New Exact Solutions of the Modified Benjamin-Bona-Mahony Equation Yun-jie YANG and Li YAO
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1 06 International Conference on Artificial Intelligence and Computer Science (AICS 06) ISBN: New Exact Solutions of the Modified Benamin-Bona-Mahony Equation Yun-ie YANG and Li YAO Department of Mathematics, Kunming University, Economic and Technological Development Zone, Kunming, China Keywords: The extended simplest equation method, The mbbm equation, Exact solutions. Abstract. The extended simplest equation method is used to construct exact solutions of the modified Benamin-Bona-Mahony (mbbm) equation. The results obtained are in the form of hyperbolic, trigonometric, and rational functions. Throughout the paper, all the calculations are made with the aid of the Maple packet program. The method is more effective and simple than other method and can be used for many other nonlinear evolution equations. Introduction Many nonlinear evolution equations (NLEEs) play a significant role in physical applications. The study of the exact solutions of NLEEs has long been one of the central themes of perpetual in mathematics and physics. Over the last years, many powerful methods have been developed to find exact solutions of NLEEs, such as the bifurcation method of dynamic system[-3],the Hirota bilinear method[4], the sine cosine method[5], the tanh-function method[6,7], Fan-expansion method[8], the homogeneous balance method[9-] and so on. Recently, Bilige et al. introduced a method called the extend simplest equation method to look for the exact solutions of NLEEs[-4]. The obective of this paper is to apply the extended simplest equation method to construct the exact solutions for the mbbm equation. The Extended Simplest Equation Method We consider the proective Riccati equation f( ) f( ) g( ), g( ) g ( ) f( ), where and are constants. We suppose solution f( ), g( ) of Eqs. () in the form () f, g, () the function ( ) satisfies the second order linear ordinary differential equation (ODE) in the form. (3) We obtain the solutions of Eq. (3) are more easily Eq. (). It has three types of general solution as follows: 3
2 Acosh( ) Asinh( ), when 0, ( ) Acos( ) Asin( ), when 0, A A, when 0, (4) A A, when 0, A A, when 0, A A, when 0, (5) where A and A are arbitrary constants. Suppose that a nonlinear equation, say in two independent variables x and t, is given by Puu (, x, ut, uxx, uxt, utt, ) 0, (6) where u u( x, t) is an unknown complex-valued function, P is a polynomial on u and its various partial derivatives. The main steps of the extended simplest equation method are the following: Step. Look for its travelling wave solution in the form of u u( ), xct where c is constant. We reduce Eq.(6) to the following ODE: Puu cu u cu cu (,,,,,, ) 0. Step. We look for that Eq. (8) has the formal solution: i M M i i0 0 u( ) a b, where a, b ( i 0,,, M; 0,,, M ) i are constants and a b 0. The function ( ) satisfies the second order linear ODE (3). Step 3. We determine the positive integer M in Eq. (9) by considering the homogeneous balance between the highest order derivatives and the nonlinear terms in Eq. (8). Step 4. By substituting Eq. (9) into Eq. (8) and using the second order linear ODE (3) and (5), collecting all terms with the same order of i M M (7) (8) (9) and together, the left-hand side of Eq. (8) is converted into another polynomial in i and. Equating each coefficient of these different power terms to zero yields a set of algebraic equations for ai, b( i 0,,, M; 0,,, M ), c, and. Step 5. Assuming constants ai, b( i 0,,, M; 0,,, M ), c, and can be determined by solving the algebraic equations in Step 4. Then substituting these terms and the general solutions (4) of Eq. (3) into (9), we can obtain more exact travelling wave solutions of (6). 4
