International Conference on Advanced Computer Science and Electronics Information (ICACSEI 2013) equation. E. M. E. Zayed and S. A.

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1 Internatonal Conference on Advanced Computer Scence and Electroncs Informaton (ICACSEI ) The two varable (G'/G/G) -expanson method for fndng exact travelng wave solutons of the (+) dmensonal nonlnear potental Yu-Toda-Sasa-Fukuyama equaton E. M. E. Zayed and S. A. Hoda Ibrahm Mathematcs Department Faculty of Scence Zagazg Unversty P.O. Box 59 Zagazg Egypt. e.m.e.zayed@hatmal.com dhoda_jsa@yahoo.com dfferental equaton G ( ) G ( ) where and are constants. The degree of ths polynomal can be determned by consderng the homogeneous balance between the hghest order dervatves and the nonlnear terms appearng n the gven nonlnear PDEs. The coeffcents of ths polynomal can be obtaned by solvng a set of algebrac equatons resulted from the process of usng ths method. Recently L et al [] have appled the G / G / G - Abstract - The two varable G / G / G -expanson method s employed to construct exact travelng wave solutons wth parameters of the ( + )-dmensonal nonlnear potental Yu-TodaSasa-Fukuyama (YTSF) equaton. When the parameters are replaced by specal values the well-known soltary wave solutons of ths equaton redscovered from the travelng waves. Ths method can be thought of as the generalzaton of the well-known orgnal G / G expanson method proposed by M. Wang et al. It s shown that the two varable G / G / G -expanson method provdes a more expanson method and determned the exact solutons of Zakharov equatons whle Zayed et al [] have used ths method to fnd the exact solutons of the combned KdVmKdV equaton. The objectve of ths paper s to apply the two varable G / G / G -expanson method to fnd the exact travelng powerful mathematcal tool for solvng many other nonlnear PDEs n mathematcal physcs. Index Terms - The two varable G / G / G -expanson method ; The ( + )-dmensonal potental YTSF equaton ; Exact travelng wave solutons; Soltary wave solutons.. Introducton wave solutons of the followng nonlnear (+)-dmensonal potental YTSF equaton [5]: In the recent years nvestgatons of exact solutons to nonlnear partal dfferental equatons (NPDEs) play an mportant role n the study of nonlnear physcal phenomena. Many powerful methods have been presented such as the nverse scatterng method [] the Hrota blnear transform [] the truncated panleve expanson method [-6] the Backlund transsform method [78] the exp-functon method [9-] the tanh functon method [-7] the Jacob ellptc functon expanson method [8-] the G / G -expanson u xt u xxxz u x u xz u z u xx u yy. Ths equaton s a potental-type counterpart of the (+)dmensonal nonlnear equaton [ v t (v )v z ]x v yy x v v x v x x method [-] the modfed G / G -expanson method (.) ntroduced by Yu el al [5 ] whle makng the (+)dmensonal generalzaton from the ( + )- dmensonal Calogero-Bogoyavlenk-Schff equaton [6]: [] the G / G / G -expanson method [] the frst ntegral method [] and so on. The key dea of the one varable G / G -expanson method s that the exact v t (v )v z x v v x x solutons of nonlnear PDEs can be expressed by a polynomal n one varable G / G n whch G G ( ) satsfes the (.) as dd for the KP equaton from the KdV equaton. Takng v u x the equaton (.) transforms nto the potental-ytsf second order lnear ODE G ( ) G ( ) G ( ) where and are constants and ' d / d. The key dea of the two equaton (.). We also remark that the equaton (.) tself becomes the potental KP equaton f z x and reduces to the potental KdV equaton whle further takng u y. varable G / G / G -expanson method s that the exact travelng wave solutons of nonlnear PDEs can be expressed by a polynomal n the two varables G / G and /G n Therefore varous applcatons of the KP and KdV equatons show great potental for applcatons of (.) n the physcal scences. Recently Zayed [9] have dscussed ths equaton whch G G ( ) satsfes the second order lnear ordnary. The authors - Publshed by Atlants Press (.) 88

