THE ROSENAU-HYMAN K(2,2) EQUATION
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1 THE ROSEA-HYMA K EATIO. ITRODCTIO Mos of wly non-lnr nd lnr dsprsv quons sudd so fr d solry wvs clld solons r nfn n n []. Wll nown prl dffrnl quons PDEs w solon soluon nclud sn-gordon SG quon cuc non-lnr Scordngr LS quon nd Korwg-d Vrs KdV quon. Rosnu nd Hy [8] rpord clss of PDEs n n.. wc s gnrlzon of KdV quon. Ts quons w vlus of nd n r dnod y K n. I s n sown nrcon of non-lnr dsprson w non-lnr convcon gnrs cly copc srucurs clld copcons fr of ponnl ls. T copcon soluons so gnrd v d pplcons n sudy of prn forons s osrvd sonry nd dyncl prns n nur r usully fn n n. T copcon spd dpnds on s g u unl solon s wd s ndpndn of s spd. Bsd copcon srucur nd unusul spd-wd rlon copcons v rrl solon l propry y colld lsclly. T PDEs.. s n gnrl wo consrvd quns gvn y P u d ; u n d... Vrous fors of B-spln ss ogr w fn ln ods v n nsvly usd n solvng so prl dffrnl quons. For nsnc qunc B-spln collocon fn ln od for nurcl soluon of KdV Burgrs nd KdVB quons [] s n succssfully plnd. Idrs Dg nd
2 s co-uors [ 7] v lso solvd Burgr s nd RLW quons usng qunc B-spln ss w fn ln ods. T rsuls of clculons sowd ccurcy of soluon s provd f Glrn forulon ogr w qunc B-spln funcons s usd n gng nurcl soluon of prl dffrnl quons u copuonl cos of B-spln Glrn lgor ncrss. T Rosnu Hyn K n quon v n solvd nurclly y vrous ods. Fruos l. [8] usd fn dffrnc od o solvd K quon. Isl nd T [] v sudd K nurclly y vrous fn dffrnc nd fn ln cnqus. Ty lso provd condonl nd uncondonl sly of proposd scs. A fn dffrnc od s n consrucd y Isl [] for soluon of K quon. Isl nd Al-Soly [] v usd fn ln od w usd of lnr funcon nd cuc B- spln funcon s rl nd s funcons for nurcl soluon of K quon. Solry-wv soluon of K n vng copc suppor s n sudd y Wzwz [9] usng Adon dcoposon od. In s ppr w wll dscr nurcl soluon of on of prl dffrnl quons.. vz. K y Glrn s od usng qunc B- spln funcons. T consrvd quns r clculd usng Spson s rul. Fnlly coprson wn nlycl nd nurcl soluon of K quon for purposd lgor s d.. ITIC B-SPLIE GALERKI METHOD T Rosnu-Hyn K quon s for..
3 wr suscrs nd dno dffrnon. Boundry condons r slcd fro oognous oundry condons ] [ T.. nd nl condon ]. [ f Applyng Glrn cnqu o Eq... w wg funcon W ylds ngrl quon d W... W consdr s s unfor pron of soluon don y nos nd... rougou ppr. L... orws ] [ ] [ ] [ ] [ ] [ ] [..
4 qunc B-splns w nos.... T s of qunc B-splns fors ss ovr rgon. T glol pproon dfnd usng qunc B-splns.. wll soug o nlycl soluon. In s ppro soluon s dpndn prr o drnd fro qunc Glrn for of Eq.... T nodl vlus of nd s drvvs up o four ordr r gvn n rs of prr fro us of splns.. nd rl soluon..... A locl coordn sys cn dfnd usng ppng rlon o rnsfor fn ln ] [ no nrvl ]. [ T prssons of qunc B-spln sp funcons r ndpndn of ln poson r ond w rlon of glol nd locl coordns rlon ovr ] [ s
5 ...7 Fro qunc B-splns covrng s succssv fn lns ypcl fn lns r covrd y s qunc B-spln sp funcons. So pproon rducd ovr ln ] [ s..8 wr... c s ln prrs. Tng wg funcons W w qunc B-spln sp funcons nd susung ln rl funcon n ngrl quon.. ovr ln ] [ lds o d d d..9 wr nd only vlus nd... nd dnos drvv w rspc o wc n r for s T T M L A..
