Travelling Wave Solutions for the Gilson-Pickering Equation by Using the Simplified G /G-expansion Method

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1 ISSN (print, (online International Journal of Nonlinear Science Vol8(009 No3,pp Travelling Wave Solutions for the ilson-pickering Equation by Using the Simplified /-expansion Method Xinghua Fan, Shouxiang Yang, Dan Zhao Department of Mathematics, Jiangsu University, Zhenjiang, Jiangsu,1013,PRChina (Received 3 March 009, accepted 30 June 009 Abstract: We simplify the expression of / in the / expansion method to tanh, coth, cot and rational forms under some conditions The simplified /-method can be thought as a combination of tanh-coth method and cot method Travelling wave solutions for the ilson- Pickering equation are considered The equation contains the Fornberg-Whitham equation, the Rosenau-Hyman equation and the Fuchssteiner-Fokas-Camassa-Holm equation as its special cases We construct its travelling wave solutions involving parameters by using the simplified / expansion method The resulting solutions are expressed by hyperbolic, trigonometric functions and rational functions Keywords: the /-expansion method; traveling wave solutions; solitary wave solutions; homogeneous balance; hyperbolic function solutions; trigonometric function solutions 1 Introduction We consider the class of fully nonlinear third-order partial differential equations (PDEs u t εu xxt + κu x uu xxx αuu x βu x u xx = 0, (1 where ε, κ, α and β are arbitrary constants The equation was introduced by ilson and Pickering (P in [11] and was called the ilson-pickering equation in [3] where solutions for β = 1 were studied Our results are corresponding to β = Three special cases of Eq(1 have appeared in the literature Up to some rescalings, these are:the Fornberg-Whitham (FW equation [9, 0, 1],the Rosenau-Hyman (RH equation [15],the Fuchssteiner-Fokas-Camassa-Holm (CH equation [, 10] Studies to the P equation (1 include [4, 11] For β = 1, the qualitative behavior and exact travelling wave solutions of Eq(1 were studied by using the qualitative theory of polynomial differential system[3] We are interested to find the explicit solutions of Eq (1 With the fast development of symbolic computation systems, directly searching for exact solutions of PDEs by symbolic computation has attracted much attention Various methods have appeared such as the homogenous balance method [16], the Sine-Cosine method [8], the Jacobi elliptic function method [7], the sine-gordon equation expansion method [4], the F-expansion method [6], the Exp-function method[1, 13,, 3] and so on In the present paper, we shall use the /-expansion method [17] It is based on the explicit linearization of nonlinear differential equations for traveling waves with a certain substitution which leads to a second-order differential equation with constant coefficients The /-expansion method have been applied to many kinds of PDEs [1, 5, 18] It is also applied to lattice equation in [5] It is pointed out in [14] that the /-expansion method coincides with the truncated expansion method if we use the travelling wave solutions The /-expansion method is equivalent to application of the simplest equation method with the Riccati equation, to the tanh-function method and to the truncated expansion method Corresponding author address: fan131@ujseducn Copyright c World Academic Press, World Academic Union IJNS009115/91

