Integrability and Exact Solutions for a (2+1)-dimensional Variable-Coefficient KdV Equation

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Ailble hp://pm.ed/m Appl. Appl. Mh. ISSN: 9-966 Vol. 9 Isse December pp.

1 Ailble hp://pm.ed/m Appl. Appl. Mh. ISSN: Vol. 9 Isse December pp Applicios d Applied Mhemics: A Ieriol Jorl AAM Iegrbili d Ec Solios for +-dimesiol Vrible-Coefficie KdV Eqio Zhg Y Deprme of Mhemics College of Scieces Shghi Uiersi Shghi P.R. Chi ig87@si.c X Gi-Qiog Deprme of Iformio Mgeme College of Mgeme Shghi Uiersi Shghi P.R. Chi gq@sff.sh.ed.c Receied: Ocober ; Acceped: Ags Absrc B sig he WTC mehod d smbolic compio we ppl he Pileé es for +- dimesiol rible-coefficie Korweg-de Vries KdV eqio d he cosidered eqio is fod o possess he Pileé proper wiho prmeric cosris. The o-bǎckld rsformio d seerl pes of ec solios re obied b sig he Pileé rced epsio mehod. Fill he Hiro s bilier form is preseed d mli-solio solios re lso cosrced. Kewords: +-dimesiol rible-coefficie KdV eqio; Pileé proper; Hiro s bilier form; solio solio; smbolic compio AMS MSC No.: 5Q5 5C8 F 68W. Irodcio De o he poeil pplicios of solio heor i mhemics phsics biolog commicios d srophsics he sdies o he olier eolio eqios NLEEs 66

2 AAM: Ier. J. Vol. 9 Isse December 67 he rced reserchers eio. M scieiss he cocered heir reserches o sors of ec solios d remrkble properies of NLEEs which ribes o he beer dersdig for hose olier mechism [Ablowiz d Ssm 978 Y 996 Peg 5]. Up o ow rios powerfl mehods he bee preseed o obi he ec solios for he NLEEs sch s he ierse scerig rsformio [Ablowiz d Clrkso 99] he Drbo rsformio [Mee d Slle 99] he Bǎckld rsformios [Rogers d Shdwick 98] he Hiro s bilier mehod [Hiro.] he Pileé lsis [Weiss Tbor d Crele 98] homoop perrbio mehod[gi d Rfei 6] h fcio mehod [M d Fchsseier 996 Wzwz 5] sie-cosie mehod [Abdo 7] he homogeos blce mehod [F d Zhg 998] riiol ierio mehod [Tg d Lig 6 Peg 6] ep-fcio mehod [He d W 6] Jcobi ellipic fcio epsio mehod [Che e l. 5] F-epsio mehod [Abdo 7] he sb-ode mehod [Li d Wg 7] d so o. Cosiderig h coefficie fcios re ble o reflec he slowl-rig ihomogeeiies oiformiies of bodries d eerl forces recel olier eqios wih rible coefficies he rced cosiderble eio i he lierre. Alhogh he rible coefficies icrese he difficl of or iesigio m reserchers he iesiged he iegrble proper d ec solios of he rible-coefficie olier eolio eqios [Deg 6 X 9 Lü e l. Krekel e l. ]. I is esil fod h solios c be compressed d heir dmics effeciel corolled hrogh hese rible prmeers. The Korweg-de Vries eqio is oe of he mos impor ssems i mhemicl phsics d m phsicl siios re goered b rible-coefficie KdV eqio. To or kowledge he sd of he +-dimesiol rible-coefficie KdV eqio wih differe form hs bee pid eio b some hors d he hors he obied he iegrble proper d ec solios of hese eqios [F Y 8 Zh e l. ]. Frhermore here re some reserches bo he pplicios d meros ieresig properies of he +-dimesiol rible-coefficie KdV eqio. For emple Emmel Yomb e l. cosrced he ec solios of he followig eqio: 6 e b mes of he h fcio mehod he homogeos blce mehod d oher mehods [Yomb Moss d M. El-Shiekh ]. Ver recel Peg e l. [ Peg ] deried ew +-dimesiol KdV eqio wih rible coefficies i he L pir geerig echiqe he eqio is gie s follows: d b 6 where d b re wo rbirr fcios of idiced ribles. To or kowledge eqio ws cosidered for specil choices wih b. Therefore i his pper we will sd he Pileé proper d lic solios of i he geerl form. The res of his pper is rrged s follows. I Secio we perform he Pileé lsis of. I is proe h eqio possesses he Pileé proper wiho prmeric cosris. I

3 68 Z. Y d X. Gi-Qiog Secio seerl pes of ec solios will be cosrced i erms of he Ao-Bǎckld rsformio. I Secio he Hiro s bilier form of will be preseed d he mlisolio solios will be deried for eqio wih specil choice of prmeers. Fill or coclsios d discssios will be gie i Secio 5.. Pileé lsis The iegrble clsses of KdV eqios irige reserchers for he ps few decdes de o heir rich rie of solios. The Pileé lsis is oe of he powerfl mehods for ideifig he iegrble properies of olier pril differeil eqios. I his secio we ppl he Pileé es for iegrbili o sig he well-kow WTC mehod d smbolic compio [X d Li Q e l. ]. I order o ppl he Pileé lsis we irodce he rsformio d eqio redces o he followig copled ssem:. b 6 Eqios is sid o possess he Pileé proper if is solios re sigle-led bo rbirr o-chrcerisic moble siglri mifolds. I oher words ll solios of c be epressed s Lre series wih sfficie mber of rbirr fcios mog i ddiio o d re lic fcios of. Moreoer he ledig orders shold be egie iegers. Firs fid he ledig order d coefficies. To rech his im we iser io eqios. Upo blcig he domi erms we obi 5. 6 Ne i order o fid he resoces h re powers which he rbirr coefficies eer io he Lre series we sbsie 7 5 io eqios. Iserig eqios 6 d ishig he coefficies of ields

4 AAM: Ier. J. Vol. 9 Isse December b 8 The eigeles of he boe mri gies he followig resoce eqio for he epoe :. 6 b 9 Ths he resoces occr 6. As sl he resoce correspods o he rbirriess of he siglr mifold. I order o check he eisece of sfficie mber of rbirr fcios he resoce les he rced Lre epsios 6 6 ~ ~ re sbsied io eqios. To simplif he clclios we mke se of he Krskl sz d is rbirr fcio of d. The he coefficies fcios d i eqios will be fcios of d oo. Eqig he coefficies of o zero we obi lier ssem wih respec o d 5 b from which we he. Seig he coefficies of o zero we obi lier ssem wih respec o d. 8 8 b b Solig i we ge b

5 65 Z. Y d X. Gi-Qiog where is rbirr fcio which correspods o he resoce. Collecig he coefficies of d sig he les of d we he b. 5 Solig he boe eqios wih respec o ields. 6 I his w b proceedig frher d collecig he coefficies of d oe c obi oher coefficies of Lre series b b b [6 6b 6 5 5b 6 6 6b 6 6 b b 6 6b 5 5b 6 6b 6b b b b ] 7 6 ] / [6 7 where b b d 6 re rbirr fcios which correspod o he resoces 6. Up o ow we esblish he reqired mber of rbirr fcios correspodig o d 6 wiho ddiiol resricios o he prmeers. Ths we c coclde h eqios hs Pileé proper d hece is epeced o be iegrble.. Ao-Bǎckld rsformio d ec solios I order o ge o-bǎckld rsformio we m rce he Lre series he cos erms mel

6 AAM: Ier. J. Vol. 9 Isse December 65 8 Where re lic fcios of. The sbsiig 8 io d seig he coefficies of wih differe powers o zero ields b b 6 6 9b 6 9b 6 6 b 9b b b b b 6b b b b b b 6b 6b b b 6b b 6b b 6b b 6b From 9 oe c esil obi 6b b. 9. From 8 d we obi o-bǎckld rsformio s follows

7 65 Z. Y d X. Gi-Qiog l l. From he ls wo eqios of 9 i is esil see h is se of solios of so we m ke he riil cm solio s he seed solio he re redced o where sisfies he followig cosri codiios l l b b 8 b 5b b. 5 Oce Eqios -5 re soled we c ge ec solios of b mes of he rsformio. Eqios -5 re homogeeos differeil eqios we m sppose h he solios of -5 re i he form f ep f k l ep m 6 where k re coss l m re he rbirr lic fcios wheres fcios f m be sie cosie hperbolic sie hperbolic cosie d so o. I his secio we ol cosider some specil cses. Cse. f I his cse he solios of -5 red where he prmeers sisf he codiio ep m 7 m m b 8 Cse. f cos or f si I his cse we obi he solios of -5 s follows where he prmeers sisf he codiios cos l ep m 9 si l ep m

8 AAM: Ier. J. Vol. 9 Isse December 65 Cse. f cosh or f sih Similrl we c ge he solios of -5 l ml m l b. l l cosh l ep m sih l ep m 5 where he prmeric codiios re gie b. Iserig eqio 7 eqio 9 eqio eqio d eqio io eqio oe c obi fie pes of ec solios of. For ske of simplici we ol lis oe solio of. Sbsiig he epressio 7 io eqio we obi he followig solio m sec h m sec h m where b sisfies he cosri codiio. Hiro s Bilier form d Solio Solios m m b 5 As he firs sep oe shold rsform io he bilier forms wih he help of depede rible rsformio. To his prpose we cosider he sdrd Pileé rced epsio l f 6 l f for simplici we m ke he seed solios. Wih he rsformio 6 we ge he followig bilier form of D D D D f f s D D D D D s bd f f 7 where d b re he fcios wih respec o d s is he ilir rible he wellkow Hiro bilier operors D D D re defied b [Hiro ]

9 65 Z. Y d X. Gi-Qiog D m ' m ' ' k ' ' ' D D b b. 8 k ' ' ' For he simplici of compio we will cosider eqio wih specil choice of prmeers wih = b =. I his cse eqio becomes Accordigl is bilier form reds d 6. 9 D D D D f f D D D s D D s D f f. The e sep is he sl oe. Le s represe f b forml series f f f f where seres s prmeer. Proceedig s i he Hiro mehod we sbsie io d eqe o zero he differe powers of o : D D D D f D D D s D D s D f o : D D D f D D D f f s s DD D D DDs D f D D D D D D D f f s o : D D D f D D D f f s s DD D D DDs D f D D D D D D D f f s Firs we cosider oe-solio solio sppose h f e k l 5 where k l re coss o be deermied. Iserig i o leds o he dispersio relio k l k. 6

10 AAM: Ier. J. Vol. 9 Isse December 655 We fid i fc h he righ-hd side of is eql o zero d so we c se f for. Therefore he series rces he he ec solio o reds k l k f e k l. 7 Applig he rsformio 6 ields he oe-solio solio of 9 for simplici we ke l k k l f l e sec h 8 k l k where. We ow proceed o serch for wo-solio solio. Sppose h f e e k l i 9 i i i i i from oe ges ki li ki i i. 5 Togeher he les of f solig wih respec o f ields f k k k e e k. 5 Wih he help of Mple he righ-hd side of is redced o zero. Coseqel we c se f for i. Therefore he series rces he ec solio for 9 is obied i k k f e e e k k 5 where ki li ki i i ki li i. For simplici oe m le sig he rsformio 6 he eplici wo-solio solio of 9 reds

11 656 Z. Y d X. Gi-Qiog k k l e e e. 5 k k Geerll he N-solio solio of 9 c be epressed s l ep i k l k k l i e i i k k i i k k 5 where he firs mes smmio oer ll possible combiios of d i mes smmio oer ll possible pirs i. 5. Coclsios I his work we he cosidered +-dimesiol rible-coefficie KdV eqio. Throgh he Pileé lsis he cosidered eqio is fod o possess he Pileé proper wiho prmeric cosris. Usig he Pileé rced epsio mehod he o- Bǎckld rsformio d fie pes of ec solios re obied. Moreoer he Hiro s bilier form of he +-dimesiol rible-coefficie KdV eqio is cosrced. The mli-solio solios re cosrced for he specil choice of prmeers. The obied ec solios m be sefl for describig he correspod phsicl pheome. I is desered o mke cosiderios o obiig oher iegrble properies of his eqio sch s he L pir Bǎckld rsformio coserio lws d so o. REFERENCES Ablowiz M.J. d Ssm J Solios d riol solios of olier eolio eqios J. Mh. Phs Vol. 9 pp Y C. T A simple rsformio for olier wes Phs. Le. A Vol. pp Peg Y. Z. 6. Ec periodic d solir wes d heir iercios for he +- dimesiol KdV eqio Phs. Le. A Vol. 5 pp. -7. Ablowiz M. J. d Clrkso P. A. 99. Solios Nolier Eolio Eqios d Ierse Scerig Trsform Cmbridge Uiersi Press. Mee V. B. d Slle M. A. 99. Drbo Trsformio d Solio Spriger. Rogers C. d Shdwick W. F. 98. Bǎckld rsformios Acdemic Press. Hiro R.. The Direc Mehod i Solio Theor Cmbrige Uiersi Press. Weiss Joh Tbor M. d Crele George. 98 The Pileé proper for pril differeil eqios J. Mh. Phs. Vol. pp

12 AAM: Ier. J. Vol. 9 Isse December 657 M W. X. d Fchsseier B Eplici d ec solios o Kolmogoro- Peroskii-Pisko eqio I. J. No-Lier Mech. Vol. pp Wzwz A.M. 5. The h-fcio mehod: Solios d periodic solios for he Dodd- Bllogh-Mikhilo d he Tzizeic-Dodd-Bllogh eqios Chos Solios d Frcls Vol. 5 No. pp F E.G.. Eeded h-fcio mehod d is pplicios o olier eqios Phs. Le. A Vol. 77 pp. -8. Gi D. D. d Rfei M. 6. Solir we solios for geerlized Hiro Ssm copled KdV eqio b homoop perrbio mehod Phs. Le. A Vol. 56 pp Abdo M. A. 7. New solir we solios o he modified Kwhr eqio Phs. Le. A Vol. 6 pp F E. G. d Zhg H. Q A oe o he homogeeos blce mehod Phs. Le. A Vol. 6 pp. -6. Wg M. L Solir we solios for ri Bossiesq eqios Phs. Le Vol. 99 pp Tg X. Y. d Lig Z. F. 6. Vrible sepred solios for he +-dimesiol Jimbo-Miw eqio Phs. Le. A Vol. 5 pp Peg Y. Z. 6. The Vrible Seprio Mehod d Ec Jcobi Ellipic Fcio Solios for he Nizhik-Noiko-Veselo Eqio Ac Phsic Poloic A Vol. pp. -9. He H. J. d W H. X. 6. Ep-fcio mehod for olier we eqios Chos Solios d Frcls Vol. pp Che Y. Wg Q. d Li B.5. Ellipic eqio riol epsio mehod d ew ec rellig solios for Whihm roer p eqios Chos Solios d Frcls Vol. 6 pp. -6. Abdo M. A. 7. The eeded F-epsio mehod d is pplicio for clss of olier eolio eqios Chos. Solios d Frcls Vol. pp Li X. Z. d Wg M. L. 7. A sb-ode mehod for fidig ec solios of geerlized KdV-MKdV eqio wih high-order olier erms Phs. Le. A Vol. 6 pp Deg S. F. 6. Ec Solios for Noisospecrl d Vrible-Coefficie Kdomse Peishili Eqio Vol. No. 7 pp X G. Q. 9. A oe o he Pileé es for olier rible-coefficie PDEs Comp. Phs. Comm Vol. 8 pp.7-. Lü X. Ti B. Zhg H. Q. X T. d Li H.. Iegrbili sd o geerlized +-dimesiol rible-coefficie Grder model wih smbolic compio Chos Vol. 5. Krekel R. A. Nkkeer K. d Chow K. W.. Iegrble NLS eqio wih imedepede olier coefficie d self-similr rcie BEC Comm. Nolier. Sci. Nmer. Siml Vol. 6 pp F E. G.. The iegrbili of oisospecrl d rible-coefficie KdV eqio wih bir Bell polomils Phs Le A Vol. 75 pp Y Z. Y. 8. The modified KdV eqio wih rible-coefficies: eci/bi-rible rellig we-like solios Appl. Mh. Comp Vol. pp. 6-. Zh S. H. Go Y. T. Y X. S Z. Y. Gi X. L. Meg D. X.. Pileé proper

13 658 Z. Y d X. Gi-Qiog solio-like solios d compleios for copled rible-coefficie modified Koreweg de Vries ssem i wo-ler flid model Appl. Mh. Comp Vol. 7 pp Yomb E.. Cosrcio of ew solio-like solios for he + dimesiol KdV eqio wih rible coefficies Chos Solios d Frcls Vol. pp Moss M. H. M. d M. El-Shiekh Rehb.. Direc Redcio d Ec Solios for Geerlized Vrible Coefficies D KdV Eqio der Some Iegrbili Codiios Comm. Theor. Phs. Vol. 55 pp Peg Y. Z.. A ew +-dimesiol KdV eqio d is loclized srcres Comm. Theor. Phs Vol. 5 pp X G. Q. d Li Z. B.. A mple pckge for he Pileé es of oelier pril differeil eqios Chi. Phs. Le Vol. pp X G. Q. d Li Z. B.. Smbolic compio of he Pileé es for olier pril differeil eqios sig Mple Comp. Phs. Comm Vol. 6 pp Q Q. X. Ti B. Li W. J. Li M. S K.. Pileé iegrbili d N-solio solio for he rible-coefficie Zkhro-Kzeso eqio from plsms Nolier D Vol. 6 pp. 9-5.

Finding Formulas Involving Hypergeometric Functions by Evaluating and Comparing the Multipliers of the Laplacian on IR n

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