Elsayed M. E. Zayed 1 + (Received April 4, 2012, accepted December 2, 2012)

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1 ISSN , England, UK Journal of Information and Computing Science Vol. 8, No., 03, pp A modified (G'/G)- expansion method and its application for finding hyperbolic, trigonometric and rational function solutions of nonlinear evolution equations Elsayed M. E. Zayed + Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt (Received April 4, 0, accepted December, 0) Abstract. A modified (G'/G)- expansion method is proposed to construct hyperbolic, trigonometric and rational function solutions of nonlinear evolution equations which can be thought of as the generalization of the (G'/G)- expansion method given recently by M.Wang et al. To illustrate the validity and advantages of this method, the (+)-dimensional Hirota-Ramani equation and the (+)-dimensional Breaing soliton equation are considered and more general traveling wave solutions are obtained. It is shown that the proposed method provides a more general powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Keywords: Nonlinear evolution equations, modified (G'/G)-expansion method, hyperbolic Function solutions, Trigonometric Function solutions, Rational function solutions.. Introduction Nonlinear evolution equations are often presented to describe the motion of isolated waves, localized in a small part of space, in many fields such as hydrodynamics, plasma physics, and nonlinear optics. Seeing exact solutions of these equations plays an important role in the study of these nonlinear physical phenomena. In the past several decades, many effective methods for obtaining exact solutions of these equations have been presented,such as the inverse scattering method [], the Hirota bilinear method [8,9], the Baclund transformation [8,8], the Painleve expansion method [,3,4,],the Sine-Cosine method [,5], the Jacobi elliptic function method [5,6], the tanh-function method [6,3,7,3], the F-expansion method [3], the exp-function method [4,5,7], the ( G / G) -expansion method [3,0,,7,9,0,4,6,8-30], the modified ( G / G) -expansion method [,7,30] and so on. Wang et.el [0] introduced the ( G / G) - expansion method to loo for traveling wave solutions of nonlinear evolution equations. This method is based on the assumption that these solutions can be expressed by a polynomial in ( G / G), and that G = G( ξ) satisfies a second order linear ordinary differential equation G ( ξ) + λg ( ξ) + μg( ξ) () where λ, μ are constants and ' = d / dξ, while ξ = x + ωt, and,ω are constants. The degree of this polynomial can be determined by considering the homogeneous balance between the highest-order derivatives and nonlinear terms appearing in the given nonlinear evolution equations. The coefficients of this polynomial can be obtained by solving a set of algebraic equations resulted from the process of using the method. The present paper is motivated by the desire to propose a modified ( G / G) -expansion method for constructing more general exact solutions of nonlinear evolution equation. To illustrate the validity and advantages of the proposed method, we would lie to employ it to solve the (+)- dimensional Hirota Ramani equation [,9] and the (+)-dimensional Breaing soliton equation [6,9]. The rest of this paper is organized as follows: In Sec., we describe the modified ( G / G) -expansion method. In Sec.3, we use this method to solve the two nonlinear equations indicated above. In Sec.4, conclusions are given. + address: e.m.e.zayed@hotmail.com. Published by World Academic Press, World Academic Union

2 4 Elsayed M. E. Zayed:A modified (G /G)-expansion method and its application for finding hyperbolic trigonometric and ration al function solutions of nonlinear evolution equation. Description of the modified (G'/G)-expansion method For a given nonlinear evolution equation in the form where ux (, yt, ), we use the wave transformation Puu (, x, uy, ut, uxx, u xt,...), () ux (, yt, ) = uξ ( ), ξ = x + y+ ωt, (3) where,, ω are constants, then Eq.() is reduced into the ordinary differential equation ( r) ( r ) ( ) Q u, u +,... (4) r ( r ) du ( r ) where u =, r 0and r is the least order of derivatives in the equation. Setting u = V ( ξ ), where r dξ V ( ξ) is a new function of ξ, we further introduce the following anstaz : m i ( r ) u ( ξ) = V ( ξ) = i +, i = 0 0, m (5) where G = G( ξ) satisfies Eq.() while i ( i,..., m) are constants to be determined later. To determine u( ξ ) explicitly, we tae the following four steps : Step. Determine the positive integer m in Eq. (5) by balancing the highest-order nonlinear terms and the highest-order derivatives, in Eq. (4). Step. Substitute Eq.(5) along with Eq.() into Eq. (4) and collect all terms with the same i powers of +, i = (0,,..., m) together, thus the left-hand side of (4) is converted into i a polynomial in +. Then set each coefficient of this polynomial to zero, to derive a set of algebraic for i,,, ω, i = (0,,..., m). Step 3. Solve these algebraic equations by use of Mathematica to find the values ofi,,, ω, i = (0,,..., m). Step 4. Use the results obtained in above steps to derive a series of fundamental solutions V ( ξ) of Eq.(4) depending on +. Since the solutions of Eq.() have been well nown for us as follows: (i) If λ 4μ > 0, then ξ ξ csinh λ 4μ ccosh λ 4μ G λ + λ 4 μ + =. ξ ξ ccosh λ 4μ + csinh λ 4μ (6) (ii) If λ 4μ < 0, then JIC for contribution: editor@jic.org.u

