Compacton Solutions and Peakon Solutions for a Coupled Nonlinear Wave Equation

Transcription

1 ISSN (print), (online) International Journal of Nonlinear Science Vol 4(007) No1,pp31-36 Compacton Solutions Peakon Solutions for a Coupled Nonlinear Wave Equation Dianchen Lu, Guangjuan Yang Faculty of Science, Jiangsu University Zhenjiang, Jiangsu, 1013, PRChina (Received 5 November 006, accepted 4 April 007) Abstract: A coupled nonlinear wave equation is studied in the present paper With the aid of Mathematica Wu elimination method, through different ansatze, more solitary solutions, including compacton solutions, peakon solutions, as well as traveling solutions are found in this paper Key word: compacton solutions; coupled nonlinear wave equation; peakon solutions; solitary solutions; ansatze method 1 Introduction The investigation of exact solutions for nonlinear evolution equations has important academic actual value For a long time, this work is of special interest to mathematicians physicists A number of methods were presented, such as inverse scatting theory, Hirota s bilinear methods, the truncated Painlere expansion, homogeneous balance method, the sine-cosine method other methods [1]-[3] Recently, there are lots of results about coupled nonlinear equations Guha-Roy [4]-[6] analyzed the following coupled nonlinear wave equation ut + αv v x + βu u x + λuu x + γu xxx = 0, (11) v t + δ(uv) x + vv x = 0 where α, β, γ, λ, δ are any arbitrary constants When v = 0 is correct, Eq(11) turns to kdv mkdv, which is generally used in solid-state physics, plasma physics, hydro physics, quanta field theory so on Guha-Ray [4]-[5] suppose ξ = x ct + with u(ξ), u (ξ), u (ξ) 0, by transformation u(ξ) = 1 φ(ξ) Weierstrass ellipse function, some exact solitary wave solutions Cnoidal solution in special conditions are obtained Shang [7], some new explicit exact traveling wave solutions to the coupled nonlinear wave equation are present through two different ansatze Those solutions include the bell-shaped solitary wave solutions which have non-zero asymptotic value, the kink-shaped antikink-shaped solitary wave solutions, singular traveling wave solutions periodic wave solutions of the triangular function type Xu[8], new exact explicit solitary solutions are obtained for the coupled nonlinear wave equation by using a simple way of making the ansatze u(x, t) = A sec h n (ξ) + A 0 with ξ = k(x + ωt) + ξ 0 Compacton solutions are zero outside a finite domain of space variable x Peakon solutions have discontinuous first derivative on the peak To those kinds of solitary wave solutions of the above coupled nonlinear wave equation, fewer results have been carried out The main aim of the paper is to present such special solitary wave solutions Our method is a direct way similar as Tian [9]-[11] In the past few years, function-series method [9]-[11] has been systematized to obtain solitary solutions of nonlinear equations Functions in the method were chosen to be hyperbolic secant, sine, cosine, etc We use this method to give more new solitary solutions for the coupled nonlinear wave equations Corresponding author: dclu@ujseducn Copyright c World Academic Press, World Academic Union IJNS /091

