New Exact Travelling Wave Solutions for Regularized Long-wave, Phi-Four and Drinfeld-Sokolov Equations

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1 ISSN print), online) International Journal of Nonlinear Science Vol.008) No.1,pp.4-5 New Exact Travelling Wave Solutions for Regularized Long-wave, Phi-Four and Drinfeld-Sokolov Euations Ahmet BEKIR Mathematics Department, Dumlupinar University, Kutahya, Turkey Received 8 July 007, accepted 18 September 007) Abstract: In this paper, by using the solutions of an auxiliary ordinary differential euation, a direct algebraic method is described to construct the exact travelling wave solutions for RLW, PF euations and Drinfeld-Sokolov system. It is the method which can be adapted to solve nonlinear partial differential euations. Key words: travelling wave solutions; regularized long-wave euation RLW); Phi-Four euation PF); Drinfeld-Sokolov system DS) 1 Introduction The investigation for the travelling wave solutions of nonlinear partial differential euations plays an important role in the study of nonlinear physical phenomena. Nonlinear phenomena appears in a wide variety of scientific applications such as plasma physics, solid state physics, optical fibers, biology, fluid dynamics and chemical kinetics. Nonlinear wave phenomena of dispersion, dissipation, diffusion, reaction and convection are very important in nonlinear wave euations. Because of the increased interest in the theory of solitary waves, a broad range of analytical methods was used in the analysis of these scientific models. Mathematical modeling of many physical systems leads to nonlinear ordinary or partial differential euations in various fields of physics and engineering. An effective method is reuired to analyze the mathematical model which provides solutions conforming to physical reality. Common analytic procedures linearize the system or assume that nonlinearities are relatively insignificant. Such assumptions, sometimes strongly, affect the solution with respect to the real physics of the phenomenon. In recent years, new exact solutions may help to find new phenomena. Thus seeking exact solutions of nonlinear ordinary or partial differential euation is great of importance. Various powerful mathematical methods are such as inverse scattering method [1], Backlund transformation [-3], homogeneous balance method [4], tanh method [5-7], extended tanh method [8-10], sine-cosine method [11-13], pseudo spectral method [14], Jacobi elliptic method [15-1] and F-expansion method [17-19]. The auxiliary euation method Let us now simply describe the auxiliary euation method. Suppose we are given nonlinear partial differential euation for ux, t) in the form: We seek its wave solutions of the following form: P u, u t, u x, u tt, u xx,...) = 0 1) u = uξ), ξ = kx ct) + ξ 0, ) Corresponding author. address: abekir@dumlupinar.edu.tr Copyright c World Academic Press, World Academic Union IJNS /11

2 Ahmet BEKIR: New Exact Travelling Wave Solutions for Regularized Long-wave, Phi-Four 47 where ξ 0 is an arbitrary constant, k and c are the wave number and wave speed, respectively. Under the transformation ), E. 1) becomes an ordinary differential euation as Qu, u, u, u,...) = 0 3) By virtue of the above method we assume that the solution of E. 3) is of the form uξ) = m a i F i ξ), 4) i=0 and F ξ) is the solution of the auxiliary ordinary differential euation F ξ) = F ξ) + 3 F 3 ξ) + 4 F 4 ξ), 5) where, 3, 4 are real parameters, and hence holds for F ξ), F F = F F + 3 3F F + 4 F 3 F, F = F + 3 3F + 4 F 3, F = F F F + 4 F F. ) Integer m in 4) can be determined by considering homogeneous balance between the nonlinear terms and the highest derivatives of uξ) in E. 3). In this paper, we present new types solitary wave solutions of E. 5), we shall seek the explicitly solitary wave solutions of some nonlinear evolution euations by using the following new solitary wave solutions of E. 5): F ξ) = 1 ± sinh ) ξ) 4 4 cosh,, 4 > 0, = 0, 7) ξ) ± 1 and [0] F ξ) = 3 sec h ξ) 3 ), > 0, 4 1 tanh ξ) sec h ξ) sec h ξ), > 0, > 0. And then by using some significant special solutions of the auxiliary ordinary differential euation, some famous euations are investigated and exact solutions are explicitly obtained with the aid of symbolic computation [17]. As a result, we obtain new explicit solitary wave solutions of some nonlinear evolution euations. 8) 3 Exact travelling wave solutions of RLW euation The RLW euation is given by u t + au x uu x bu xxt = 0 a, b > 0, 9) where a, b are real constants [11]. Making the transformation ux, t) = uξ), ξ = kx ct) + ξ 0, E. 9) becomes a c)u uu + bck u = 0. 10) Following the balancing act procedure we balance the highest order of derivative term u with the highest power nonlinear term uu, yields m =. Therefore we may choose the solution of E. 10) in the form uξ) = a 0 + a 1 F + a F, 11) IJNS homepage:

3 48 International Journal of Nonlinear Science,Vol.008),No.1,pp. 4-5 where a 0, a 1 and a are constants to be determined later. It is easy to deduce that u = a 1 F + a F F, 1) u = a 4 F 4 + a a 3 )F a a )F + a 1 F, 13) u = 4a 4 F 3 F + 3a a 3 )F F + 3 a a )F F + a 1 F, 14) uu = a 0 a 1 F + a 0 a + a 1)F F + 3a 1 a F F + a F 3 F. 15) Substituting 1)-15) into 10), setting each coefficient of F i F i = 0, 1,, 3) to zero, yields a set of euations for a i, a c)a 1 a 0 a 1 + bck a 1 = 0, 1) We obtain a c)a a 0 a + a 1) + bck 3a a 4 ) = 0, 17) 18a 1 a + 3bck a a 3 ) = 0, 18) 1a + 4bck a 4 = 0. 19) a 0 = a c + bck, a 1 = bck 3, a = bck 4, 14 = ) Substituting 0) into 11) with 7) and 8), we obtain new solutions of E. 9) u 1 ξ) = a c + bck bck ± sinh ) ξ) cosh + bck 4 1 ± sinh ) ξ) ξ) ± 1 cosh ξ) ± 1 where, 4 > 0, and u ξ) = a c + bck bck 3 sec h ξ) tanh ξ) + bck 4 3 sec h ξ) tanh ξ) ) where > 0, and u 3 ξ) = a c + bck where > 0, > 0. bck sec h ξ) sec h sec h ξ) ξ) +bck sec h ξ) ) 4 Exact travelling wave solutions of PF euation We consider the Phi-four euation u tt au xx u + u 3 = 0, a > 0, 1) where a is real constant [11]. After making transformation ux, t) = uξ), ξ = kx ct) + ξ 0, E. 1) becomes k c a)u u u 3 = 0, ) Using the method mentioned above, we balance the highest order of derivative term u with the highest power nonlinear term u 3, yields m = 1. Therefore we may choose the solution of E. ) in the form uξ) = a 0 + a 1 F, 3) IJNS for contribution: editor@nonlinearscience.org.uk

4 Ahmet BEKIR: New Exact Travelling Wave Solutions for Regularized Long-wave, Phi-Four 49 where a 0 and a 1 are constants to be determined later. It is easy to deduce that u = a 1 F, 4) u = a 1 F = a 1 F + 3 3F + 4 F 3 ), 5) Substituting 4)-5) into ), setting each coefficient of F i i = 0, 1,, 3) to zero, yields a set of euations for a 0, a 1, a 0 + a 3 0 = 0, ) k c a)a 1 a 1 + 3a 0a 1 = 0, 7) 3 k c a)a a 1a 0 = 0, 8) k c a)a a 3 1 = 0. 9) We obtain Case A: Case B: a 0 = ±1, a 1 = 3, k c a) + = 0. 30) 4 a 0 = ±1, a 1 = ±, k c a) + = 0. 31) Substituting 30) and 31) into 3) with 7) and 8), we obtain new solutions of E. 1) Case A: u 1 ξ) = ±1 1 ± sinh ) ξ) cosh ξ) ± 1 where > 0, = 0, and u ξ) = ±1 3 sec h ξ) 3 ) 4 1 tanh ξ) where > 0, and u 3 ξ) = ±1 + 3 sec h ξ) sec h ξ) where > 0, > 0. Case B: u 1 ξ) = ±1 ± 1 ± sinh ) ξ) cosh ξ) ± 1 where > 0, and where, 4 > 0, and 4 3 sec h u ξ) = ±1 ± ξ) 3 ) 3) 4 1 tanh ξ) 4 sec h ξ) u 3 ξ) = ±1 ± sec h ξ) where > 0,, 4 > 0. IJNS homepage:

5 50 International Journal of Nonlinear Science,Vol.008),No.1,pp Exact travelling wave solutions of DS system We next consider the Drinfeld-Sokolov system u t + v ) x = 0, v t av xxx + 3bu x v + 3duv x = 0. 33) where a, b and d are real constants [1]. Using the wave variable ξ = kx ct) + ξ 0, the system 33) is carried to a system of ODEs cku + v ) = 0, cv + ak v 3bvu 3duv 34) = 0. Integrating the first euation in the system and neglecting the constant of integration we find cku = v. 35) Substituting 35) into the second euation of the system and integrating we find c kv + ack 3 v b + d)v 3 = 0. 3) Balancing v with v 3 in 34) gives so that n + = 3n, m = n. m =, n = 1. The auxiliary euation method 4) admits the solution of E. 33) in the form 37) 38) vξ) = b 0 + b 1 F, 39) where b 0 and b 1 are constants to be determined later. It is easy to deduce that v = b 1 F, 40) v = b 1 F = b 1 F + 3 3F + 4 F 3 ), 41) Substituting 40)-41) into 3), setting each coefficient of F i i = 0, 1,, 3) to zero, yields a set of euations for b 0, b 1, kc b 0 b + d)b 3 0 = 0, 4) We obtain kc b 1 3b + d)b 1 b 0 + ack 3 b 1 = 0, 43) 3 ack3 b 1 3 3b + d)b 1b 0 = 0, 44) ack 3 b 1 4 b + d)b 3 1 = 0. 45) b 0 = ± c k, b 1 = ± k ack 4, 3 = ) b + d b + d Substituting 4) into 3) with 35) and 39), we obtain new solutions of E. 33) u 1 ξ) = ± c b + d 1 ± sinh ) ξ) cosh, ξ) ± 1 v 1 ξ) = ± c k 1 ± sinh ) ξ) b + d cosh. ξ) ± 1 where > 0. 3 = 4 4. Due to this there is no solution providing E. 8). IJNS for contribution: editor@nonlinearscience.org.uk

