From th SlctdWork of Ji-Hn H 8 Exp-fnction mthod to olv th nonlinr dipriv K(mn qtion Xin-Wi Zho Yi-Xin Wn Ji-Hn H Dongh Univrity Avilbl t: http://work.bpr.com/ji_hn_h/9/
Frnd Pblihing Ho Ltd. Intrntionl Jornl of Nonlinr Scinc nd Nmricl Simltion9(- 8 Exp-fnction mthod to olv th nonlinr dipriv K(mn qtion Xin-Wi Zho Yi-Xin Wn Ji-Hn H Modrn Txtil Intitt Dongh Univrity 88 Yn-n Xil Rod Shnghi P.R. Chin Dprtmnt of Mthmtic Knming Collg Kn-hi Rod Ynnn P.R. Chin Abtrct Som nw xct oltion r obtind for th nonlinr dipriv K(m n qtion ing th xpfnction mthod. Th rlt how tht th mthod i trightforwrd nd conci nd it ppliction r promiing. Kyword: Exp-fnction mthod; Nonlinr dipriv K(mn qtion; Highr ordr nonlinrity.. Introdction In thi ppr w conidr th following nonlinr dipriv K(mn qtion: t m n + + ( x firt propod by Ron nd Hymn []. For crtin vl of m nd n th K(mn qtion h olitry wv which r compctly pportd. It w hown in Rf.[] tht Eq.( prnt compctly pportd oltion with nonmooth front. A lrg nmbr of mthod wr ggtd rcntly to tdy th nonlinr qtion ch th vritionl itrtion mthod [-8] th homotopy prtrbtion mthod [9-] th vritionl mthod [] nd th prmtr-xpnion mthod[] complt rviw on rcntly dvlopd nlyticl mthod i vilbl in Rf[7]. In thi ppr w will pply th Exp-fnction mthod[8-] to K(+ qtion in th form []. xxx + + t x xxx Exp-fnction mthod ( Th Exp-fnction mthod[8-] i widly d to rch for gnrlizd olitry oltion nd priodic oltion[-]. To illtrt th gnrl oltion procdr w conidr th following gnrl prtil diffrntil qtion: F. ( t x tt xx W im t it wv oltion o w introdc complx vribl η dfind η kx + ωt. ( W thrfor cn convrt th prtil diffrntil qtion Eq.( into n ordinry diffrntil qtion : ( ω ω G k k. ( Th Exp-fnction mthod i to rch for it oltion in th form ( η d jc q i q j xp b xp i ( jη ( iη ( cd p nd q r poitiv intgr nknown to b frthr dtrmind j nd bi r nknown contnt. Soltion Procdr Uing th trnformtion ( Eq. ( bcom ω k k + + (7
X.W. Zho Y.-X. Wn J.-H. HExp-fnction mthod to olv th nonlinr dipriv K(mn qtion Intgrting (7 nd tting th contnt of intgrtion to b zro w obtin k + ω+ + k (8 + th prim dnot th diffrntil with rpct to η. Mking th trnformtion Eq.(8 bcom / v (9 ( + ωv + k v + k v + k ( + vv ( W m tht th oltion of Eq.( cn b xprd in th form v ( η ( η + + xp( η ( η + b + b xp( η xp xp Sbtitting Eq. ( into Eq. ( w hv ( { C + C xp xp(8 } η + + C 8 η ( A Eqting th cofficint of xp( nη in Eq.( to b zro yild t of lgbric qtion: C C C C C C C C C 7 8 Solving th bov lgbric ytm with th hlp of ymbolic compttion ytm w obtin th following rlt C : A+ B A+ B b A B b b k( A+ B k ω ; ( + b C : A B b A+ B b b ( k A B k ω + b A B ; ( A ( + k b nd B A b. C nd : ( + k A B b ( + b k A B ( + k B ± ( 9 9 b B b b b ± ( B kb C ω k ; ( B A b + b B b + b b ( 779k 9 C b k + 9 C nd ( + k A B b + +. ( + k B ± k b A ( + B ( 9 9 b B b ± ( B kb C ω k ( B