3 Journal of Information and Computing Science, Vol. 8 (03) No., pp ξ ξ csin λ 4μ ccos λ 4μ G λ + λ 4 μ + =. ξ ξ ccos λ 4μ + csin λ 4μ (7) (iii) If λ 4μ then c + =. c+ cξ (8) where c and c are constants, we can obtain exact solutions of Eq.() by integrating each of the obtained fundamental solutions V ( ξ) with respect to ξ and r times as follows: ξ ξ ξ r r = r r + j j = r j u( ξ)... V ( ξ ) dξ... dξ dξ d ξ, (9) where d ( j =,,..., r) are constants. Remar j It can easily be found that when r = 0, u( ξ) = V ( ξ) then (5) becomes the anstaz solutions obtained in [0]. When r,the solution (9) can be found in [30] and cannot be obtained by the methods in [0], see the next section for more details. 3. Applications Example. Nonlinear Hirota-Ramani equation In this subsection, we will use the proposed method of Sec., to find the solutions to Hirota-Ramani equation [,9]: u u + u ( u ) t xxt x t (0) where 0 is a constant. To this end, we use the wave transformation where, ω are constants, to reduce Eq.(0) to the following ODE: uxt (, ) = uξ ( ), ξ = x + ωt, () + = () ( ω ) u ωu ωu 0. Setting r = and u = V, we have u( ξ) = V ( ξ) dξ + d, where V ( ξ) satisfies the equation + = ( ω V ) ωv ωv 0. (3) According to Step, we get m+ = m, and hence m =. We then suppose that Eq.(3) has the formal solution JIC for subscription: publishing@wau.org.u

4 6 Elsayed M. E. Zayed:A modified (G /G)-expansion method and its application for finding hyperbolic trigonometric and ration al function solutions of nonlinear evolution equation λ V ( ξ ) = , (4) It is easy to see that 3 G λ G λ λ G λ λ V ( ξ ) = μ + + μ, 4 4 (5) 4 3 G λ G λ λ G λ λ G λ V ( ξ) = μ + μ λ + μ. (6) 4 Substituting (4)-(9) into Eq. (3) and collecting all terms with the same powers of + together, the left-hand side of Eq.(3) is converted into a polynomial in +. Setting each coefficient of this polynomial to zero, we get the following algebraic equations: 0:8 ( ω+ ) ω( λ 4 μ) 8ω 0 0 : ( ω+ ) + ω( λ 4 μ) 4ω 0 : ( ω+ ) + ω( λ 4 μ) ω ω 0 3: ω ω 4: 6 ω ω = 0. On solving the algebraic equations (7)-() we have two cases: (7) (8) (9) (0) () Case. 6 =, where λ μ ( 4 ). 3 ( 4 ), 0 = λ μ ω, = ( λ 4 μ) () Case. 6 =, 0 = ( λ 4 μ), ω, = + ( λ 4 μ) (3) where ( λ 4 μ). Consequently, we deduce the following exact solutions of Eq.(0): (i) If λ 4μ > 0( Hyperbolic solutions ) ξ When λ 4μ > 0, we set φ = λ 4 μ. Then for Case we get csinhφ+ ccoshφ 3 ( λ 4 μ) 3 ( λ 4 μ) u( ξ) = dξ ξ d ccoshφ+ csinhφ (4) JIC for contribution: editor@jic.org.u

