The tanh - coth Method for Soliton and Exact Solutions of the Sawada - Kotera Equation

Transcription

1 Volume 117 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu The tanh - oth Method for Soliton and Exat Solutions of the Sawada - Kotera Equation R. Asokan 1 and D. Vinodh 2 1,2 Department of Mathematis, Madurai Kamaraj Univerisy, Madurai , India. 1 rasokanmkuniversity@gmail.om 2 dvinothmaths@gmail.om Abstrat In this paper, we onsider Sawada - Kotera equation to obtain expliit exat soliton, ompaton, periodi and travelling wave solutions by using tanh oth method with the help of Mathematia. This signifiant method for handling many integrable nonlinear evolution equations to produe various types of solitary wave solutions. AMS Subjet Classifiation: 35C08, 35Q51. Key Words: Sawada - Kotera equation, tanh oth method, soliton, exat travelling wave solutions. 1 Introdution In past deades many nonlinear evolution equations are extensively studied by various types of powerful methods[1, 2, 3]. Nonlinearity play the entral role for mathematis, physis, engineering and sientifi fields to produe various types of solutions of nonlinear partial differential equations(nlpde) for different methods. The following methods, that is the inverse sattering method[1], Miura transformation[3], Auto - Bäklund transformation[4], similarity transformation[4, 5], Painlevé analysis[6], (G /G) Expansion method[7], homogeneous balane method[8], Hirota bilinear formalism[2, 3, 9, 10, 11], homotopy analysis method[19], variational iteration method[20], et. are ommonly used to find solutions of NLPDE. We onsider the Sawada - Kotera[11] equation, u t + 45u 2 u x 15u x u xx 15uu xxx + u xxxxx = 0 (1) 19

2 Equation (1) represents a nonlinear physial model equation. This signifiant model equation was studied in referene[9, 10, 11, 12, 13, 14, 15] for different approahes. In setion (3), we apply the tanh oth[8, 9, 10, 13, 14, 15, 16, 17] method for Sawada - Kotera equation (1), formally obtain soliton, ompaton, periodi and travelling wave solutions and given some of their illustrations. The new expliit soliton solution adequate previously known[20] numerial soliton solution and also produed new expliit solitary wave solutions. 2 The tanh-oth Method The tanh - oth method disovered by Wazwaz [16]. A wave variable ξ = x t onverts any Partial Differential Equation (PDE) to an Ordinary Differential Equation (ODE) P (u, u t, u x, u xx, u xxx,...) = 0, (2) Q(u, u, u, u,...) = 0. (3) Eq. (3) is then integrated as long as all terms ontains derivatives where integration onstants are onsidered zeros. The standard tanh method is developed by Malfliet[18], where the tanh is used as a new variable, sine all derivatives of a tanh are represented by tanh itself. Introduing a new independent variable, Y = tanh(µξ), ξ = x t, (4) where, µ is the wave number, leads to the hange of derivatives: d dξ = µ(1 Y 2 ) d dy, d 2 dξ = 2 2µ2 Y (1 Y 2 ) d dy + µ2 (1 Y 2 ) 2 d2 dy. 2 (5) The tanh oth method admits the use of the finite expansion u(µξ) = s(y ) = M M a k Y k + b k Y k, (6) k=0 k=1 and Y = µ(1 Y 2 ). (7) where, M is a positive integer, in most ases, that will be determined by Homogeneous Balane Method (HBM). Substituting (6) and (7) into the redued ODE results in an algebrai equation in powers of Y. Balaning the highest order the linear term with the highest order nonlinear term to determine the parameter M. We then ollet all oeffiients of powers of Y in the 20

3 resulting equation where these oeffiients have to vanish. This will give a system of algebrai equations involving the parameters a k, b k, µ, and. Having determined these parameters, we obtain an analyti solution u(x, t) in a losed form. The solutions we obtain may be solitons, kink, ompaton, peakon, uspon, travelling wave and periodi solutions as well. 3 Sawada Kotera Equation The Sawada Kotera Equation is u t + 45u 2 u x 15u x u xx 15uu xxx + u xxxxx = 0 (8) u(x, t) = u(ξ), ξ = x t (9) using (9) into (8) and integrating we get an ODE u + 45 u3 3 15uu + u = 0 (10) From Eq. (10) balaning the nonlinear term u 3 with the higher order derivative u we get the value M = 2 and hene, u(x, t) = s(y ) = a 0 + a 1 Y + a 2 Y 2 + b 1 Y + b 2 Y 2. (11) Substituting (11) into (10), olleting the oeffiients of eah power of Y i, setting eah oeffiient to zero. 21

