New Exact Traveling Wave Solutions of Nonlinear Evolution Equations with Variable Coefficients

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1 Studies in Nonlinear Sciences (: 33-39, ISSN -39 IDOSI Publications, New Exact Traveling Wave Solutions of Nonlinear Evolution Equations with Variable Coefficients M.A. Abdou, E.K. El-Shewy and H.G. Abdelwahed Theoretical Physics Group, Department of Physics, Faculty of Science, Mansoura University, 3556 Mansoura, Egypt Abstract: The extended, (G/G-expansion method with a computerized symbolic computation is used for constructing the exact traveling wave solutions of nonlinear evolution equations with variable coefficients arising in physics. The obtained travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and rational functions. The applied method will be used in further wors to establish more entirely new solutions for other inds of two nonlinear evolution equations with variable coefficients arising in physics, namely, the variable coefficients nonlinear Sharma-Tasso-Olver (STO equation and the generalized Zaharov Kuznetsov equation with variable coefficients. Key words: Extended (G/G-expansion method nonlinear evolution equations with variable coefficients travelling wave solutions INTRODUCTION In recent years, nonlinear evolution equations in mathematical physics play a major role in various fields, such as fluid mechanics, plasma physics, optical fibers, solid state physics, chemical inematics, chemical physics and geochemistry. Nonlinear wave phenomena of dispersion, dissipation, diffusion, reaction and convection are very important in nonlinear wave equations. The investigation of exact solutions of nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena and gradually becomes one of the most important and significant tass. In the past several decades, many effective methods for obtaining exact solutions of NLEEs have been presented [-6]. The rest of this paper is arranged as follows. In section, we describe the extended (G/G-expansion method. In section 3, in order to illustrate the method two models of special interest in physics are chosen, namely, the variable coefficients nonlinear Sharma- Tasso-Olver (STO equation and the generalized Zaharov Kuznetsov equation and aboundant exact solutions are obtained which included the hyperbolic functions, the trigonometric functions and rational functions. Finally, conclusion and discussion are given in section. THE METHOD AND ITS APPLICATIONS Here, we give a brief description of the extended (G/G-expansion method [5-]. For a given a nonlinear equation, say in two independent variables x and t as: φ (u,u t,u,u,u x tt xx,... = ( where u = u(x,t is an unnown function φ is a polynomial in u = u(x,t and its various partial derivatives, in which the highest order derivatives and nonlinear term are involved. Combining the independent variables x and t into one variable ξ = (xct, we suppose that Eq. (, becomes u(x,t = u( ξ, ξ= x ct ( ϕ(u, cu,u,cu,u,... = (3 The solution of Eq. (3 can be expressed by a polynomial in G ( ( ξ G( ξ G( ξ u( ξ = a( m i i ( i= m G( ξ Corresponding Author: Dr. M.A. Abdou, Theoretical Physics Group, Department of Physics, Faculty of Science, Mansoura University, 3556 Mansoura, Egypt 33