6 Ahmet BEKIR: New Exact Travelling Wave Solutions for Regularized Long-wave, Phi-Four 51 Conclusion The auxiliary euation method was succesfully used to establish solitary wave solutions. Many well known nonlinear wave euations were handled by this method to show the new solutions compared to the solutions obtained in [1],[1]. The performance of the auxiliary euation method is reliable and effective and gives more solutions. The applied method will be used in further works to establish more entirely new solutions for other kinds of nonlinear wave euations. The availability of computer systems like Mathematica or Maple facilitates the tedious algebraic calculations. The method which we have proposed in this letter is also a standard, direct and computerizable method, which allows us to do complicated and tedious algebraic calculation. References [1] M. J. Ablowitz, P. A. Clarkson: Solitons, nonlinear evolution euations and inverse scattering transform.cambridge University Press, Cambridge.1990) [] M. Wadati: Introduction to solitons, Pramana.J. Phys.575-): ) [3] Lu, D., Hong, B., Tian, L.: Backlund transformation and n-soliton-like solutions to the combined KdV- Burgers euation with variable coefficients.international Journal of Nonlinear Science.:3-1000) [4] Wang, L., Zhou, J., Ren, L.: The exact solitary wave solutions for a family of BBM euation.international Journal of Nonlinear Science.1: ) [5] W. Malfliet: The tanh method: I. Exact solutions of nonlinear evolution and wave euations. Physica Sprica. 54: ) [] A.M. Wazwaz: The tanh method for travelling wave solutions of nonlinear euations.applied Mathematics and Computation.154: ) [7] E. Yusufoglu, A. Bekir: Exact solutions of coupled nonlinear evolution euations.chaos, Solitons & Fractals.373): ) [8] Fan, E.: Extended tanh-function method and its applications to nonlinear euations.phys. Lett. A.77:1000) [9] S.A. El-Wakil, M.A. Abdou: New exact travelling wave solutions using modified extended tanhfunction method. Chaos, Solitons & Fractals. 314): ) [10] E. Yusufoglu, A. Bekir: On the extended tanh method applications of nonlinear euations. International Journal of Nonlinear Science. 41): ) [11] A.M. Wazwaz: A sine-cosine method for handling nonlinear wave euations. Math. and Comput. Modelling. 40: ) [1] Wazwaz, A.M: The sine-cosine method for obtaining solutions with compact and noncompact structures. Applied Mathematics and Computation. 159): ) [13] E. Yusufoglu, A. Bekir: Solitons and periodic solutions of coupled nonlinear evolution euations by using sine-cosine method. International Journal of Computer and Mathematics. 831): ) [14] P. Rosenau, J.M Hyman.: Compactons: solitons with finite wavelengths. Phys. Rev. Lett.. 705): ) [15] E. Fan, Y. C. Hon: A series of travelling wave solutions for two variant Boussines euations in shallow water waves. Chaos Soliton & Fractals. 153): ) IJNS homepage:

7 5 International Journal of Nonlinear Science,Vol.008),No.1,pp. 4-5 [1] Zhenya Yan: Abundant families of Jacobi elliptic function solutions of the +1)-dimensional integrable Davey Stewartson-type euation via a new method. Chaos Soliton & Fractals.18): ) [17] Cai, G., Wang, Q., Huang, J.: A modified f-expansion method for solving breaking soliton euation. International Journal of Nonlinear Science.:1-1800) [18] D. Zhang: Doubly periodic solutions of modified Kawahara euation. Chaos Soliton & Fractals. 5: ) [19] Zhang, H.: New exact travelling wave solutions for some nonlinear evolution euations. Chaos Soliton & Fractals.: ) [0] Zhang, H.: New exact travelling wave solutions for some nonlinear evolution euations, Part II. Chaos, Soliton & Fractals.375): ) [1] Wazwaz, A.M.: Exact and explicit travelling wave solutions for the nonlinear Drinfield-Sokolov system. Commun. In Nonlinear Sci. and Num. Simulation.11: ) IJNS for contribution: editor@nonlinearscience.org.uk