4 } 60a 1 b 1 µ a 2 b 2 µ a 2 b a 0 a 1 b a 2 1b a 0 a 2 b 2 a 0 16a 2 µ 4 30a 0 b 2 µ 2 30a 0 a 2 µ a b 2 µ 4 = 0 (12) } 150a 2 b 1 µ 2 60a 1 b 2 µ a 2 1b a 0 a 2 b a 1 a 2 b 2 a a 1 µ 4 +30a 0 a 1 µ 2 30a 1 a 2 µ a 2 = 0 0a 1 (13) } 30a 1 b 1 µ 2 120a 2 b 2 µ a 1 a 2 b a 2 2b 2 a a 2 1µ 2 30a 2 2µ a 0 a 2 µ a 2 µ a 0 a a 2 = 0 0a 2 (14) 90a 2 b 1 µ a 2 2b 1 40a 1 µ 4 30a 0 a 1 µ a 1 a 2 µ a a 0 a 1 a 2 = 0 (15) 240a 2 µ 4 30a 2 1µ a 2 2µ 2 90a 0 a 2 µ a 0 a a 2 1a 2 = 0 (16) 24a 1 µ 4 120a 1 a 2 µ a 1 a 2 2 = 0 (17) 120a 2 µ 4 90a 2 2µ a 3 2 = 0 (18) } 30a 0 b 1 µ 2 60a 2 b 1 µ a 1 b 2 µ a 1 b a 2 0b 1 b a 0 a 1 b 2 +90a 2 b 1 b b 1 µ 4 30b 1 b 2 µ 2 = 0 (19) } 30a 1 b 1 µ a 0 b 2 µ 2 120a 2 b 2 µ a 0 b a 2 b a 2 0b 2 b 2 +90a 1 b 1 b 2 30b 2 2µ b 2 µ b 2 1µ 2 = 0 (20) 30a 0 b 1 µ 2 90a 1 b 2 µ a 1 b a 0 b 1 b 2 40b 1 µ b 1 b 2 µ b 3 1 = 0 (21) 90a 0 b 2 µ a 0 b b 2 µ 4 30b 2 1µ b 2 2µ b 2 1b 2 = 0 (22) 24b 1 µ 4 120b 1 b 2 µ b 1 b 2 2 = 0 (23) 120b 2 µ 4 90b 2 2µ b 3 2 = 0 (24) Solving the resulting algebrai systems (12-24) with omputer algebra software suh as Mathematia, we obtain the following sets: 22

5 a 0 = 2, a 1 = 0, a 2 = 2, b 1 = 0, b 2 = 0, µ = 1 4 (25) 2 a 0 = 2, a 1 = 0, a 2 = 0, b 1 = 0, b 2 = 2, µ = 1 4 (26) 2 a 0 = 4, a 1 = 0, a 2 = 8, b 1 = 0, b 2 = 8, µ = 1 4 (27) 4 1 a 0 = 2, a 1 = 0, a 2 = 2, b 1 = 0, b 2 = 0, µ = (28) 1 a 0 = 2, a 1 = 0, a 2 = 0, b 1 = 0, b 2 = 2, µ = (29) 1 a 0 = 4, a 1 = 0, a 2 = 8, b 1 = 0, b 2 = 8, µ = (30) Substituting Eqs.(4) and (25-30) into Eq. (11), we obtain the following soliton, ompaton, periodi and travelling wave solutions [ ] 1 u 1 (x, t) = 2 seh (x t), > 0 (31) [ ] 1 u 2 (x, t) = 2 sh (x t), > 0 (32) [ ] [ ]) 14 u 3 (x, t) = (2 tanh (x t) oth 2 14 (x t), > 0 (33) [ ] 1 u 4 (x, t) = 2 se (x t), > 0 (34) [ ] 1 u 5 (x, t) = 2 s (x t), > 0 (35) [ ] [ ]) 14 u 6 (x, t) = (2 + tan (x t) + ot 2 14 (x t), > 0 (36) 23