2 Studies in Nonlinear Sci., (: 33-39, where G = G(ξ satisfies where G ( ξ G( ξµ G( ξ = (5 G G = = dg( ξ dξ dg( ξ a i, and µ are constants to be determined later, a i, the unwritten part in ( is also a polynomial in (G/G, but the degree of which is generally equal to or less than m-, the positive integer m can be determined by balancing the highest order derivative terms with nonlinear term appearing in Eq. (3. It is worth noting that the solutions of Eq. (5 for (G/G can be written in the form of hyperbolic, trigonometric and rational functions as given below. The first type: when -µ> G( ξ µ Csinh µξ C cosh µξ ( = G( ξ Ccosh µξ C sinh µξ dξ The second type: when -µ< (6 equation and the generalized Zaharov Kuznetsov equation with variable coefficients Example []. The generalized Zaharov Kuznetsov equation with variable coefficients: Let us first consider the generalized Zaharov Kuznetsov equation with variable coefficients as [7, 8]. u α (tuu β (tu u u γ (tu = (9 t x x xxx xyy where α(t, β (t and γ (t are arbitrary function of t. Eq. (9 includes considerable interesting equations, such as KdV equation, mkdv equation, ZK equation and mzk equation. The Zaharov-Kuznetsov equation with dualpower law nonlinearity is of interest in this paper. This equation is a generalization of the ZKE that is studied in the context of plasma physics. Recently, there have been results for the ZKE with power law nonlinearity [9]. The ZKE describes wealy nonlinear ion-acoustic modes in strongly magnetized plasma. Thus, the ZKE with dual-power law nonlinearity is a generalization of the ZKE with power law nonlinearity or the modified ZKE. To study the travelling wave solutions of Eq. (9,we tae a plane wave transformation in the form u = u( ξ, ξ= x ly τ(tdt ( G( ξ µ Csin µξ C cos µξ ( = G( ξ C cos µξ C sin µξ The third type: when -µ = G( ξ C ( = G( ξ C C ξ where C and C are integration constants. Inserting Eq. ( into (3 and using Eq. (5,collecting all terms with the same order (7 (8 G ( ξ G( ξ ( together, the left hand side of Eq. (3 is converted into another polynomial in G ( ξ G( ξ (. Equating each coefficients of this polynomial to zero, yields a set of algebraic equations for a i, and µ.with the nowledge of the coefficients a i and general solution of Eq. (5 we have more travelling wave solutions of the nonlinear evolution Eq. (. where, l are constants, τ(t is an integrable function of t to be determined later. Substituting Eq. ( into (9, we have τ(tu( ξ α(tu( ξu ( ξ 3 (tu (u( [ (tl ]u ( β ξ ξ γ ξ = ( where the prime denotes the differential with respect to ξ. In view of the technique of solution, we introduce the anstaz G u( a( N i ξ = ( i i=n G where a i are constants to be determined later. Our main goal is to solve Eq. ( by means illustrated above. Considering the homogeneous balance between u (ξ and u (ξ u(ξ in Eq. (, yields N =, we suppose that the solution of Eq. (9 can be expressed by NEW APPLICATIONS To illustrate the effectiveness and convenience of the proposed method, we consider the variable coefficients nonlinear Sharma-Tasso-Olver (STO 3 G G u( a a( a ( G G ξ = (3 where a, a, a - are constants to be determined later. Substituting Eq. (3 with Eq. (5 into Eq. (,

3 Studies in Nonlinear Sci., (: 33-39, G ( ξ G( collecting the coefficients of ( we obtain a set of algebraic equations for a ξ, a, a - and τ(t and solving this system with the aid of Maple Pacage we obtain the two sets of solutions as Case ( ² - ι ι β (² γ(t² ι a (a ((ta α 6 γ(t ι² (6(² γ(t² ι a=(, a =aµ, =, a =a, =, (t= - µ γ ι γ ιµ γ ι γ ιµα ( 9 (t 9 (t (t 96 (t (t a τ(t= (² γ (t² ι ( Case ( a ( (ta 6(tl 6( (tl α γ γ =,l= l,a =,a = a,a =, β (t = ( γ(tl a ( (tl (tl 96(tl 8(tl (t a 8 γ γ γ µ γ µα µ τ (t = ( γ(tl (5 By using Eqs. (, Eq. (3 can written as G( ξ a ( (ta 6(tl α γ G( ξ u( a[ ] [ ] a[( ] ξ = G( ξ ( γ(tl G( ξ (6 ( 9 (tl 9 (tl (t l 96 (t l (t a µ γ γ µ γ γ µα ξ= x ly [ ]dt (7 ( γ(tl With the aid of Eqs. (5, Eq. (3 can written as G ( ξ a ( (ta 6(tl α γ u ( ξ = a[( ] G( ξ ( γ(tl (8 ( (tl (t l 96(tl 8 (t l (t a 8 γ γ γ µ γ µα µ ξ= x ly [ ]dt (9 ( γ(tl with the nowledge of the solution of Eq. (5 and Eqs. (6-9, we have three types of travelling wave solutions of generalized Zaharov Kuznetsov equation with variable coefficients Eq. (9 as: The first type: when -µ> µ Csinh µξ Ccosh µξ a ( (ta 6(tl α γ u a( ξ = a[ ] ( (tl Ccosh µξ Csinh µξ γ µ Csinh µξ Ccosh µξ a[ ] Ccosh C sinh µξ µξ 35 (

