New Exact Traveling Wave Solutions for Two Nonlinear Evolution Equations

Transcription

1 Internatonal Conference on Computer Technology and Scence (ICCTS ) IPCSIT vol. 47 () () IACSIT Press, Sngapore DOI:.7763/IPCSIT..V47.66 New Exact Travelng Wave Solutons for Two Nonlnear Evoluton Equatons Qnghua Feng + School of Scence, Shandong Unversty of Technology, Zhangzhou Road, Zbo, Shandong, Chna, 5549 Abstract. In ths paper, a generalzed sub-ode method s proposed to construct exact solutons of two nonlnear equatons. As a result, some new exact travelng wave solutons for them are found. Keywords:sub-ODE method, travelng wave soluton, exact soluton, nonlnear equaton,. Introducton In scentfc research, seekng the exact solutons of nonlnear equatons s a hot topc. Many approaches have been presented so far. Some of these approaches are the homogeneous balance method [,], the hyperbolc tangent expanson method [3,4], the tral functon method [5], the tanh-method [6-8], the nonlnear transform method [9], the nverse scatterng transform [], the Backlund transform [,], the Hrotas blnear method [3,4], the generalzed Rccat equaton [5,6], the theta functon method [7-9], the sne-ccosne method [], the Jacob ellptc functon expa-nson [,], the complex hyperbolc functon method [3-5], and so on.. In ths paper, we proposed a sub-ode method to construct exact travelng wave solutons for NLEES. The rest of the paper s organzed as follows. In Secton, we descrbe the sub-ode method for fndng travelng wave solutons of nonlnear evoluton equatons, and gve the man steps of the method. In the subsequent sectons, we wll apply the method to fnd exact travelng wave solutons of the Boussnesq equaton and (+) dmensonal Boussnesq equaton. In the last Secton, some conclusons are presented.. Descrpton of the Bernoull Sub-ODE method In ths secton we present the solutons of the followng ODE: G' G G + = μ, (.) where, G = G( ξ ) When μ, Eq. (.) s the type of Bernoull equaton, and we can obtan the soluton as G = μ + de ξ, (.) where d s an arbtrary constant. Suppose that a nonlnear equaton, say n two or three ndependent varables x, y and t, s gven by + Correspondng author. Tel.: E-mal address: fqhua@sna.com 354

2 Puuu (,, u, u, u, u, u, u...) (.3) t, x y tt xt yt xx yy where u = u(x, y, t) s an unknown functon, P s a polynomal n u = u(x, y, t) and ts varous partal dervatves, n whch the hghest order dervatves and nonlnear terms are nvolved. By usng the solutons of Eq. (.), we can construct a serals of exact solutons of nonlnear equatons:. Step.We suppose that uxyt (,, ) = u( ξ ), ξ = ξ( xyt,, ) (.4) the travelng wave varable (.4) permts us reducng Eq. (.3) to an ODE for u = u( ξ ) Puu (, ', u '',...) (.5) Step. Suppose that the soluton of (.5) can be expressed by a polynomal n G as follows: u = G + G + (.6) m m ( ξ) αm αm... where G = G( ξ ) satsfes Eq. (.), and αm, αm... are constants to be determned later, αm. The postve nteger m can be determned by consderng the homogeneous balance between the hghest order dervatves and nonlnear terms appearng n (.5). Step 3. Substtutng (.6) nto (.5) and usng (.), collectng all terms wth the same order of G together, the left-hand sde of Eq. (.5) s converted nto another polynomal n G. Equatng each coeffcent of ths polynomal to zero, yelds a set of algebrac equatons for αm, αm,..., μ. Step 4. Solvng the algebrac equatons system n Step 3, and by usng the solutons of Eq. (.), we can construct the travelng wave solutons of the nonlnear evoluton equaton (.5). In the subsequent sectons we wll llustrate the proposed method n detal by applyng t to Boussnesq equaton and (+) dmensonal Boussnesq equaton. 3. Applcaton for Boussnesq Equaton In ths secton, we wll consder the followng Boussnesq equaton: u + u + u + u = < (3.) (4) tt α xx ( ) xx γ x, α Suppose that u( ξ), ξ = k( x ct) (3.) where the constants ckcan, be determned later. By usng (3.), (3.) s converted nto an ODE 3 (5) ( α c ) u'' ( u ) '' γk u = (3. 3) Integratng (3.3) twce, and take the ntegraton constant for zero, then we have 3 ( α c ) u u γk u''' = (3. 4) 355

