International Conference on Computer Technology and Science (ICCTS ) IPCSIT vol. 47 () () IACSIT Press, Singapore DOI:.776/IPCSIT..V47.59 New Analytical Solutions For () Dimensional Kaup-Kupershmidt Equation Qinghua Feng School of Science, Shandong University of Technology, Zhangzhou Road, Zibo, Shandong, China, 5549 Abstract. In this paper, we derive exact traveling wave solutions of Kaup-Kupershmidt equation by a proposed Bernoulli sub-ode method. The method appears to be efficient in seeking exact solutions of nonlinear equations. We also make a comparison between the present method and the known (G /G) expansion method. Keywords: Bernoulli sub-ode method, traveling wave solutions, exact solution, evolution equation, Kaup-Kupershmidt equation. Introduction Research on nonlinear equations is a hot topic. The powerful and efficient methods to find analytic solutions and numerical solutions of nonlinear equations have drawn a lot of interest by a diverse group of scientists. Many efficient methods have been presented so far. During the past four decades or so searching for explicit solutions of nonlinear evolution equations by using various different methods have been the main goal for many researchers, and many powerful methods for constructing exact solutions of nonlinear evolution equations have been established and developed such as the homogeneous balance method, the hyperbolic tangent expansion method, the trial function method, the tanh method, the nonlinear transform method, the inverse scattering transform, the Backlund transform, the Hirotas bilinear method, the generalized Riccati equation, the theta function method, the sine-ccosine method, the Jacobi elliptic function expansion, the complex hyperbolic function method [-5], and so on. In this paper, we proposed a Bernoulli sub-ode method to construct exact traveling wave solution-ns for NLEES. The rest of the paper is organized as follows. In Section, we describe the Bernoulli sub-ode method for finding traveling wave solutions of nonlinear evolution equations, and give the main steps of the method. In the subsequent sections, we will apply the Bernoulli Sub-ODE method and the known (G /G) expansion method to find exact traveling wave solutions of the Kaup-Kupershmidt equation. In the last Section, some conclusions are presented.. Description of the Bernoulli Sub-ODE method In this section we present the solutions of the following ODE: G' G G =, (.) where, G = G( ξ ) When, Eq. (.) is the type of Bernoulli equation, and we can obtain the solution as Corresponding author. Tel.: 86-5664 E-mail address: fqhua@sina.com
G =, (.) de ξ where d is an arbitrary constant. Suppose that a nonlinear equation, say in two or three independent variables x, y and t, is given by Puuu (,, u, u, u, u, u, u...) (.) t, x y tt xt yt xx yy where u = u(x, y, t) is an unknown function, P is a polynomial in u = u(x, y, t) and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved. By using the solutions of Eq. (.), we can construct a serials of exact solutions of nonlinear equations:. Step.We suppose that uxyt (,, ) = u( ξ ), ξ = ξ( xyt,, ) (.4) the traveling wave variable (.4) permits us reducing Eq. (.) to an ODE for u = u( ξ ) Puu (, ', u '',...) (.5) Step. Suppose that the solution of (.5) can be expressed by a polynomial in G as follows: u = G G (.6) m m ( ξ) αm αm... where G = G( ξ ) satisfies Eq. (.), and αm, αm... are constants to be determined later, αm. The positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (.5). Step. Substituting (.6) into (.5) and using (.), collecting all terms with the same order of G together, the left-hand side of Eq. (.5) is converted into another polynomial in G. Equating each coefficient of this polynomial to zero, yields a set of algebraic equations for αm, αm,...,. Step 4. Solving the algebraic equations system in Step, and by using the solutions of Eq. (.), we can construct the traveling wave solutions of the nonlinear evolution equation (.5). In the subsequent sections we will illustrate the propo-sed method in detail by applying it to