2 3 International Journal of Nonlinear Science,Vol4(007),No1,pp31-36 This paper is organized in five sections In section, we study compacton solutions of the coupled nonlinear wave equation by direct sine cosine method We find peakon solutions in section 3 Some solitary solutions are obtained in section 4 The conclusion is given in section 5 Compacton solutions First we assume that v = Au + B, (1) where A B are constants to be determined later We seek compacton solutions for Eq(11), that is solution whose energy is limited in a finite domain We set u(x, t) = u(ξ), v(x, t) = v(ξ), ξ = k(x Dt), () where k denotes wave numbers D is velocity Substituting (1) () into Eq(11), Eq(11) is now ( D + αb A)u + (A αb + λ)uu + (αa 3 + β)u u + γk u = 0 ( DA + δb + BA)u + uu (A + A ) = 0, (3) where d dξ In order to find compacton solutions, we suppose that Eq(11) has the following traveling wave form solutions: u(ξ) = R cos m (ξ), ξ π 0, ξ > π u(ξ) = R sin m (ξ), ξ π 0, ξ > π, (4), (5) where R m are constants to be determined Substituting (4) into system (3) collecting all terms with the same power in cos(ξ), we get Rm( D + αb A k m γ) cos m 1 (ξ) + (αa 3 + β)r 3 m cos 3m 1 (ξ) +(A αb + λ)r m cos m 1 (ξ) γk Rm(m 1)(m ) cos m 3 (ξ) = 0, (6) ( AD + δb + AB)Rm cos m 1 (ξ) + R m cos m 1 (ξ)(a + A ) = 0 (7) Letting all coefficients of cos(ξ) in Eq(6) (7) to zero we can get a set of algebraic polynomials with respect to the unknown variables A, B, D, m, R, k, δ, α, β, m 3m + = 0 D αb A + k m γ = 0 A Bα + λ = 0 αa 3 + β = 0 DA + δb + BA = 0 Aδ + A = 0 (8) Using the Wu elimination method yields the following solutions which include two different cases when the parameters satisfy: We find the following two cosine-type compacton solutions for Eq(11): β 3 = 8αδ 3 (9) u 1 = cos(k(x Dt)), (x Dt) π k 0, (x Dt) > π k, (10) IJNS for contribution: editor@nonlinearscienceorguk

3 DLu, GYang: Compacton Solutions Peakon Solutions for a Coupled 33 v 1 = β, λδ cos(k β, (x Dt) π k (x Dt) > π k, (11) cos u = ( k (x Dt)), (x Dt) π k 0, (x Dt) > π k, (1) v = cos ( k λδ β, (x Dt) π k β, (x Dt) > π k, (13) where k = 1 λ 4Dβ βγ Figure 1: Graphs of solution u 1 v 1 Figure : Graphs of solution u v We can get a view of u 1, v 1 u, v in Fig1 Fig with = 4, δ =, λ = 4, γ =, β = 1, d = 1 According to formula (5), taking the same steps we obtain sin-type compacton solutions for Eq(11) u 3 = v 3 = sin(k(x Dt)), (x Dt) π k 1, (x Dt) > π k 1, (x Dt) < π k sin ( k β, β, λδ β, (x Dt) π k (x Dt) > π k (x Dt) < π k, (14), (15) u 4 = sin ( k (x Dt)), (x Dt) π k 1, (x Dt) > π k, (16) IJNS homepage:

4 34 International Journal of Nonlinear Science,Vol4(007),No1,pp31-36 v 4 = sin ( k λδ β, (x Dt) π k β,, (x Dt) > π k, (17) where k = 1 λ 4Dβ βγ Figure 3: Graphs of solution u 3 v 3 Figure 4: Graphs of solution u 4 v 4 Given the same parameters = 4, δ =, λ = 4, γ =, β = 1, d = 1 Fig3 shows that u 3 v 3 are kink solutions while Fig4 shows that u 4 v 4 are one-peak compacton solutions 3 Peakon solutions Integrating (3), with respect to the variable ξ, taking the integrating constant as zero yields ( D + αb A)u + (A αb + λ) 1 u + (αa 3 + β) 1 3 u3 + γk u = 0 ( DA + δb + BA)u + 1 u (A + A (31) ) = 0 Letting u = P e ξ, where P is constant need to be determined, substituting this into (31), we obtain ( D + αb A + γk )P e ξ + 1 (A αb + λ)p e ξ (αa3 + β)p 3 e 3 ξ = 0 ( DA + δb + BA)e ξ + 1 (A + A )P e ξ = 0 Setting the coefficients of e n ξ (n = 1,, 3) to zero yields a set of algebraic polynomials with respect to unknowns A, B k, D + αb A + γk = 0 A αb + λ = 0 αa 3 + β = 0 DA + δb + BA = 0 Aδ + A = 0 (3) IJNS for contribution: editor@nonlinearscienceorguk