2 usng the G / G -expanson method. The rest of ths paper s organzed as follows : In Sec. we gve the descrpton of the two varable G / G/ G -expanson method. In Secs. we apply ths method to solve Eq. (.). In Sec. some conclusons are gven.. Descrbton of the Two Varable (G'/G/G) -Expanson Method Before we descrbe the man steps of ths method we need the followng remarks ( see [ ] ): Remark. If we consder the second order lnear ODE: G( ) G( ) (.) and set G then we get G G. (.) Remark. then the general solutons of Eq. (.) has the form: G ( ) Asnh( ) Acosh( ) (.) where A and A are arbtrary constants. Consequently we have where ( ) A A. (.) Remark. If then the general solutons of Eq. (.) has the form: G ( ) Asn( ) Acos( ) (.5) and hence where ( ) A A. (.6) Remark. If then the general solutons of Eq. (.) has the form: and hence G ( ) A A (.7) ( ). A A (.8) Suppose we have the followng nonlnear evoluton equaton F( u ut ux u y uz uxx...) (.9) where F s a polynomal n u( x y z t ) and ts partal dervatves. In the followng we gve the man steps of the -expanson method [ ]: G / G/ G Step.The travelng wave transformaton u( x y z t ) u( ) x y z t (.) where s a constant reduces Eq.(.9) to an ODE n the form: P( u u u...) (.) where P s a polynomal of u ( ) and ts total dervatves wth respect to Step. Assumng that the soluton of Eq.(.) can be expressed by a polynomal n the two varables and as follows: N N ( ) (.) u a b (... N ) and b (... N ) constants to be determned later. Step. Determne the postve nteger N n Eq.(.) by usng the homogeneous balance between the hghest-order dervatves and the nonlnear terms n Eq.(.). Step. Substtute Eq.(.) nto Eq.(.) along wth (.) and (.) the left- hand sde Eq.(.) can be converted nto a polynomal n and n whch the degree of s not longer than. Equatng each coeffcents of ths polynomal to zero yelds a system of algebrac equatons whch can be solved by usng the Maple or Mathematca to get the values of a b A A and where. Step 5. Smlar to step substtute Eq.(.) nto Eq.(.) along wth (.) and (.6) for (or (.) and (.8) for ) we obtan the exact solutons of Eq.(.) expressed by trgonometrc functons (or by ratonal functons) respectvely... Applcatons In ths secton we wll apply the method descrbed n Sec. to fnd the exact travelng wave solutons of the nonlnear (+)-dmensonal potental YTSF equaton (.). To ths end we see that the travelng wave transformaton (.) permts us convertng Eq.(.) nto the followng ODE: u u ( ) u (.) are 89

3 wth zero constant of ntegraton. By balancng between u wth N. Consequently we get u n Eq. (.) we get u( ) a a b (.) a and b are constants to be determned later. There are three cases to be dscussed as follows: Case. Hyperbolc functon solutons ) (.) the left-hand sde of Eq. (.) becomes a polynomal n and. Settng the coeffcents of ths polynomal to be If substtutng (.) nto (.) and usng (.) and zero yelds a system of algebrac equatons n a a b and as follows: b : 6a a b ab : 6 6 6b 6ba : 6b :a 6a a a : 8a 6a b a ( ) b ab : 5b 6ab b ( ) 6 b 6 ba : 6 a 6 a :5a 6a a () a a : a a a (). On solvng the above algebrac equatons usng the Maple or Mathematca we get the followng results: a b ( ). In ths case the exact soluton of Eq. (.) has the form : Acosh( ) Asnh( ) u( ) a Asnh( ) Acosh( ) Asnh( ) Acosh( ) s an arbtrary constant x y z ( ) t. A A and (.) If A A and then we have the soltary wave soluton u( ) a [tanh( ) sec h( )]. (.) If A A and then we have the soltary wave soluton u( ) a [coth( ) co sec h( )]. (.5) Case. Trgonometrc functon solutons ) (.6) the left- hand sde of Eq. (.) becomes a polynomal n and. Settng the coeffcents of ths polynomal to be If substtutng (.) nto (.) and usng (.) and zero yelds a system of algebrac equatons n a a b and as follows: b : 6a a b ab : 6 6 6b 6ba : 6b :a 6a a a : 8a 6a b a ( ) 9

4 b ab : 5b 6ab b ( ) 6 b 6 ba : 6 a 6 a :5a 6a a () a a : a a a (). On solvng the above algebrac equatons usng the Maple or Mathematca we get the followng results: a b ( ). In ths case the exact soluton of Eq. (.) has the form : b : 6 a a A A b ab : 6 6 6b 6b a : A A A A 6b : 6 a a A A a a : a ( ) A A A A b a b : b( ) A A A A 6 a 6 a : a ( ). A A A A On solvng the above algebrac equatons usng the Maple or Mathematca we get the followng results: Acos( ) Asn( ) u( ) a Asn( ) Acos( ) Asn( ) Acos( ) s an arbtrary constant A A and (.6) a b A A. In ths case the exact soluton of Eq. (.) has the form : A A A u( ) a A A s an arbtrary constant and (.9) x y z ( ) t. If A A and then we have the soltary wave soluton u( ) a [tan( ) sec( )]. (.7) If A A and soluton then we have the soltary wave u( ) a [cot( ) co sec( )]. (.8) Case. Ratonal functon solutons ) (.8) the left-hand sde of Eq. (.) becomes a polynomal n and. Settng the coeffcents of ths polynomal to the zero yelds a system of algebrac equatons b and as follows: If substtutng (.) nto (.) and usng (.) and n a a x y z t. Remark 5. All solutons of ths paper have been checked wth Maple by puttng them back nto the orgnal equaton (.).. Conclusons The two varable G / G/ G -expanson method s used n ths artcle to obtan some new as well as some known solutons of a selected nonlnear evoluton equaton namely the ( + )- dmensonal potental YTSF equaton (.). As the two parameters A and A take specal values we obtan the soltary wave solutons. When and b n Eqs. (.) and (.) the two varable G / G/ G -expanson method reduces to the G / G expanson method. So the two varable G / G/ G -expanson method s an extenson of the G / G expanson method. The proposed method n ths paper s more effectve and more general than the G / G - 9

5 expanson method because t gves exact solutons n more general forms. In summary the advantage of the two varable G / G/ G G / G -expanson -expanson method over the method s that the solutons usng the frst method recover the solutons usng the second one. References [] M. J. Ablowtz and P. A. Clarkson Soltons Nonlnear Evoluton Equaton and Inverse Scatterng Cambrdge Unversty press New York 99. [] R. Hrota Exact solutons of the KdV equaton and multple collsons of soltons Phys. Rev.Lett. 7(97) 9-9. [] J.Wess M. Tabor and G. Carnvalle The Panleve property for PDEs. J. Math. Phys. (98) [] N. A. Kudryashov Exact solton solutons of the generalzed evoluton equaton of wave dynamcs J. Appl. Math. Mech. 5 (988) [5] N.A. Kudryashov Exact solutons of the generalzed Kuramoto- Svashnsky equaton Phys. Lett. A 7 (99) [6] N.A. Kudryashov On types nonlnear nonntegrable dfferental equatons wth exact solutons Phys. Lett. A 55 (99) [7] M. R. Mura Backlund transformaton. Sprnger Berln 978. [8] C. Rogers and W. F. Shadwck Backlund transformaton Academc Press New York 98. [9] J. H. He and X. H. Wu Exp-functon method for nonlnear wave equatons Chaos soltons & Fractals (6) [] E. Yusufoglu New soltary solutons for the MBBM equatons usng the exp-functon method.phys. Lett. A 7 (8) -6. [] S. Zhang Applcaton of the exp-functon method to hgh dmensonal evoluton equatons Chaos Soltons and Fractals 8 (8) [] A. Bekr The exp-functon method for Ostrovsky equaton Int. J. Nonlnear Sc. Num. Smul. (9) [] A. Bekr Applcaton of the exp-functon method for nonlnear dfferental-dfference equatons Appl. Math. Comput. 