ISSN: -9 Intrntionl Jornl of Nonlinr Scinc nd Nmricl Simltion 9( - 8 A b b B b b b ( 779k 9 C b k + 9 C 7 nd 8 A C C +. Bb ACb ± ( 9 Bb b ± b b A k ω. (7 C 9 nd : A C C Bb ACb ± ( + 9 Bb b ± b b A k ω. (8 A + B + A B k ( + C. For ch c w obtin th corrponding oltion follow: ωt ( ωt ( A+ B + b + b ( A+ B kx t ( kx t + ω + ω b + b ( A B + b (9 kx ωt ( ωt + ( A B + b + b ( AB ωt ( ω b + b ( A+ B + b A ( + k b ( B ( + b k A + B A b ω ( ( k A B nd ω + b. ( kx ω t ωt ( ωt ( k A B k B b k A B + ± + ωt + ( B ± b 9B 9b + b B ωt + ( k A B k B b k A B + ± + ωt + ( B ± b 9B 9b + b B ( ( kx ω ( ( kx ω A b + b B b + b A b b b kc b kc B b b ω k ω k B B
X.W. Zho Y.-X. Wn J.-H. HExp-fnction mthod to olv th nonlinr dipriv K(mn qtion C b k + 9 + b 779k + 9 C b k + 9 b 779k + 9. 78 9 ωt ( ωt 8 AC ± 8 Bb C+ 8 ACb ωt ( ωt ( 9 ( ± Bb + b ωt ( ω 8 AC ± 8 Bb C+ 8 ACb ωt ( ω ( 9 ± + Bb + b ( + A B A k A k k ( + ω ω nd C. A B + Whn k i n imginry nmbr th obtind olitry oltion cn b convrtd into priodic oltion [8]. Stting k ik nd ω iω w obtin th following trnformtion: nd ωt co[ Kx+Ω t] + iin[ Kx+Ω t] ( ( ω co[ Kx+Ωt] iin[ Kx+Ω t]. ( Sbtitting ( nd ( into Eq. (9-( rpctivly ld to th following priodic oltion: ( δ δ + co[ Kx+Ω t] + co[ Kx +Ω t] + ( δ+ δ (7 ( δ δ co[ Kx+Ω t] + co[ Kx t] (8 +Ω + K + K ( δδ δ k ( + δ δ Ω Ω. + ( δ δ ( δ δ nd ( + ( k co[ Kx t] ( k + δ δ +Ω ± + δ ( δ co[ Kx +Ω t] ± ( 9δ 9 (9
ISSN: -9 Intrntionl Jornl of Nonlinr Scinc nd Nmricl Simltion 9( - 8 ( k co[ Kx t] ( k + δ δ +Ω ± + δ ( δ co[ Kx +Ω t] ± ( 9δ 9 ( Kα δ + δ + δ δ Ω K δ Kα Ω K α ( ( k + 9 + 779k + 9 nd δ α ( ( k + 9 779k + 9. 78 9 δδ co[ Kx +Ω t] ± 8 δδ 7 8 8 co[ Kx +Ω t] ± 9 δ8 δδ co[ Kx +Ω t] ± 8 δ δ 9 8 co[ Kx +Ω t] ± + 9 δ ( ( + δ7 δ 8 k ( + δ 7K δ 9K δ Ω nd Ω. δ9 + δ. Conclion Rfrnc Th xp-fnction mthod itlf i of ttr implicity. Uing th mthod w cn obtin ri of xct oltion with om fr prmtr which cn b dtrmind ccording to th bondry/initil condition. Acknowlgmnt Th work i pportd by th Fondtion of Ynnn Provinc (7C8 nd th Progrm for Nw Cntry Excllnt Tlnt in Univrity ndr grnd No. NCET--7. [] P. Ron J.M. Hymn Compcton: oliton with finit wvlngth. Phy Rv Ltt. 7(99 7. [] P. Ron Compct nd noncompct dipriv trctr. Phy. Ltt. A. 7( 9. [] M. Inc Nmricl imltion of KdV nd mkdv qtion with initil condition by th vritionl itrtion mthod Cho Soliton. Frct. (77-8 [] M. Moghimi F.S.A. Hjzi Vritionl itrtion mthod for olving gnrlizd Brgr-Fihr nd Brgr qtion Cho Soliton. Frct. (77-7