6 (a) = 2.5, 2 t 2, 5 x 5 (b) different values of with t = 0 and 5.9 x 5.9 () different values of x with = 2.5 and 6 t 10 Figure 1: Soliton solution of u 1 (x, t) 4 Conlusion In this paper, we studied Sawada - Kotera equation for briefly. Many as possible solutions suh as soliton, ompaton, periodi and travelling wave solutions are obtained analytially by tanh - oth method with the help of Mathematia. Referenes [1] R. Rajaraman, Solitons and instantons, Elsevier Siene Publishers, (1982). [2] A. M. Wazwaz, Partial differential equations and solitary waves theory, Springer Siene & Business Media, (2010). [3] Inna K.Shingareva, Carlos Lizárraga-Celaya, Maple and Mathematia: a problem solving approah for mathematis, Springer Siene & Business Media, (2011). [4] E. Fan, Auto-Bäklund transformation and similarity redutions for general variable oeffiient KdV equations, Physis Letters A., 294(1) (2002), [5] B. Mayil Vaganan, R. Asokan, Diret similarity analysis of generalized Burgers equations and perturbation solutions of Euler-Painlevé transendents, Studies in Appl. Math., 111(4) (2003), [6] M. Lakshmanan, K. M. Tamizhmani, Painlevé Analysis and Integrability Aspets of Nonlinear Evolution Equations, Solitons, Springer, Berlin, Heidelberg, (1988), [7] J. Manafian, L. Mehrdad, A. Bekir, Comparison between the generalized tanh - oth and the (G /G) -expansion methods for solving NPDEs and NODEs, Pramana - J. Phys., 87(6) (2016),

7 (a) = 2, 2 t 2, 4 x 4 (b) = 2, 2 t 2, 4 x 4 () = 0.2, 2 t 2, 0.4 x 0.4 (d) = 1, t = 0, 20 x 20 (e) = 5, t = 0, 15 x 15 Figure 2: (a), (b) are Travelling wave solutions of u 2 (x, t), u 3 (x, t), () is Compaton solution of u 4 (x, t) and (d), (e) are Periodi solutions of u 5 (x, t), u 6 (x, t). [8] E. Fan, Z. Hongqing, A note on the homogeneous balane method, Physis Letters A, 246(5) (1998), [9] A. M. Wazwaz, The Hirotas diret method and the tanh-oth method for multiple-soliton solutions of the Sawada-Kotera-Ito seventh-order equation, Appl. Math. and Comp., 199(1) (2008), [10] A. M. Wazwaz, The Hirotas bilinear method and the tanh-oth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equation, Appl. Math. and Comp., 200(1) (2008), [11] Ö. Ünsal,A. Bekir, F. Taan, M. Nai Özer, Complexiton solutions for two nonlinear partial differential equations via modifiation of simplified Hirota method, Waves in Random and Complex Media, 27(1) (2017), [12] K. Sawada, T. Kotera, A method for finding N-soliton solutions of the KdV equation and KdV-like equation, Prog. of Theo. Phy., 51(5) (1974), [13] A. M. Wazwaz, Abundant solitons solutions for several forms of the fifth-order KdV equation by using the tanh method, Appl. Math. and Comp., 182(1) (2006), [14] A. M. Wazwaz, The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Appl. Math. and Comp., 184(2) (2007), [15] C. A. Gómez, A. H. Salas, The variational iteration method ombined with improved generalized tanh-oth method applied to Sawada-Kotera equation, Appl. Math. and Comp., 217(4) (2010),

8 [16] A. M. Wazwaz, The tanh-oth method for solitons and kink solutions for nonlinear paraboli equations,appl. Math. and Comp., 188(2) (2007), [17] E. J. Parkes, B. R. Duffy, An automated tanh-funtion method for finding solitary wave solutions to non-linear evolution equations, Compu. phy. ommun., 98(3) (1996), [18] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Ame. J. of Phy., 60(7) (1992), [19] A. K. Gupta, and S. Saha Ray, Numerial treatment for the solution of frational fifth-order SawadaKotera equation using seond kind Chebyshev wavelet method, Appl. Math. Mod. 39(17) (2015), [20] M. T. Darvishi, and F. Khani, Numerial and expliit solutions of the fifthorder Korteweg-de Vries equations, Chaos, Solitons & Fratals 39(5) (2009),

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