3 Suppose that the soluton of (3.4) can be expressed by a polynomal n G as follows: m u( ξ ) = ag (3.5) = where a are constants, and G = G( ξ ) satsfes Eq. (.). Balancng the order of obtan that m = m+ 3 m = 3.So Eq. (3.5) can be rewrtten as u( ξ ) = a G + a G + ag+ a, a u and u ''' n Eq. (3.5), we where a 3, a, a, a are constants to be determned later. Substtutng (3.5) nto (3.3) and collectng all the terms wth the same power of G together, the left-hand sde of Eq. (3.3) s converted nto another polynomal n G. Equatng each coeffcent to zero, yelds a set of smultaneous algebrac equatons as follows: G : a + αa + c a G :a a + αa + c a + γk aμ G : 7γk aμ + a a + ( c + α) a + 8γk a μ + a 3 G : 38 γk a μ + ( c + α) a + a a + aa γk aμ+ 7γa k μ 3 3 G :54γa μk + a + aa γa k μ 6γak G :44γa μk + a a 4γa k G : 6γa k + a 6 Solvng the algebrac equatons above, yelds: Case : a3 =, a = a= a, k= k, c = α, =, μ (3.6) Substtutng (3.6) nto (3.5), we have u ( ξ ) 3 = G, ξ kx ( αt) = (3. 7) Combnng wth Eq. (.) and we can obtan the travelng wave solutons of (3.) as follows: u( x, t) [ de α ] k( x t) 3 = (3.8) where d s an arbtrary constant. Case : 356

4 a3 =, a = a= a, k= k, c = α, =, μ (3.9) Substtutng (3.6) nto (3.5), we have 3 u( ξ ) = G, ξ = kx ( + αt) (3. ) Combnng wth Eq. (.) and we can obtan the travelng wave solutons of (3.) as follows: u( x, t) [ de α ] k( x+ t) 3 = (3.) where d s an arbtrary constant 4. Applcaton for (+) dmensonal Boussnesq Equaton In ths secton, we wll consder the followng (+) dmensonal Boussnesq equaton: u u u u u = (4.) tt xx yy ( ) xx xxxx Suppose that uxyt (,, ) = u( ξ ), ξ = kx+ ly+ mt+ d (4.) where lkmd,,, are constants that to be determned later. By (4.), (4.) s converted nto an ODE 4 ( m k l ) u'' k ( u' uu'') k u'''' + = (4.3) Integratng (4.3) once we obtan 4 ( ) ' ' ''' m k l u k uu k u = g (4.4) where g s the ntegraton constant. Suppose that the soluton of (4.4) can be expressed by a polynomal n G as follows: m u( ξ ) = ag (4.5) = where a are constants, and G = G( ξ ) satsfes Eq.(.). Balancng the order of uu ' and u ''' n Eq.(4.4), we have m+ = m+ 3 m=.so Eq. (4.5) can be rewrtten as u( ξ ) = a G + ag+ a, a (4.6) where a, a, a are constants to be determned later. 357

5 Substtutng (4.6) nto (4.4) and collectng all the terms wth the same power of G together, equatng each coeffcent to zero, yelds a set of smultaneous algebrac equatons as follows: G : g G :( k + l m ) a+ k a + k a a 4 3 G : 7k μa + 8k a + ( k + l m ) a + ( m k l ) μa a + k a + 4k a a k a aμ G :( m k l ) a μ 4k a a μ + 6k aa 38a μk k a μ+ ak μ 3 G : 6k aa μ+ 54a k μ 6ak μ + 4k a G : 4a k μ 4k a μ Solvng the algebrac equatons above, yelds: 4 l + k m + k a = 6 k μ, a = 6 k μ, a = k = k, l = l, m= m, d = d (4.7) k where klmd,,, are arbtrary constants. Substtutng (4.7) nto (4.6), we get that 4 l + k m + k ( ) = 6 6 ξ = kx + ly + mt + d (4.8) k u ξ k μ G k μg Combnng wth Eq. (.), we can obtan the travelng wave solutons of (4.) as follows: u k k + de + de ( ξ) = 6 μ ( ) 6 μ( ) μ ξ μ ξ 4 l + k m + k (4.9) k Remark : Our result (4.9) s new exact travelng wave solutons for Eq. (4.). 5. Conclusons In the present work, we propose a new sub-ode method, and then test ts power by fndng some new travelng wave solutons of Boussnesq equaton and (+) dmensonal Boussnesq equaton. Ths method s one of the most effectve approaches handlng nonlnear evoluton equatons. One can see the method s concse and effectve. Also ths method can be used to many other nonlnear problems. 6. References [] M. Wang, Soltary wave solutons for varant Boussnesq equatons, Phys. Lett. A 99 (995) [] E.M.E. Zayed, H.A. Zedan, K.A. Gepreel, On the soltary wave solutons for nonlnear Hrota-Satsuma coupled KdV equatons, Chaos, Soltons and Fractals (4) [3] L. Yang, J. Lu, K. Yang, Exact solutons of nonlnear PDE nonlnear transformatons and reducton of nonlnear PDE to a quadrature, Phys. Lett. A 78 () [4] E.M.E. Zayed, H.A. Zedan, K.A. Gepreel, Group analyss. and modfed tanh-functon to fnd the nvarant solutons and solton soluton for nonlnear Euler equatons, Int. J. Nonlnear Sc. Numer. Smul. 5 (4) -34 [5] M. Inc, D.J. Evans, On travelng wave solutons of some nonlnear evoluton equatons, Int. J. Comput. Math. 8 (4) 9-358

6 [6] M.A. Abdou, The extended tanh-method and ts applcatons for solvng nonlnear physcal models, Appl. Math. Comput. 9 (7) [7] E.G. Fan, Extended tanh-functon method and ts applcatons to nonlnear equatons, Phys. Lett. A 77 () -8. [8] W. Malflet, Soltary wave solutons of nonlnear wave equatons, Am. J. Phys. 6 (99) [9] J.L. Hu, A new method of exact travelng wave soluton for coupled nonlnear dfferental equatons, Phys. Lett. A 3 (4) -6. [] M.J. Ablowtz, P.A. Clarkson, Soltons, Nonlnear Evoluton Equatons and Inverse Scatterng Transform, Cambrdge Unversty Press, Cambrdge, 99. [] M.R. Mura, Backlund Transformaton, Sprnger-Verlag, Berln, 978. [] C. Rogers, W.F. Shadwck, Backlund Transformatons, Academc Press, New York, 98. [3] R. Hrota, Exact envelope solton solutons of a nonlnear wave equaton, J. Math. Phys. 4 (973) [4] R. Hrota, J. Satsuma, Solton soluton of a coupled KdV equaton, Phys. Lett. A 85 (98) [5] Z.Y. Yan, H.Q. Zhang, New explct soltary wave solutons and perodc wave solutons for Whtham-Broer-Kaup equaton n shallow water, Phys. Lett. A 85 () [6] A.V. Porubov, Perodcal soluton to the nonlnear dsspatve equaton for surface waves n a convectng lqud layer, Phys. Lett. A (996) [7] E.G. Fan, Extended tanh-functon method and ts applcatons to nonlnear equatons, Phys. Lett. A 77 () -8. [8] E.G. Fan, Multple travelng wave solutons of nonlnear evoluton equatons usng a unfex algebrac method, J. Phys. A, Math. Gen. 35 () [9] Z.Y. Yan, H.Q. Zhang, New explct and exact travelng wave solutons for a system of varant Boussnesq equatons n mathematcal physcs, Phys. Lett. A 5 (999) [] S.K. Lu, Z.T. Fu, S.D. Lu, Q. Zhao, Jacob ellptc functon expanson method and perodc wave solutons of nonlnear wave equatons, Phys. Lett. A 89 () [] Z. Yan, Abundant famles of Jacob ellptc functons of the (+)-dmensonal ntegrable Davey-Stawartson-type equaton va a new method, Chaos, Soltons and Fractals 8 (3) [] C. Ba, H. Zhao, Complex hyperbolc-functon method and ts applcatons to nonlnear equatons, Phys. Lett. A 355 (6) -3.. [3] E.M.E. Zayed, A.M. Abouraba, K.A. Gepreel, M.M. Horbaty, On the ratonal soltary wave solutons for the nonlnear Hrota-Satsuma coupled KdV system, Appl. Anal. 85 (6) [4] K.W. Chow, A class of exact perodc solutons of nonlnear envelope equaton, J. Math. Phys. 36 (995) [5] M.L.Wang, Y.B. Zhou, The perodc wave equatons for the Klen-Gordon-Schordnger equatons, Phys. Lett. A 38 (3)