2 X Fan, S Yang, D Zhao: Travelling Wave Solutions for the ilson-pickering Equation by Using 369 We simplify the expression of / in the / expansion method to tanh, coth, cot and rational forms under some conditions and call it the simplified / method We say that the simplified /-method can be thought as a combination of tanh-coth method and cot method Hyperbolic, trigonometric and rational function solutions can be found The layout of this paper is organized as follows In Section, we give the description of the simplified /-expansion method In Section 3, we apply this method to the P equation Conclusions are given in the last section Description of the ( /-expansion method We first describe the /-expansion method [17] with two independent variables For a given nonlinear PDE P (u, u t, u x, u tt, u xt, u xx, = 0, ( where u = u(x, t is an unknown function, P is a polynomial in u and its partial derivatives, in which the highest order derivatives and nonlinear terms are involved There are mainly four steps in the /expansion method Step 1 The travelling wave variable permits us to reduce Eq ( to an ODE for u = φ(ξ in the form u(x, t = φ(ξ, ξ = x V t, (3 P (φ, V φ, φ, V φ, V φ, φ, = 0, (4 where denotes derivative about ξ Step Suppose that the solution of ODE (4 can be expressed by a polynomial in / as follows: φ(ξ = m i=0 ( i a i, (5 where = (ξ satisfies the following second order linear ODE + λ + μ = 0, (6 where a i (i = 0, 1,,, m, λ and μ are constants to be determined later, a m = 0 The degree of the polynomial m can be determined by balancing the highest order derivative with nonlinear terms Step 3 Substituting (5 into (4 and using the second order linear ODE (6 and then equating each coefficient of the resulted polynomial to zero, yields a set of algebraic equations with respect to a i (i = 0, 1,,, m, V, λ, and μ Solving the algebraic system, we may find the values of unknowns Step 4 Substituting a i (i = 0, 1,,, m, V, λ, μ gotten in Step 3 and the general solutions of Eq (6 into (5 we can obtain more traveling wave solutions of the nonlinear PDE ( Solutions to Eq(?? depending on whether λ 4μ > 0, λ 4μ > 0, λ 4μ = 0, = λ 4μ cosh( 1 λ 4μ ξ+c sinh( 1 λ 4μ ξ sinh( 1 λ 4μ ξ+c cosh( 1 λ 4μ 4μ λ cos( 1 4μ λ ξ C sin( 1 4μ λ ξ sin( 1 4μ λ ξ+c cos( 1 4μ λ ξ λ, λ 4μ > 0, ξ λ, λ 4μ < 0, C +C ξ λ, λ 4μ = 0, As a result, solutions in sinh-cosh, or sin-cos forms are given in [17] and others Next we will give the simplified method (7 IJNS homepage:

3 370 International Journal of Nonlinear Science,Vol8(009,No3,pp As pointed in [14], some authors do not simplify the solutions of differential equations We find that these form should be simplified to ( λ 4μ λ tanh 4μ ξ + ξ 0 λ, λ 4μ > 0, tanh ξ 0 = C, C > 1, ( λ 4μ λ coth 4μ ξ + ξ 0 λ, λ 4μ > 0, coth ξ 0 = C, C < 1, = ( 4μ λ ξ + ξ 0 λ, λ 4μ < 0, tan ξ 0 = C 4μ λ cot C +C ξ λ, λ 4μ = 0 Simplifying (7 with (8, solutions can be written in more strait form Now we call the method the simplified /-method We will use (8 to get travelling wave solution of Eq(1 3 Application to the ilson-pickering equation In this section, we apply the simplified /-expansion method to construct the traveling wave solutions of the ilson-pickering equation (1 Combining the independent variables x and t into one variable ξ = x V t, we suppose that Then Eq (1 is converted into an ODE for u = φ(ξ Integrating Eq (10 with respect to ξ once yields u(x, t = φ(ξ, ξ = x V t (9 (κ V φ + εv φ φφ αφφ βφ φ = 0 (10 (κ V φ + εv φ φφ α φ + 1 β (φ = C (11 where C ia an integration constant that is to be determined later Suppose that the solution of the ODE (10 can be expressed by polynomials in terms of / as follows: m ( i φ(ξ = a i, (1 i=0 while = (ξ satisfies (6 The positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in ODE (11 To determine the degree of the polynomial solutions, we can only substitute the leading term Let the degree in / of (1 be D(φ Then D(φ p ( d s φ dξ s q = pd(φ + (s + D(φq So we get m = and ( ( φ(ξ = a + a 1 + a 0, (13 By substituting (13 and its derivatives into Eq (11 and collecting all terms with the same power of / together, the left-hand side of Eq (11 is converted into another polynomial in / Equating each coefficient of this polynomial to zero, yields a set of simultaneous algebraic equations for a, a 1, a 0, V, λ, μ and C For the length of th epaper, we just omit it We can see that only when β = can we get some solutions In the rest text we will take β = Solving the algebraic equations,yields three sets of solutions: Set A: a = 4 a 0 + α ε a 0 κε (1 + α ε (λ α, a 1 = a λ, a 0 = a 0, V = κ 1 + α ε, ( κε C = α, μ = λ α, 1 + αε 4 (8 (14 IJNS for contribution: editor@nonlinearscienceorguk