4 Studies in Nonlinear Sci., (: 33-39, Csinh C cosh µ µξ µξ u ( a[ a a ξ = ] ( α (ta 6(tl γ Ccosh C sinh µξ µξ ( γ(tl ( The second type: when -µ< Csin C cos µ µ ξ µ ξ u ( a[ a b ξ = ] ( α (ta 6(tl γ [ ] Ccos C sin µ ξ µ ξ ( γ(tl Csin C cos µ µ ξ µ ξ a[ ] Ccos C sin µ ξ µ ξ ( Csin C cos µ µ ξ µ ξ u ( a[ b ξ = ] Ccos C sin µ ξ µ ξ a ( (ta 6(tl α γ ( γ(tl (3 The third type: when -µ = C a ( (ta 6(tl α γ C u ( a[ ] a[ ] c ξ = C C ξ ( γ(tl C Cξ C a ( (ta 6(tl α γ u c( ξ = a[ ] C C ξ ( γ(tl ( (5 where C and C are integration constants. Example []. The variable coefficients nonlinear Sharma-Tasso-Olver (STO equation: A second instructive model is the Sharma-Tasso-Olver equation with its fission and fusion which can be written as [3]. u f(t(uu u 3 t x x g(tuxxx = (6 3 where the coefficients ƒ(t, g(t and are both functions of variable t only. This equation is well nown as a model equation describing the propagation of nonlinear dispersive waves in inhomogeneous media. Since those variable coefficients are of practical important, it is meaningful to construct various exact analytic solutions. Obtaining exact solutions for nonlinear differential equations have long been one of the central themes of perpetual interest in mathematics and physics. To investigate the travelling propagation solution of STO Eq. (6, we first introduce the following transformation w t u(x,t = u( ξ, ξ= x g(t dt α (7 where w is the wave speed and α is a constant. Using transformation (7, STO (6 in terms of the new variable ξ reads w u u 3(uu u 3 ξ ξξξ ξ ξ = α 3 (8 where the functions ƒ(t and g(t in Eq. (6 should satisfy the condition ƒ(t = 3g(t. Upon integration Eq. (8, reduces to w u u 3(uu u 3 ξξ ξ α 3 = (9 Balancing the linear term of the highest order u ξξ with the nonlinear term u 3 (ξ, yields N =. Then the solution are given by G ( ξ G( ξ u( ξ = a a[( ] a [( ] G( ξ G( ξ (3 36

5 Studies in Nonlinear Sci., (: 33-39, where a, a, a - are constants to be determined later. Substituting Eq. (3 with Eq. (5 into Eq. (8, collecting the coefficients of G ( ξ G( ( we obtain a set of ξ algebraic equations for a, a, a -, w and α and solving this system with the aid of Maple, we have Case ( a,a,,a,w =µ = α=α = =ααµ (3 Case ( a =,w = ααµ,a =,a =, α=α (3 According to case (, Eq. (3 yields G( ξ G( ξ u( ( [( ] ξ = µ G( ξ G( ξ (33 ααµ t ξ= x g(t dt α (3 For case (, Eq. (3 becomes G( ξ u ( ξ = ( G( ξ (35 ααµ t ξ= x g(t dt α (36 using the solution of Eq. (5 into Eqs. (33-36, we have three types of travelling wave solutions as The first type: when -µ> Csinh C cosh µ µξ µξ u ( [ a ξ = ] Ccosh C sinh µξ µξ Csinh C cosh µ µξ µξ [ µ ] Ccosh C sinh µξ µξ Csinh C cosh µ µξ µξ u ( [ a ξ = ] Ccosh C sinh µξ µξ (37 (38 The second type: when -µ< Csin C cos µ µ ξ µ ξ u ( [ b ξ = ] Ccos C sin µ ξ µ ξ Csin C cos µ µ ξ µ ξ [ µ ] Ccos C sin µ ξ µ ξ (39 Csin C cos µ µ ξ µ ξ u ( [ b ξ = ] Ccos C sin µ ξ µ ξ ( The third type: when -µ = C C u ( [ ] [ ] c ξ = µ C Cξ C Cξ ( 37

6 C u ( [ c ξ = ] C Cξ where C and C are integration constants. CONCLUSION Studies in Nonlinear Sci., (: 33-39, ( In this study, we implement a new analytical technique, namely, (G/G-expansion method with a computerized symbolic computation Maple to establish the new exact travelling wave solutions for nonlinear evolution equations arising in mathematical physics, namely, Sharma-Tasso-Olver equation and Zaharov Kuznetsov equation with variable coefficients. As results, many exact travelling wave solutions are obtained which include the hyperbolic functions, trigonometric functions and rational functions. Finally, it is worthwhile to mention that the proposed method is reliable and effective and gives more solutions. The applied method will be used in further wors to establish more entirely new exact travelling wave solutions for other inds of nonlinear evolution equations with variable coefficients arising in physics. ACKNOWLEDGEMENT The authors would lie to express sincerely thans to the referees for their useful comments and discussions. REFERENCES. Ablowitzm, M. and P.A. Clarson, 99. Soliton, nonlinear evolution equations and inverse scattering. New Yor; Cambridge University Press.. El-Wail, S.A., S.K. El-Labany, M.A. Zahran, and R. Sabry, 5. Modified extended tanh-function method and its applications to nonlinear equations. Appl. Math. and Comput., 6: El-Wail, S.A. and M.A. Abdou, 7. Modified extended tanh function method for solving nonlinear partial differential equations. Chaos, Solitons and Fractals, 3: El-Wail, S.A. and M.A. Abdou, 8. New exact travelling wave solutions of two nonlinear physical models. Nonlinear Analysis, 68: Ping, Liu, C., 9. (G/G-expansion method equivalent to extended tanh function method. Commun. Theor. Phys., 5: Yanze Peng, et al., 8. Traveling wave-lie solutions of the Zaharov-Kuznetsov equation with variable coefficients. J. Phys., 7: Wazwaz, A.M., 7. New solitons and ins solutions to the Sharma-Tasso-Olver equation. Appl. Math. and Comput., 88: Mali, A., F. Chand, and S.C. Mishra,. Exact travelling wave solutions of some nonlinear equations by G/G]-expansion method. Appl. Math. and Comput. Doi:.6/J/amc Zayed, E.M.E. and A. Gepreel Khaled, 9. Some applications of the G/G]-expansion method to non-linear partial differential equations. Appl. Math. and Comput., : -3.. Zhang, H., 9. New application of the G/G]- expansion method. Comm.in Non.Sci. and Numer. Simul., :.. Ling, Lu, H. Qing Liu and Xi. Lei Niu, 9. A generalized G/G]-expansion method and its applications to nonlinear evolution equations. Appl. Math. and Comput., DOi..6/J.AMC Ling-Xi, Li. and Ming-Liang Wang, 9. The G/G]-expansion method and travelling wave solutions for a higher-order nonlinear Schrödinger equation. Appl. Math. and Comput., 8: Ugurlu, Y. et al., 7. Analytic method for solitary solutions of some partial differential equations. Phys. Lett. A, 37: Lia, Zitian and Zhangb Xiufeng,. New exact in solutions and periodic form solutions for a generalized Zaharov-Kuznetsov equation with variable coefficients. Commun Nonlinear Sci Numer Simulat, 5: Abulwafa, E.M., M.A. Abdou, and A.A. Mahmoud, 6. The solution of nonlinear coagulation problem with mass loss. Chaos, Solitons and Fractals, 9: He, J.H. and Xu-Hong Wu, 6. Exp-function method for nonlinear wave equations. Chaos Solitons and Fractals, 3: Abdou, M.A., 7. Further improved F-expansion and new exact solutions for nonlinear evolution equations. Nonlinear Dynamics, 5 (3: Abdou, M.A., 8. Exact periodic wave solutions for some nonlinear evolution equations. International Journal of Nonlinear Science, 5: -9.. Abdou, M.A. and S. Zhang, 9. New periodic wave solutions via extended mapping method. Communication in Nonlinear Science and Numerical Simulation, : -.. El-Wail, S.A., M.A. Madour and M.A. Abdou, 7. Application of Exp -function method for nonlinear evolution equations with variable coefficients. Physics Letter A, 369:

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