Fractals; 20: 765-769.
15. [15]. Wazwaz, AM. 2018. Two-mode Sharma-Tasso-Olver equation and two-mode fourth-order Burgers equation: Multiplekink solutions. Alexandria Engineering Journal;57: 1971–1976.
16. [16]. Chen, Y, Yan, Z. 2006. The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations. Chaos Solitons and Fractals; 29: 948-964.
17. [17]. Wazwaz, AM. 2010. Burgers hierarchy: Multiple kink solutions and multiple singular kink solutions. Journal of the Franklin Institute; 347: 618–626.
18. [18]. 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.
19. [19]. Inan, IE, Ugurlu, Y, Bulut H. 2016. Auto-B cklund transformation for some nonlinear partial differential equations. Optik;127: 10780-10785.
20. [20]. Manafian J, Lakestain, M. 2016. Application of tan - expansion method for solving the Biswas-Milovic equation for Kerr law nonlinearity. Optik; 127: 2040-2054.
21. [21]. Manafian, J, Aghdaei, MF, Khalilian M, Jeddi, R.S. 2017. Application of the generalized -expansion method for nonlinear PDEs to obtaining soliton wave solution. Optik; 135: 395–406.
22. [22]. Zhou, Q, Ekici, M, Sonmezoglu, A, Mirzazadeh, M. 2016. Optical solitons with Biswas–Milovic equation by extended - expansion method. Optik; 127: 6277–6290.
23. [23]. Ebadi, G, Biswas, A. 2010. Application of the -expansion method for nonlinear diffusion equations with nonlinear source. Journal of the Franklin Institute; 347: 1391–1398.
24. [24]. Kudryashov, NA. 2018. Exact solutions and integrability of the Duffing–Van der Pol equation, Regul. Chaotic Dyn; 23 (4): 471–479.
25. [25]. Inan, IE, Kaya, D. 2007. Exact solutions of some nonlinear partial differential equations. Physica A; 381: 104-115.
26. [26]. Biswas, A, Ekici, M, Sonmezoglu, A, Belic, M.R. 2019. Highly dispersive optical solitons with Kerr law nonlinearity by F– expansion. Optik ;181: 1028–1038
27. [27]. Biswas, A, Ekici, M, Sonmezoglu, A, Belic, M.R. 2019. Highly dispersive optical solitons with quadratic–cubic law by Fexpansion. Optik ;182: 930–943.
28. [28]. Liu, J, Yang, L, Yang, K. 2004. Jacobi elliptic function solutions of some nonlinear PDEs. Physics Letters A; 325: 268-275.
29. [29]. Pandir, Y. 2017. A New Type of Generalized F-Expansion Method and its Application to Sine-Gordon Equation. Celal Bayar University Journal of Science; 13: (3) 647-650.
30. [30]. Yaşar, E, Giresunlu, İB. 2017. Symmetry Reductions, Exact Solutions and Conservation Laws for the Coupled Nonlinear Klein-Gordon System. Celal Bayar University Journal of Science; 13: (3) 593-599.