4 X Fan, S Yang, D Zhao: Travelling Wave Solutions for the ilson-pickering Equation by Using 371 Set B: a = 1 a 0 + α ε a 0 κε (1 + α ε (3 λ + α, a 1 = a λ, a 0 = a 0, V = κ 1 + α ε, ( αa0 (1 + αε + εκ(3λ α ( 3εκ(λ + α αa 0 (1 + αε C = α μ = λ + α, 4 (1 + α ε (3 λ + α, (15 Set C: a = 4 ε ( ε κ α ε a 0 a 0 (1 λ ε (1 + α ε, a 1 = λa, a 0 = a 0, V = α ε a 0 ε κα + 6 ε λ κ 4 ε κ a 0 3ε (1 λ ε (1 + α ε C = κ ( α ε a 0 + a 0 ε κ ε κα + 3 ε 3 κα λ 3 (1 λ ε (1 + α ε, μ = λ ε 1 4ε If the integration constant C in Eq(11 is correspondingly taken as that in (14, (15 and (16, we can write (13 in expression of / and the parameters ε, κ, α, β Substituting (8 into it we can derive the travelling wave solutions of Eq (1 31 First travelling solution set To Set A, substituting (14 and (8 into (1, we deduce the following three types of traveling wave solutions of the P equation (1: Case 1-1 When λ 4μ = α > 0, we have the hyperbolic function travelling wave solution (16 u + 1 (x, t = ( α ε a 0 + ε κ a 0 α α (1 + α ε (λ sech ( α ξ + ξ ε κ α ε κ 1+α ε t, ξ 0 = tanh 1 C, C / > 1, and u + (x, t = (α ε a 0 ε κ + a 0 α (1 + α ε (λ csch ( α α ξ + ξ 0 + ε κ 1 + α ε (17 (18 κ 1+α ε t, ξ 0 = coth 1 C C / < 1, Solution (17 is a well-known bounded solitary wave solution Case 1- When λ 4μ = α < 0, we have the trigonometric solutions u 1 (x, t =(a 0 + α ε a 0 ε κ α (1 + α ε (λ α κ 1+α ε t, ξ 0 = tan 1 C Case 1-3 When λ 4μ = α = 0, the rational solutions can be found: α cot ( ξ + ξ 0 (a 0 + α ε a 0 ε κ λ (1 + α ε (λ + a 0, (19 α u 0 1(x, t = 4 a 0 ε κ λ C ( + C ξ a 0 + ε κ (0 κt, and C are arbitrary constantsthe integration constants C in (11 is now zero We can see that all travelling wave solutions are independent of variable t when κ = 0 3 Second travelling solution set Substituting (15 and (8 into (1, three types of traveling wave solutions of the P equation (1 are obtained: IJNS homepage:

5 37 International Journal of Nonlinear Science,Vol8(009,No3,pp Case -1 When λ 4μ = α > 0 If C / > 1, we get a solitary solution u + 3 (x, t =3 ( α ε a 0 + ε κ a 0 α α (1 + α ε (3 λ sech ( + α ξ + ξ 0 + α ε a 0 3 α ε κ + α a ε λ κ (1 + α ε (3 λ, + α (1 κ 1+α ε t, ξ 0 = tanh 1 C If C / < 1, letting C / = coth ξ 0, we have u + 4 (x, t =3 (α ε a 0 ε κ + a 0 α α (1 + α ε (3 λ + α csch ( ξ + ξ 0 + α ε a 0 3 α ε κ + α a ε λ κ (1 + α ε (3 λ ( + α κ 1+α ε t, ξ 0 = coth 1 C, Case - When λ 4μ = α < 0,we can get a periodic solution u (x, t = 3α (α ε a 0 ε κ + a 0 (1 + α ε (3 λ + α cot ( α ξ + ξ 0 3 (α ε a 0 ε κ + a 0 λ (1 + α ε (3 λ + a 0, (3 + α κ 1+α ε t, ξ 0 = tan 1 C When λ 4μ = α = 0, we have rational solution just the same as (0 33 Third travelling solution set Substituting (16 and (8 into (1, we deduce the following traveling wave solutions of the P equation (1: Case 3-1: When λ 4μ = 1 ε > 0, we can get a solitary solution u + 5 (x, t = ε (α ε a 0 ε κ + a 0 α ( 1 + λ ε (1 α ε sech ( ξ ε + ξ 0 α ε a 0 ε κα + ε λ κ a 0 (1 λ ε (1 + α ε (4 where ξ = x + α ε a 0 ε κα+6 ε λ κ 4 ε κ a 0 3ε (1 λ ε(1+α ε t, ξ 0 = tanh 1 C, C / > 1, and u + 6 (x, t = ε (α ε a 0 ε κ + a 0 α (1 λ csch ( ξ ε (1 α ε ε + ξ 0 α ε a 0 ε κα + ε λ κ a 0 (1 λ ε (1 + α ε (5 where ξ = x + α ε a 0 ε κα+6 ε λ κ 4 ε κ a 0 t, ξ 3ε (1 λ ε(1+α ε 0 = coth 1 C, C / < 1 Case 3-: When λ 4μ = 1 ε < 0, we have u 3 =ε ( ε κ α ε a 0 a 0 α (1 λ ε (1 + α ε cot 1 ( ε ξ + ξ 0 ε ( ε κ α ε a 0 a 0 λ (1 λ ε (1 + α ε + a 0 (6 where ξ = x + α ε a 0 ε κα+6 ε λ κ 4 ε κ a 0 t, ξ 3ε (1 λ ε(1+α ε 0 = tan 1 C There is no rational solution for Set C 4 Conclusions We successfully obtained exact and explicit analytic solutions with arbitrary parameters to fully nonlinear third-order ilson-pickering equation via the simplified /-expansion method The procedure is simple, direct and constructive with the help of a computer algebra system These travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions and rational functions We foresee that our results can be found potentially useful for applications in mathematical physics and applied mathematics including numerical simulation Acknowledgements Research was supported by the Post-doctoral Foundation of China ( No , the Post-doctoral Foundation of Jiangsu Province(No080073c and the High- level Talented Person Special Subsidizes of Jiangsu University (No08JD013 IJNS for contribution: editor@nonlinearscienceorguk