Kaup-Kupershmidt equation.. Application Of the Bernoulli Sub-ODE Method For Kaup-Kupershmidt Equation In this section, we will consider the following Kaup-Kupershmidt equation: 75 uxxxxx ut 45uxu uxxux 5uuxxx (.) Suppose that uxyt (,, ) = u( ξ ), ξ = kx ωt (.) where k, ω are constants that to be determined later. By (.), (.) is converted into an ODE 5 (5) 75 ku ωu' 45 kuu ' kuu ' '' 5 kuu''' (.) Integrating (.) once we obtain 5 (4) 45 ku ωu 5 ku ku' 5 kuu'' = g (.4) 4 where g is the integration constant. Suppose that the solution of (.4) can be expressed by a polynomial in G as follows: m i u( ξ ) = ag (.5) i= i 4
where a i are constants, and G = G( ξ ) satisfies Eq. (.). Balancing the order of u (4) and u in Eq.(.4), we have m = m 4 m =.So Eq.(.5) can be rewritten as u( ξ ) = a G ag a, a (.6) where a, a, a are constants to be determined later. Substituting (.6) into (.4) and collecting all the terms with the same power of G together, equating each coefficient to zero, yields a set of simultaneous algebraic equations as follows: G : g ωa 5ka G : 5ak a ωa k a 45ka a 5 G :45kaa 6kaa 45kaa 45aa k 5 ka 5ak a 5a k 4 ω = 5 G : 9ka aa k a k a a a k 5 5ka 5k a a a k 5a a 5 4 5 5 65 G : 85kaa 45ka a 6k a 9aak 5ka ka 45kaa ka 4 G : 4k a 4k a 6k a 45ka a 65k a a 5 5 G :k a 5ka 5k a 6 Solving the algebraic equations above, yields: Case : 7 6 a = k, a = k, a = k, k = k, ω = k, g = k (.7) 6 88 where k is an arbitrary constant. Substituting (.7) into (.6), we obtain u( ξ ) k G k G k =, ξ = kx k t (.8) 6 Combining with Eq. (.), we can obtain the traveling wave solutions of (.) as follows: u( x, t) = k [ ] k [ ] k ( kx k t ) 6 de ( kx k t) 6 de Case : 7 6 a = 8 k, a = 8 k, a = k, k = k, ω = k, g = 9 k (.) 9 where k is an arbitrary constant. Substituting (.7) into (.6), we obtain u( ξ ) 8k G 8k G k =, (.9) ξ = kx k t (.) Combining with Eq. (.), we can obtain the traveling wave solutions of (.) as follows: u( x, t) = k [ ] k [ ] k ( kx k t ) de ( kx k t) de 4. Application Of (G /G) expansion Method For Kaup-Kupershmidt Equation (.) 5
In this section, we apply the (G /G) expansion method to obtain the traveling wave solutions of Kaup-Kupershmidt equation (.). Suppose that the solution of (.4) can be expressed by a polynomial in G as follows: G' G' u( ξ ) = a( ) a a, a G G (4.) where a i are constants, and G = G( ξ ) satisfies G'' G' G (4.) Substituting (4.) into (.4) and collecting all the terms with the same power of G together, equating each coefficient to zero, yields a set of simultaneous algebraic equations. Solving these equations we obtain 5 4 a= 8 k, a= 8 k, a= k ( 8 ), k = k, ω = k ( 8 6 ), 664 7 4 7 4 46 7 g = k k k (4.) 9 where k is an arbitrary constant. Substituting (4.) into (4.), we obtain G' G' u( ξ ) = 8 k ( ) 8 k k ( 8 ) G G 5 4, ξ = kx k ( 8 6 ) t (4.4) Combining (4.) and (4.4) we obtain the following solutions. When 4 > Csinh 4ξ Ccosh 4ξ u( ξ ) = k ( 8 ) k k ( 4 ) ( ) Ccosh 4ξ Csinh 4ξ When 4 < u ξ = k k k ( ) ( 8 ) (4 ) Csin 4 ξ Ccos 4 ξ ( ) Ccos 4 ξ Csin 4 ξ When 4 8kC u ( ξ) = k ( 8 ) k ( C C ξ ) Remark: As one can see from Section III and Section IV, the traveling wave solutions obtained by the Bernoulli Sub-ODE method are different from those by the known (G /G) expansion method. 5. Conclusions We have seen that some new traveling wave solutions of Kaup-Kupershmidt equation are successfully found by using the Bernoulli sub-ode method. The main points of the method are that assuming the solution of the ODE reduced by using the traveling wave variable as well as integrating can be expressed by an m -th degree polynomial in G, where G = G( ξ ) is the general solutions of a Bernoulli sub-ode equation. The positive integer m can be determined by the general homogeneous balance method, and the coefficients of the polynomial can be obtained by solving a set of simultaneous algebraic equations. Also we make a comparison between the proposed method and the known (G /G) expansion method. The Bernoulli Sub-ODE method method can be applied to many other nonlinear problems. 6
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