THE EXTENDED TANH METHOD FOR SOLVING THE -DIMENSION NONLINEAR DISPERSIVE LONG WAVE EQUATION

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1 Jornl of Mhemil Sienes: Adnes nd Appliions Volme Nmer 8 Pes 99- THE EXTENDED TANH METHOD FOR SOLVING THE ( ) -DIMENSION NONLINEAR DISPERSIVE LONG WAVE EQUATION SHENGQIANG TANG KELEI ZHANG nd JIHONG RONG Shool of Mhemis nd Compin Siene Gilin Uniersi of Eleroni Tehnolo Gilin Gni 5 P. R. Chin e-mil: nsq@e.ed.n Asr This pper is deoed o sdin he ( ) -dimension nonliner dispersie lon we eqion. The eended nh mehod is sed o ond he nlsis. The ( ) -dimension NDLW eqion ie rise o he ompons solions: kink nd periodi wes. The work onfirms he power of he proposed mehod.. Inrodion In his pper we onsider he ( ) -dimension nonliner dispersie lon we eqion whih ws inroded Boii e l. [] s follow: ( ) () Mhemis Sje Clssifiion: 5Q5. Kewords nd phrses: he eended nh mehod kink we periodi we ()- dimension nonliner dispersie lon we eqion. This reserh ws sppored he Siene Fondion of Edion Deprmen of Gni proine (D87). Reeied M 8 8 Sienifi Adnes Plishers

2 SHENGQIANG TANG KELEI ZHANG nd JIHONG RONG ( ). () If we le e eql o he ( ) -dimension nonliner dispersie lon we eqion n e reded o he ( ) -dimension nonliner dispersie lon we eqion h desries he rel of he shllow wer we. [] s ompiili for wek L pir. Reenl onsiderle effor hs een deoed o he sd of his ssem. In [] Pqin nd Winerniz showed h he smmer ler of ( ) - dimensionl dispersie lon we eqion () is infinie-dimensionl nd possesses K-Mood-Virsoro srre. Some speil similri solions re lso ien in [] sin smmer ler nd he lssil heoreil nlsis. Lo [] hs ien nine pes of he wodimensionl pril differenil eqion redions nd pes of he ordinr differenil eqion redions mens of he dire nd nonlssil mehod. The ssem () he no Pinleé proper hoh he re L or IST inerle []. More reenl Tn e l. [6] mens of he rile seprion pproh he ndn lolized oheren srres of he ssem () re ien o. In [7] he possile hoi nd frl lolized srres re reeled for he ssem (). [5] sin new eended Rii eqion rionl epnsion mehod o onsrin mliple e solions. In his pper we will onsider he kink nd periodi wes for () sin he eended nh mehod [8]. To find he rellin we solions of () we firs onsider he rellin we solions in he form ( ) ( ξ) ( ) ( ξ) ξ () where denoes he we speed. Therefore () redes o e ( ( ) ) ( ) () () where is he deriie wih respe o ξ. Inerin () one ( ) ()

3 THE EXTENDED TANH METHOD FOR SOLVING where is inerion onsn. Now we onsider he followin rnsformions: β γ (5) where () β γ re onsns o e deermined ler. Ssiin (5) ino ( β γ( ( ) )) ( ( ) ) (6) Oiosl when β γ (7) ssiin (5) nd (7) ino () herefore () redes o e. (8) In his pper we lws ssme h Eq. () sisfies (7). Therefore we onl onsider he ssem (8).. Usin he Eended Tnh Mehod Applin he lnin ehniqe of wih ies M. The eended nh mehod ssies he finie epnsion ( ) S( Y ) Y Y nh( ξ) (9) Y ino (8) ollein he oeffiiens of eh power of Y we oin ssem of leri eqions for nd nd sin n smoli ompion prorm sh s Mple solin his ssem ies he followin for ses of solions (i) The firs se: ( ) >. ()

4 SHENGQIANG TANG KELEI ZHANG nd JIHONG RONG (ii) The seond se: i. i () (iii) The hird se:. > () (i) The forh se: i i. () This in rn ies he followin kink we solions

5 THE EXTENDED TANH METHOD FOR SOLVING β nh. oh γ β Howeer for we oin he periodi solions γ β n for we oin he periodi solions γ β. o. Disssion The ()-dimension nonliner dispersie lon we eqion ws inesied sin he eended nh mehod. Seerl e disin rellin we solions were formll deried sin he mehod. The oined resls lerl demonsre he effiien of he mehod sed in his work. Moreoer he mehod is ple of rel minimizin he size of ompionl work ompred o oher eisin ehniqes. Unlike wh ws hoh efore h he ()-dimension nonliner dispersie lon we eqion nno e soled he eended nh mehod he inroded nlsis onfirm h he eended nh mehod n e effeiel sed for hese eqions.

6 SHENGQIANG TANG KELEI ZHANG nd JIHONG RONG Referenes [] P. Boii J. J. P. Leon M. Mnn nd F. Pempinelli Inerse Prol. (987) 5. [] S. Y. Lo Phs. Le. A 76 (99) 9. [] S. Y. Lo Mh. Meh. Appl. Si. 8 (995) 789. [] G. Pqin nd P. Winerniz Phsi D 6 (99). [5] Li-N Son e l. Chos Solions Frls (7) [6] X. Y. Tn C. L. Chen nd S. Y. Lo J. Phs. A 5 () L9. [7] X. Y. Tn nd S. Y. Lo Chos Solions Frls () 5. [8] A. M. Wzwz New solir we solions o he modified forms of Desperis- Proesi nd Cmss-Holm eqions Appl. Mh. Comp. 86() (7) -.