5 Journal of Information and Computing Science, Vol. 8 (03) No., pp Substituting the formulas (8), (0), () and (4) obtained in [9] into (4) we have respectively the following in-type traveling wave solutions: () If c > c, then 3 ( λ 4 μ) 3 ( λ 4 μ) u( ξ) = tanh [ φ sgn( cc ) ψ ] dξ ξ d = 3 λ 4 μ tanh[ φ+ sgn( cc ) ]. ψ+ d (5) () If c > c 0, then 3 ( λ 4 μ) 3 ( λ 4 μ) u( ξ) = coth [ φ sgn( cc ) ψ ] dξ ξ d = 3 λ 4 μ coth[ φ+ sgn( cc ) ]. ψ + d (3) If c > c then 3 ( λ 4 μ) 3 ( λ 4 μ) u( ξ) = coth φdξ ξ d + + = 3 λ 4 μ coth φ + d. (6) (7) (4) If c = c, then u( ξ ) = d, (8) where ψ = tanh ( c / c), ψ = tanh ( c / c ) and sgn( cc ) is the sign function, while ξ is given by t ξ = x. ( λ 4 μ) Similarly, we can find the in-type traveling wave solutions for Case, which are omitted here for simplicity. (ii) If λ 4μ < 0( Trigonometric solutions ) For Case, we have ξ ξ csin 4μ λ + ccos 4μ λ 3 ( λ 4 μ) u( ξ ) = dξ ξ ξ ccos 4μ λ + csin 4μ λ 3 (4 μ λ ) ξ + d. We now simplify (9) to get the following periodic solutions: (9) JIC for subscription: publishing@wau.org.u

6 8 Elsayed M. E. Zayed:A modified (G /G)-expansion method and its application for finding hyperbolic trigonometric and ration al function solutions of nonlinear evolution equation () c where ξ = tan and c c () c where ξ = cot and c c omitted here. (iii) If 6 4μ λ d u( ξ) = tan ξ ξ 4 μ λ, + (30) + c μ λ d u( ξ) = cot ξ ξ 4 μ λ, + + (3) c 0. λ 4μ = 0 (Rational function solutions) For both Cases,, we have + Similarly, we can find the periodic solutions for Case which are where d is an arbitrary constant. c 6c ξ + [ + ( )] 6 u( ξ) = d d d, + = + c c ξ c c x t (3) 3.. Example. Nonlinear Breaing soliton equation In this subsection, we will use the proposed method of Sec., to find the solutions to the Breaing soliton equation [6,8,9]: u 4u u u u + u = 0. xt x xy xx y xxxy (33) To this end, we use the wave transformation ux (, yt, ) = uξ ( ), ξ = x + y+ ωt, (34) where, and ω are constants, to reduce Eq.(33) to the following ODE: ω u 6 uu + u = 0. (35) Integrating (35) once w.r.t. ξ, with zero constant of integration, we get ω u 3 u + u = 0. (36) Setting r =, and u = V, we deduce that V ( ξ) satisfies the equation: ω V 3 V + V = 0. (37) According to Step, we get m =. Thus the formal solution of Eq.(37) has the same form (4). JIC for contribution: editor@jic.org.u

7 Journal of Information and Computing Science, Vol. 8 (03) No., pp Substituting (4)-(6) into Eq.(37) and collecting the coefficients of each coefficient to zero, we get the following algebraic equations: 0:8ω 4 + ( λ 4 μ) 0 0 :ω ( λ 4 μ) 0 : ω 3 6 ( λ 4 μ) 0 3: 6 + 4: = 0. On solving the algebraic equation (38)-(4) we have two cases: + j j,,3, 4. Setting (38) (39) (40) (4) (4) Case. =, Case. =, λ μ 0 = ( 4 ), λ μ 6 0 = ( 4 ), ω λ μ = ( 4 ), (43) ω λ μ = ( 4 ). (44) Consequently, we deduce the following exact solutions of Eq.(33) : (i) If λ 4μ > 0( Hyperbolic solutions ) Setting ξ = Then for Case we get φ λ 4 μ. c sinhφ+ c coshφ u = d + d ( ξ) ( λ 4 μ) ξ ( λ 4 μ) ξ.. ccoshφ+ csinhφ (45) Substituting the results (8), (0), () and (4) of [9] into (45), we have respectively, the following Kin-type traveling wave solutions: () If c > c, then u( ξ) = λ 4μ tanh[ φ+ sgn( cc ) ψ ] + d () If c > c 0, then u cc d ( ξ) = λ 4μcoth[ φ+ sgn( ) ψ] +. (3) If c > c then u d ( ξ) = λ 4μcoth φ+. (46) (47) (48) JIC for subscription: publishing@wau.org.u

8 0 Elsayed M. E. Zayed:A modified (G /G)-expansion method and its application for finding hyperbolic trigonometric and ration al function solutions of nonlinear evolution equation (4) If c = c, then where ξ = x+ y ( λ 4 μ) t. u( ξ ) = d, (49) Similarly, we can find the in-type traveling wave solutions for Case, which are omitted here for simplicity. (ii) If λ 4μ < 0( Trigonometric solutions ) For Case, we have ξ ξ csin 4μ λ + ccos 4μ λ u( ξ) = (4 μ λ ) (4 ). dξ μ λ ξ + d ξ ξ ccos 4μ λ + csin 4μ λ We now simplify (50) to get the following periodic solutions: (50) () u( ξ) = 4μ λ tan ξ ξ 4 μ λ + d, (5) c where ξ = tan and c + c 0. c () u( ξ) = 4μ λ cot ξ + ξ 4 μ λ + d, (5) c where ξ = cot and c c which are omitted here. c 0. (iii) If λ 4μ = 0 (Rational function solutions) + Similarly, we can find the periodic solutions for Case For both Cases,, we have c c ξ c+ cξ c+ c( x + y) u( ξ) = d + d = + d. where d is an arbitrary constant. (53) 4. Conclusion This study shows that the modified ( G / G )- expansion method is quite efficient and well suited for finding exact solutions to the Hirota-Ramani equation and the Breaing soliton equation. Our solutions are in more general forms, and many nown solutions to these equations are special cases of them. With the aid of Mathematica, we have assured the correctness of the obtained solutions by putting them bac into the original equations. 5. References JIC for contribution: editor@jic.org.u

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10 Elsayed M. E. Zayed:A modified (G /G)-expansion method and its application for finding hyperbolic trigonometric and ration al function solutions of nonlinear evolution equation generalized K(n,n) and BBM equations with variable coefficients, Int. J. Nonlinear Sci. (0),PP [6] E.M.E. Zayed and M. Abdelaziz and M.El-Maly, Enhanced ( G / G )-expansion method and applications to the (+)D typical Breaing soliton and Burgers equations, J. Adv. Math. Stud; 4 (0),PP 09-. [7] E.M.E. Zayed and H.M. Abdel Rahman, The extended tanh-method for finding traveling wave solutions of nonlinear evolution equations, Appl. Math. E-Notes. 0 (00),PP [8] E.M.E. Zayed and S. Al-Joudi, Applications of an extended ( G / G )-expansion method to find exact solutions of nonlinear PDEs in mathematical physics, Math. Probl. Eng. Volume 00, Article ID , 9 pages. [9] E.M.E. Zayed and S. Al-Joudi, Applications of an improved ( G / G )-expansion method to nonlinear PDEs in mathematical physics, AIP Conf. Proc. 68 (009),PP [30] S. Zhang, Y.N. Sun, J.M.Ba and L.Dong, The modified ( G / G )-expansion method for nonlinear evolution equations, Z.Naturforsch. 66a (0),PP [3] S. Zhang and T.C. Xia, A further improved tanh-function method exactly solving (+)- dimensional dispersive long wave equations, Appl. Math-E-Notes, 8 (008),PP [3] S. Zhang and T.C. Xia, A generalized F-expansion method and new exact solutions of Konopelcheno- Durovsy equation, Appl. Math. Comput. 83 (006),PP JIC for contribution: editor@jic.org.u