6 X Fan, S Yang, D Zhao: Travelling Wave Solutions for the ilson-pickering Equation by Using 373 References [1] A Bekir Application of the ( /-expansion method for nonlinear evolution equations Phys Lett A 37: (008 [] R Camsssa, D D Holm, J M Hyman An integrable shallow water equation with peaked solitons Phys Rev Lett 71: (1993 [3] A Chen, W Huang, S Tang Bifurcations of travelling wave solutions for the gilson-pickering equation Nonlinear Anal: Real World Appl doi:101016/jnonrwa (008 [4] P A Clarkson, E L Mansfield, T J Priestley New similarity reductions of the boussinesq equation Mathe Comput Model 5(8-9: 195 1(1997 [5] Ismail Aslan, T Oziş On the validity and reliability of the ( /-expansion method by using higherorder nonlinear equations Appl Math Comput 11(: (009 [6] A Elhanbaly, M Abdou Exact travelling wave solutions for two nonlinear evolution equations using the improved f-expansion method Math Comput Model 46: (007 [7] E Fan, Y C H Benny Double periodic solutions with jacobi elliptic functions for two generalized hirota-satsuma coupled kdv systems Phys Lett A 9: (00 [8] X Fan, L Tian Compacton solutions and multiple compacton solutions for a continuum toda lattice model Chaos Solitons Fractals 9(4: (006 [9] B Fornberg, Whitham A numerical and theoretical study of certain nonlinear wave phenomena Phil Trans R Soc Lond A, 89: (1978 [10] B Fuchssteiner, A S Fokas Symplectic structure, their biicklund transformations and hereditary symmetries Physica D4: 47(1981 [11] C ilson, A Pickering Factorization and painleve analysis of a class of nonlinear third-order partial differential equations J Phys A: Math en8: (1995 [1] J He, M Abdou New periodic solutions for nonlinear evolution equations using exp-function method Chaos Solitons Fractals34: (006 [13] J He, X Wu Exp-function method for nonlinear wave equations Chaos Solitons Fractals30: (006 [14] N A Kudryashov Seven common errors in finding exact solutions of nonlinear differential equations Commun Nonlinear Sci Numer Simulat 14: (009 [15] P Rosenau, J Hyman Compactons: Solitons with finite wavelength Phys Rev Lett (1993 [16] M Wang Applicatiuu of homogenous balance method to exactsolutions of nonlinear equations in mathematical physics Phys Lett A 16: 67 75(1996 [17] M Wang, X Li, J Zhang The ( -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics Phys Lett A37: (008 [18] M Wang, J Zhang, X Li Application of the ( /-expansion to travelling wave solutions of the broer-kaup and the approximate long water wave equations Appl Math Comput06: 31 36(008 [19] A-M Wazwaz Peakons, kinks, compactons and solitary patterns solutions for a family of Camassa- Holm equations by using new hyperbolic schemes Appl Math Comput41 44(006 [0] Whitham Variational methods and applications to water waves Proc R Soc Lond A 99: 6 5(1967 [1] Whitham Linear and Nonlinear Waves Wiley, New York( 1974 [] X Wu, J He Solitary solutions, periodic solutions and compacton-like solutions using the expfunction method Comput Math Appl54: (007 [3] X Wu, J He Exp-function method and its application to nonlinear equations Chaos Solitons Fractals38: (008 [4] Z Yan A new sine-gordon equation expansion algorithm to investigate some special nonlinear differential equations Chaos Solitons Fractals3: (005 [5] S Zhang, L Dong, J Ba, Y Sun The ( -expansion method for nonlinear differential-difference equations Phys Lett A373: (009 IJNS homepage: