 Applied Mathematics, 2011, 2, 752-756 doi:10.4236/am.2011.26100 Published Online June 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM On Exact Tr aveling Wave Solutions for (1 + 1) Dimensional Kaup-Kupershmidt Equation Dahe Feng1,2, Kezan Li1 1School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, China 2School of Mathematical Sciences and Computing Technology, Central South University, Changsha, China E-mail: dahefeng@hotmail.com, lkzzr@guet.edu.cn Received April 8, 2011; revised April 22, 2011; accepted April 25, 2011 Abstract In this present paper, the Fan sub-equation method is used to construct exact traveling wave solutions of the (1 + 1) dimensional Kaup-Kupershmidt equation. Many exact traveling wave solutions are successfully ob-tained, which contain solitary wave solutions, trigonometric function solutions, hyperbolic function solutions and Jacobian elliptic function periodic solutions with double periods. Keywords: Fan Sub-Equation Method, Kaup-Kupershmidt Equation, Exact Traveling Wave Solutions 1. Introduction Nonlinear partial differential equations are widely used to describe complex phenomena in vary scientific fields and especially in areas of physics such as plasma, fluid mechanics, biology, solid state physics, nonlinear optics and so on. Therefore the investigation of the exact solu-tions to nonlinear equations plays an important role in the study of nonlinear science. Up to now, many power-ful methods to seek for exact solutions to the nonlinear differential equations have been proposed. Among these are inverse scattering method , Lie group method [2,3], bifurcation method of dynamical systems [4-6], sine- cosine method [7,8], tanh function method [9-11], ho-mogenous balance method , Weierstrass elliptic function method . Recently, Fan  presented the Fan sub-equation method which is a unified algebraic method to obtain many types of traveling wave solutions based on an aux-iliary nonlinear ordinary differential equation with con-stant coefficients called Fan sub-equation. The important feature of Fan’ method is to, without much extra effort and without considering the integrability of nonlinear equations, directly get a series of exact solutions in a uniform way, which cover all results of tanh function method, extended function method, F-expansion method, etc. This method is a powerful technique to symbolically compute traveling wave solutions of nonlinear evolution equations and is widely used by many researcher such as in [15-17] and by the references therein. In this paper, we will use the Fan sub-equation method to discuss the (1+1) dimensional Kaup-Kupershmidt equ-ation  which can be shown in the form 23525552txxxxxxuuuuuuuu 0.  (1.1) 2. The Fan Sub-Equation Method For a given nonlinear partial differential equation ,, ,,,,0,t xtt xt xxFuuu uuu (2.1) where u = u(x,t) is an unknown function, F usually is a polynomial in u(x,t). To seek exact solutions of (2.1), we outline the Fan sub-equation method. The main steps are given below . Step 1. By using the traveling wave transformation ,, uxtux ct, (2.2) where c is a wave speed, we can reduce (2.1) to an ordi-nary differential equation in the form ,, ,,0,Fuuu u   (2.3) where the prime denotes the derivative with respect to . Step 2. Expand the solution of (2.3) in the form  0,niiiua (2.4) where ai (i = 1,2, ···, n) are constants to be determined D. H. FENG ET AL.753 later and the new variable  satisfies the following Fan sub-equation  40,jjjc  (2.5) where 1 and are real constants. iStep 3. Determine n in (2.4) by substituting (2.4) and (2.5) into (2.3) and balancing the highest order derivative terms with the highest order nonlinear terms. cStep 4. Substituting (2.4) and (2.5) into (2.3) again and collecting all coefficients like 40lkjjcj (l = 0, 1; k = 0, 1, ···, n), then setting these coefficients to zero will give a set of algebraic equations with respect to ai (i = 1,2, ···, n) and c. Step 5. Solve these algebraic equations to obtain c and i. Substituting these results into (2.4) yields to the gen-eral form of traveling wave solutions. aStep 6. For each solution to (2.5) which depends on the special conditions chosen for cj, it follows from (2.4) that the corresponding exact traveling wave solution of (2.1) can be constructed. 3. Exact Solutions for the (1 + 1) Dimensional Kaup-Kupershmidt Equation The fifth order Kaup-Kupershmidt Equation (1.1) is one of the solitonic equations related to the integrable cases of the Henon-Heiles system and belongs to the com-pletely integrable hierarchy of higher order KdV equa-tions. Moreover the equation has infinite sets of conser-vation laws [19-22]. Let us find the exact traveling wave solutions of the (1 + 1) dimensional Kaup-Kupershmidt equation by using the Fan sub-equation method. The traveling wave transformation (2.2) permits us to reduce (1.1) to an ODE in the form 5225552cuuuu uuuu  0. (3.1) According to Steps 1 and 2 in Section 2, by balancing and in (3.1), we obtain n + 3 = 3n − 1 and therefore give n = 2. Thus we can suppose that (3.1) has the following formal solutions 5u2uu 2012 ,uaa a   (3.2) where  satisfies (2.5). Substituting (3.2) and (2.5) into (3.1), collecting all terms with the same power in 40kjjcj etting all their coefficients to zero yields a set of tions. (0 ≤ k ≤ 5), then ssimultaneous algebraic equations omitted here for the sake of brevity. Solving these algebraic equations with the help of Maple, we get the following two sets of solu-1) The first set of parameters is given by 231241 303130, 3, , 216ccaca ca 2442342223404 324246,240c c76845256, 256ccccccccccc (3.3) where 0234, , , 0cccc The second set of paare arbitrary constants. 2)rameters is given by 2324312424, 24,cccccac42324130424 32223 43042424831612, ,211 14421768256,16ccccaca ccccccccccc   (3.4) where 0234, , , 0cccc ay obtain manyare arbitrary constants. We m kinds of exact solutions de- on the special vapending lues chosen for cj. Case 1. If 013 20, 0ccc c  and 40c, Equa-tion (2.5) admits a triangle solution 22sec .ccc4 (3.5) Substituting (3.5) along with (3.3)respectively yields two triangle solutions of (1.1) and (3.4) into (3.2) 2212222,3secuxtcccxct  (3.6) and 2222,824 sec17622 2xtc ccxct  (3.7) with c2 < 0 being an arbitrary constant. uCase 2. If 2240c0132, 0, 4ccccc  and , wo periodic solutions of (2.5) 40ce can find tw22tan .cc422c   (3.8) Substituting (3.8), (3.3) and (3.4tively yields two triangle solutions of (1.1) ) into (3.2) respec-22223223,tanccuxt c cx 224t (3.9) and 222422,812tan 442cuxtccx ct 2 (3.10) with c2 > 0 being an arbitrary constant. Copyright © 2011 SciRes. AM D. H. FENG ET AL. 754 Case 3. For , we can gain the followio013 2 40, 0, 0ccc cc  ng hyperbolic function solution t (2.5) 22sech .ccc4 (3.11) e solutions of (1.1) Similarly, we obtain two peak-shaped solitary wav2ct252222,3sechuxtc ccx an (3.12) d 2262222, 8uxtc 24 sech176.ccxct  (3.13) with c2 > 0 being an arbitrary constant. show the physical insight of these solitary wave solutions, here we take u5 as an example. Figure the wave plot of the solution u5 with c2 = 1 and thll-shaped solitary wTo 1 shows e initial status of u5. Clearly the solution is a beave with peak form and describes the traveling of wave in the negative x-direction. Case 4. For 22013244,0,0,04ccccccc, Equa- tion (2.5) admits two following hyperbolic function solu-tions 22422ctanh .cc (3.14)   turn gives two peak-shaped solitary wave solu-tions of (1.1) This in22227223,tanhccuxtccxt224 and (3.15) 222822,812 tanh442cuxtccx ct2 c2 < 0 being an arbitrary constant. Case 5. For (3.16) with 01 32424 (2.5) 0,2, 0, 0,cc ccccc   one can find the following hyperbolic solutions of2211tanh cc4.22c (3.17) wave solutions. Similar to Case 1, (1.1) has two peak-shaped solitary2222293,tanh162cccuxt xct24 32 and (3.18) 22102 22,46tanh 112cuxtccxct2  (3.19) with c2 > 0 being an arbitrary constant. Case 6. For 22 221 1013202411, 0, ck kcccccp  and 40c, Equation (2.5) admits the foelliptic function solution llowing Jacobian 212 2141 1cp pwhere 21121pkcn,,kc ck (3.20)  and 122,1k is an arbitrary constant. This in turn gives the following two wave solution of (1.1) doubly periodic 2221 1211 ,ck cuxt 222 13cn ,c qcx tk (3.21) 2111pppand 22221 21212 21,,k 211124 176,8 cnck cqcuxt cxtppp (3.22) where 421111qkk. emonstrate the phTo dysical insight of the new solu-tions, we take u11 as an example. Obviously the solution is a Jacobi elliptic function with two periods athe traveling o the negative x-direction nd de-scribes f wave inwith the wave velocity 2221 1cqp . Figure 2 shows the wave plot of the solution u11 to (1.1) with c2 = 0.5, k1 = 0.9 and the initial status of u11. Case 7. For ,0,0,0,142312242222p 0ccccckcc we can obtain one Jacobian elliptic function solution of (2.5) 22242 2cp pwheredn, ,cck (3.23 ) 2222pk and is an arbitrary con-stant. Thus we can give two correspondinging wave solutions of (1.1) 20, 1k periodic travel-222222133n ,ccqcux tk2 22,dt cxpp222p and (3.24) 2222 2214 22,,tk 222224 176,8 dncc qcuxt cxppp (3.25) where 422221qkk. Copyright © 2011 SciRes. AM D. H. FENG ET AL. Copyright © 2011 SciRes. AM 755 Figure 1. The plot of the peak-shaped solitary wave solution u5 to (1.1) with c2 = 1 and the initial status of u5. Figure 2. The plot of the periodic traveling wave solution u11 to (1.1) with c2 = 0.5, k1 = 0.9 and the initial status of u11. C2ase 8. For 23013242,0,0,0,ccccccp (2.5) 43ckmits two kinds of Jacobian elliptic doubly periodic wadave solutions 223 23sn ,ck ck   (3.26) 43 3cp pwhere and 2331pk30,1k is an arbitrastant. This in turn gives two corresponding periodic travel-olutions) ry con-ing wave s of (1.122223 32215 23233,sn ,ck qccuxt cxtkp  33pp(3.27) and 162223 322232333,24 1768sn ,uxtck qcccxppp2,tk where 4. Conclusions and Summary In this paper, the Fan sub-equation method has been successfully applied to obtain many traveling wave solu-tions of the (1 + 1) dimensional Kaup-Kuperequation. These rich results show that this method is ef-fective and simple and a lot of solutions can be obtained ame time. It is also a promising method to solve ) and Postdoctoral Science oundation of Central South University. . References  M. J. Ablowitz and P. A. Clarkson, “Solitons, Nonlinear shmidt in the sother nonlinear equations. 5. 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