5 DLu, GYang: Compacton Solutions Peakon Solutions for a Coupled 35 Solving Eq(3), we get a peakon solutions of Eq(11) as following demed β 3 = 8αδ 3, u 5 = e 1 λ +4Dβ x Dt βγ, v5 = e 1 λ +4Dβ x Dt βγ λδ β We can get a view of u 5, v 5 in Fig5 with = 5, δ = 1, λ = 1, γ = 5, β = 3, d = 1 Figure 5: Graphs of solution u 5 v 5 4 Solitary pattern solutions We suppose that Eq(11) has the following traveling wave form solutions: u(ξ) = R cosh m (ξ), (41) u(ξ) = R sinh m (ξ) (4) Substituting (41) into system (3) collecting all terms with the same power in cosh(ξ), we get Rm( D + αb A + k m γ) cosh m 1 (ξ) + (αa 3 + β)r 3 m cosh 3m 1 (ξ) +(A αb + λ)r m cosh m 1 (ξ) γk Rm(m 1)(m ) cosh m 3 (ξ) = 0, (43) ( AD + δb + AB)Rm cosh m 1 (ξ) + R m cosh m 1 (A + A ) = 0 (44) Letting all coefficients of cosh(ξ), in Eq(43) (44) to zero we can get a set of algebraic polynomials with respect to the unknown variables A, B, D, m, R, k, δ, α, β, Using the Wu elimination method yields the following solutions which include two different cases when the parameters satisfy: We find the following solitary-solutions for Eq(11): β 3 = 8αδ 3 (45) u6 = cosh(k(x Dt)) v 6 = cosh(k(x Dt)) β According to formula (4), taking the same step we obtain, u7 = cosh ( k (x Dt)) v 7 = cosh ( k λδ β where k = 1 u8 = sinh(k(x Dt)) v 8 = λ +4Dβ βγ sinh(k(x Dt)) β u9 = sinh ( k, v 9 = sinh ( k (x Dt)) λδ β IJNS homepage:

6 36 International Journal of Nonlinear Science,Vol4(007),No1,pp Conclusion We have considered the coupled nonlinear equations, by applying direct sine method, cosine method, we have succeed in given several kinds of special solitary wave solutions, such as compacton solutions, peakon solutions, as well as traveling solutions Acknowledgements This research was supported by the National Nature Science Foundation of China ( No: ) References [1] Dianchen Lu, Baojian Hong, Lixin Tian: Backlund transformation N-soliton-like solutions to the combined Kdv-Burgers equation with variable coefficients International Journal of Nonlinear Science (1), 3-10(006) [] Suping Qian: Painleve analysis aymmetry reductions to the strong dispersive DGH equation International Journal of Nonlinear Science 1(), (006) [3] Gamze Tanoglu: Hirota method for solving reaction-diffusion equations with generalize nonlinearity International Journal of Nonlinear Science 1(1), 30-36(006) [4] Guha-Roy C: Solitary wave solutions of a system of coupled nonlinear equation J Math Phys 8(6), 087(1987) [5] Guha-Roy C: Exact solutions to a coupled nonlinear equation Inter J Theor Phys 7(), 447(1988) [6] Hirota R, Satsuma J : Soliton solutions of a coupled KdV equation Phys Letter A85(8-9), 407(1981) [7] Yadong Shang: Explicit exact solutions for a coupled nonlinear wave equation Journal of Ningxia University ( Natural Science Edition) 1(), 90-94(000) [8] Xuejun Xu, Jiefang Zhang: New exact explicit solitary wave solutions to a class of coupled nonlinear equation Communications in Nonlinear Science &Numerical Simulation 3(1), (1998) [9] Xinghua Fan, Lixin Tian: Compacton solutions multiple compacton solutions for a continuum Toda lattice model Chaos, Solitons Fractals 9, (006) [10] Lixin T, Xiuying S: New peaked solitary wave solutions of the generalized Camassa-Holm equations Chaos, Solitons Fractals 19,61-637(004) [11] Lixia Wang, Jiangbo Zhou, Lihong Ren: The Exact solitary wave solutions for a family of BBM equation International Journal of Nonlinear Science 1(1), 58-64(006) IJNS for contribution: editor@nonlinearscienceorguk