6 wr ln rcs L M r r A s nd T rcs nd E so rcs A d L d M d.. L nd B M r orgnzd o n dnson s rcs L B nd B E M.. E r prssd s dpndng on ln prr. Conng ll ln rcs for c ln w on sys of nonlnr ordnry dffrnl quon wr glol prr s A B E If w usd Crn-colson dscrzon forul for vcor of ln prr nd usul fn dffrnc quon for drvvs prr n quon: n n ; n n.. w rc nonlnr rcurrnc rlon for prrs : n A ΔB ΔE A ΔB ΔE n... Ts sys s d up of quon n unnown prrs. W cn on solvl sys y posng oundry condons lf nd of rgon nd rg nd of rgon n n n n o ln prrs. T ov sd non-lnr sys s
7 7 solvd y won s od. Iron sould rpd wo or r s usng followng corrcor procdur * n n n n...7 To sr ron of rcurrnc rlon of sys.. nl prr vcor us drnd usng followng nl nd oundry condons nos Condons w r corrspondng qunc B-spln rprsnon gv r quon of sys Ts Mr sys s solvd o g nl condon prrs. On drnng nl prrs fro sys ov clculon of soluons r rd usng sys.. succssv s. By usng ond prrs fro sys.. nodl vlus nd s drvvs of ordr cn word ou fro Eqs....
8 . MERICAL APPLICATIOS In s scon w prsn so nurcl prns o fnd nurcl soluon of sngl copcon wvs nd drnng soluon of wo nd r copcons nrcons. W lso sow copcons splng fro n nl d... MOTIO OF A SIGLE COMPACTO T K quon posssss followng consrvd quns: I d nd I d.. Snc consrvd quns r pcd o rn consn durng run of lgor o v ffcn nurcl sc. Spson s rul wll usd o clcul ngrls.. dscr pons nurclly. To sur ccurcy of nurcl soluons dffrnc wn nlycl nd nurcl soluons so spcfd s s copud y usng dscr roo n squr rror nor L nd u rror nor L.... W dop sngl copcon wv soluon of K quon w nl condon cos u orws.. T c soluon of K quon.. s gvn y cos u orws.. wc dscrs copcly suppord solry wv clld copcons rvlng 8
9 w vlocy nd plud n posv -drcon. T prrs.. nd r usd. T run of lgor s crrd up o ovr prol don. T u roo n squr rrors nd consrvd quns r prsnd n l. Sngl copcon wv soluon s drwn n Fg c d Fg..: Sngl copcon profls. Soluon of K quon w nl condon. n. Soluon ; Soluon ; c Soluon ; d Spc grp of soluon up o. Tl.: Invrns nd rror nors for sngl copcon wv 9
10 I I L L ITERACTIO OF TWO COMPACTOS In s pl w sudy nrcon of wo copcons of K quon.. vng dffrn plud nd rvlng n s drcon. W consdr K quon w nl condon gvn y u u.. wr cos u orws nd. Hr w coos followng prrs T nrcon scnro s dsplyd n Fg.. nd sows ow wo wll sprd copcons nrc nd rg fr nrcon s uncngd n spd nd vlocy. 7
11 c d Fg..: Doul copcon profls. Soluon of K quon w nl condon. n. Soluon Soluon c Soluon d Spc grp of soluon up o. 7
12 c d Fg..: Trpl copcon profls. Soluon of K quon w nl condon. n 8. Soluon ; Soluon ; c Soluon ; d Spc grp of soluon up o... ITERACTIO OF THREE COMPACTOS T rpl copcon nrcon of K quon.. s nl condon gvn y 7
13 u u 8.. wr cos u orws nd c d Fg..: Trpl copcon splng profls. Soluon of K quon w nl condon. n. Soluon Soluon c Soluon d Spc grp of soluon up o. Hr w coos followng prrs nd. T nrcon of r copcons s sown n Fg... 7
14 .. SPLITTIG COMPACTOS T rpl copcon splng cs of K quon s followng nl condon gvn y[] u cos 8 orws.. T copcon splng fro or gnrl nl d r sown n Fg... Hr w coos followng prrs. nd. 8.. COCLSIO W v succssfully ppld Glrn s fn ln od usng qunc B-spln funcon for solvng gly non-lnr dsprsv prl dffrnl quon K. urcl pls r sown o llusr ccurcy of od. W lso sowd nrcon of wo nd r copcons nd splng of copcon. urcl ccurcy of od cn ncrsd y rducng spl nd porl sp szs nd rspcvly. T proposd od s or drwc of copuonl coply. I s or copuonl n or ods cd n s cpr. Bu n rs of ccurcy s od s ccpl. 7
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