5 () 9-5. [] M. A. Abdou The extended tanth-method and ts applcatons for solvng nonlnear physcal models. Appl. Math. Comput. 9 (7) [5] E. G. Fan Extended tanh-functon method and ts applcatons to nonlnear equatons Phys. Lett. A 77 () -8. [6] S. Zhang and T. C. Xa A further mproved tanh-functon method exactly solvng (+)- dmensonal dspersve long wave equatons Appl. Math. E-Notes 8 (8) [7] E. Yusufoglu and A. Bekr Exact solutons of coupled nonlnear Klen- Gordon equatons Math. Computer Modelng 8 (8) [8] Y. Chen and Q. Wang Extended Jacob ellptc functon ratonal expanson method and abundant famles of Jacob ellptc functons solutons to the (+)- dmensonal nonlnear dspersve long wave equaton. Chaos Soltons & Fractals. (5) [9] S. Lu Z. Fu S. D. Lu and Q. Zhao Jacob ellptc functon expanson method and perodc wave solutons of nonlnear wave equatons. Phys. Lett. A. 89 () [] D. Lu Jacob ellptc functon solutons for two varant Boussneq equaton Chaos Soltons & Fractals (5) [] M. L. Wang X. L and J. Zhang The G / G )-expanson method and travelng wave solutons of nonlnear evoluton equatons n mathematcal physcs Phys. Lett. A 7 (8) 7-. G / G expanson [] S. Zhang J. L. Tong and W. Wang A generalzed method for the KdV equaton wth varable coeffcents Phys. Lett. A 7 (8) [] E. M. E. Zayed and K. A. Gepreel TheG / G -expanson method for fndng travelng wave solutons of nonlnear partal dfferental equatons n mathematcal physcs. J. Math. Phys. 5 (9) 5-5. G / G -expanson method and ts applcatons [] E. M. E. Zayed The to some nonlnear evoluton equatons n the mathematcal physcs J. Appl. Math. Computng (9) 89-. G / G -expanson method for nonlnear [5] A. Bekr Applcaton of the evoluton equatons Phys.Lett. A 7 (8) -6. G / G -expanson method for the [6] B. Ayhan and A. Bekr The nonlnear lattce equatons Commu. Nonlnear Sc. Numer. Smula. 7 () G / G -expanson method Appl. [7] N. A. Kudryoshov A note on the Math. Comput. 7 () G / G -expanson method agan Appl. Math. [8] I. Aslan A note on the Comput. 7 () [9] E. M. E. Zayed Travelng wave solutons for hgher dmensonal nonlnear evoluton equatonsusng the G / G -expanson method J. Appl. Math. Informatcs 8 () [] N. A. Kudryashov Meromorphc solutons of nonlnear ordnary dfferental equatonscomm. Nonlnear Sc. Numer. Smula. 5 () G / G - [] S. Zhang Y. N. Sun J. M. Ba and L. Dong The modfed expanson method for nonlnear evoluton equatons Z. Naturforsch. 66a () -9. G / G/ G -expanson method [] X.L. L Q. E. L and L. M.Wang The and ts applcaton to travelng wave solutons of Zakharov equatons Appl. Math. J. Chnese Unv. 5 () 5-6. [] E. M. E. Zayed and M. A. M. Abdelazz The two varable G / G/ G -expanson method for solvng the nonlnear KdV-mKdV equaton Math. Prob. Engneerng. Volume Artcle ID 756 pages. [] F. Tascan and Beker Applcatons of the frst ntegral method to the nonlnear evoluton equatons Chnese Phys. B 9 () 8-. [5] S. J. Yu K. Toda N. Sasa and T. Fukuyama N-solton solutons to Bogoyavknsk-Schff equaton and a guest for the solton solutons n ( + )-dmensons J. Phys. A: Math. And Gene. (998) 7-7. [6] J. Schff Panlevé transendent ther asymptotcs and physcal applcatons Plenum New York 99. 9