X.W. Zho Y.-X. Wn J.-H. HExp-fnction mthod to olv th nonlinr dipriv K(mn qtion [] E. Yfogl Vritionl itrtion mthod for contrction of om compct nd noncompct trctr of Klin-Gordon qtion Int. J. Nonlinr Sci. 8(7-8 [] N. Bildik A. Konrlp Two-dimnionl diffrntil trnform mthod Adomin' dcompoition mthod nd vritionl itrtion mthod for prtil diffrntil qtion Int. J. Compt. Mth. 8( 97-987 [7] J.H. H X.H. W Contrction of olitry oltion nd compcton-lik oltion by vritionl itrtion mthod. Cho Soliton & Frctl 9( 8. [8] A.M.Wzwz Th vritionl itrtion mthod for rtionl oltion for KdV K( Brgr nd cbic Boinq qtion J. Compt. Appl. Mth. 7 (7 8. [9] H. Tri D.D. Gnji M. Rotmin Approximt oltion of K ( KdV nd modifid KdV qtion by vritionl itrtion mthod homotopy prtrbtion mthod nd homotopy nlyi mthod Int. J. Nonlinr Sci. 8(7 - [] T. Ozi A. Yildirim Trvling wv oltion of Kortwg-d Vri qtion ing H' homotopy prtrbtion mthod Int. J. Nonlinr Sci. 8(7 9- [] J.H. H Appliction of homotopy prtrbtion mthod to nonlinr wv qtion Cho Soliton. Frct. (9-7 [] J.H. H Homotopy prtrbtion mthod for bifrction of nonlinr problm Int. J. Nonlinr Sci. ( 7-8 [] M. Gorji D.D. Gnji S. Solimni Nw ppliction of H' homotopy prtrbtion mthod Int. J. Nonlinr Sci. 8(79-8. [] L. X Vritionl pproch to oliton of nonlinr dipriv K(mn qtion Cho Soliton & Frctl 7 (8 7. [] L. X H' prmtr-xpnding mthod for trongly nonlinr ocilltor. J. Compt. Appl. Mth. 7 ( (7 8-. [] J.H. H Som ymptotic mthod for trongly nonlinr qtion. Int. J. Modrn Phy. B ;(: 99. [7] J.H. H Non-prtrbtiv mthod for trongly nonlinr problm. Brlin: dirttion. d-vrlg im Intrnt GmbH. [8] A.M. Wzwz Nw t of olitry wv oltion to th KdV mkdvnd th gnrlizd KdV qtion Commniction in Nonlinr Scinc nd Nmricl Simltion (8 9. [9] J.H. H X.H. W Exp-fnction mthod for nonlinr wv qtion Cho Soliton & Frctl (7-78. [] X.H. W J.H. H Solitry oltion priodic oltion nd compcton-lik oltion ing th Exp-fnction mthod Comptr & Mthmtic with Appliction(79-98. [] J.H. H M.A. Abdo Nw priodic oltion for nonlinr voltion qtion ing Exp-fnction mthod Cho Soliton & Frctl (7:-9. [] S.D. Zh Exp-fnction mthod for th Hybrid-Lttic ytm Int. J. Nonlinr Sci. 8(7-. [] S.D. ZhExp-fnction mthod for th dicrt mkdv lttic Int. J. Nonlinr Sci. 8(7-8. [] Xin-Wi Zho Exp-Fnction Mthod for Solving Hxly Eqtion Mthmticl Problm in Enginring vol. 8 Articl ID 889 [] X.W. Zho Exp-fnction mthod for olving Fihr Eqtion Jornl. of Phyic: Confrnc Sri 9(8:. [] A Bkir A.Boz Exct oltion for cl of nonlinr prtil diffrntil qtion ing xp-fnction mthod Int. J. Nonlinr Sci. 8(7-.