3 Applications Let us consider mbbm equation[5] : u u u u u 0, t x x xxt where and are nonzero constants. Using the wave variable x ct, the system (0) is carried to a system of ODE 3 u cu cu 0. 3 Now balancing the highest order derivative u 3 and nonlinear term u, we get M, so the solution of Eq.() is in the form u( ) a0 a b0. () By substituting () into Eq.() and using the second order linear ODE (3) and (5), collecting all terms with the same order of i and together, the left-hand side of Eq. () is converted into another polynomial in i and zero yields a set of algebraic equations for 0 0 Maple, we obtain the following results. If 0 we obtain (0) (). Equating each coefficient of these different power terms to a, a, b c, and. Solving these equations by () 3 A b0 b0 3 3 A, a0 0, a c. 3 If 0 we obtain b0 b0 3 A 3 3 () A, a0 0, a c. 3 If 0 we obtain (3) (4) 6 6 A b, A 0, a 0, a c. 3 (5) (3) A b, a 0, a c. 3 (6) (4) 0 0 Substituting (3)-(6) into () and making use of solutions (4) of Eq. (3), we can obtain exact travelling wave solutions expressed by hyperbolic functions, trigonometric functions, and rational functions of (0): u 3/ 3 A Bsinh( ) 3 A( ) Acosh( ) b0, 3Bcosh( ) 3A sinh( ) 3 (7) where 3 A b0 b0 3 A,, x t. ( ) 3/ 3 ABsin( ) 3 3A Acos( ) 3b0 u, Bcos( ) 3A sin( ) 3 (8) 5
4 3b0 6b0 9 A 9 where A,, x t. ( ) 6 3 4b0 u3, xt. 6 3 b0 (9) 6b,. 6 b0 3A 0 u4 x t We give some computer graphs to illustrate our main results: (0) (a) A, t 0. (b) A, t Figure. D and 3D figures of solution u in (7) with,,, b0 =, and in the intervals x [ 30,30]. (a) A, t 0.6 (b) A, t Figure. D and 3D figures of solution u in (8) with,,, b =, 0 and in the intervals x [ 0,0]. 6
5 (a) t (b) t Figure 3. D and 3D figures of solution u 3 in (9) with,, b0 =, and in the intervals x [ 8,8]. (a) A, t (b) A, t Figure 4. D and 3D figures of solution u 4 in (0) with,, b0 = in the intervals x [ 8,8]. Conclusion In this paper, we employ the extended simplest equation method to obtain exact travelling wave solutions of the mbbm equation. New exact travelling wave solutions involving parameters, expressed by three types of functions which are the hyperbolic functions, the trigonometric functions and the rational functions, are obtained. For these new travelling wave solutions, we give some computer simulations to illustrate our main results. We believe that this method should play an important role for finding exact solutions in the mathematical physics. 7
6 Acknowledgments This work is supported by the Scientific Research Foundation of Kunming University (XJL50) and the Natural Science Foundation of Education Committee of Yunnan Province (05C075Y). References [] J.B. Li, Z. Liu, Travelling wave solutions for a class of nonlinear dispersive equations, Chin. Ann. Math. 3B (3) (00) [] S. Tang, W. Huang, Bifurcations of travelling wave solutions for the generalized double sinhgordon equation, Appl. Math. Comput. 89 () (007) [3] D. Feng, T. He, J. Lü, Bifurcations of travelling wave solutions for (+)-dimensional boussinesq type equation, Appl. Math. Comput. 85 () (007) [4] R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, 004. [5] S. Tang, Y. Xiao, Z. Wang, Travelling wave solutions for a class of nonlinear fourth order variant of a generalized Camassa-Holm equation, Appl. Math. Comput. 0 () (009) [6] S. Tang, J. Zheng, W. Huang, Travelling wave solutions for a class of generalized kdv equation, Appl. Math. Comput. 5 (7) (009) [7] W. Malfliet, W. Hereman, The tanh method: I. exact solutions of nonlinear evolution and wave equations, Phys. Scr. 54 (6) (996) [8] E. Fan, Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos, Solitons Fractals. 6 (5) (003) [9] M.L. Wang, Solitary wave solutions for variant boussinesq equations, Phys. Lett. A 99 (995) [0] M.L. Wang, Y.B. Zhou, Z.B. Li, Phys. Lett. A 6 (996) 67. [] E.G. Fan, H.Q. Zhang, Phys. Lett. A 46 (998) 403. [] S. Bilige, T. Chaolu, An extended simplest equation method and its application to several forms of the fifth-order KdV equation, Applied Mathematics and Computation. 6 (00) [3] S. Bilige, T. Chaolu, and X. Wang, Application of the extended simplest equation method to the coupled Schrödinger-Boussinesq equation, Applied Mathematics and Computation. 4 (03) [4] Y.M. Zhao, New Exact Solutions for a Higher-Order Wave Equation of KdV Type Using the Multiple Simplest Equation Method, Journal of Applied Mathematics. Vol. 04, Article ID , 3 pages, 04. [5] K. Khan, M.A. Akbar, and M.N. Alam, Traveling wave solutions of the nonlinear Drinfel, d- Sokolov-Wilson equation and modified Benamin-Bona-Mahony equations, Journal of the Egyptian Mathematical Society. (03)
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