Solutions of Some Nonlinear Partial Differential Equations

Transcription

1 Solons of Some onlner Prl Dfferenl Eqons Thess Smed Anl Verm Regd. o. 95 In flfllmen of he reqremen for he degree of Door of Phlosoph In Mhems Shool of Mhems Thpr Unvesr Pl-7

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5 Asr The prme oeve nd movon n rrng o he proposed sd s o demonsre he mporne nd eff of grop heore ehnqes nd dfferenl qdrre mehod o fnd nll nd nmerl solons of some nonlner eqons vz. fml of nonlner eqons wo omponen shllow wer wve ssem nd hree-opled KdV eqons Klen-Gordon eqon Fsher s pe eqons nd wo dmensonl hperol eqons wh vrle oeffens. Or hess omprses of s hpers. In he nrodor pr some mporn feres of Le grop of rnsformons nd dfferenl qdrre mehod DQM re demonsred nd he mheml fndmenls of onnos grop heor nd weghng oeffens of DQM re revewed whh re of gre mporne o he work del n Chpers -6. Chper s onerned wh fml of nonlner evolon eqons. In mheml phss he fml of nonlner evolon eqons hs een se of eensve sd. I represens lss of nonlner evolon eqons. We nvesge he smmer of he eqon mens of lssl Le smmer mehod. The smmer lgers nd grops of fml of nonlner eqon re oned. Spell he mos generl one-prmeer grop of smmeres s gven nd mos generl solons re gned. We sed G / G -epnson mehod o fnd rvellng wve solons of one ODE. ew epl solons of eqon re derved. Chper dels wh wo-omponen shllow wer ssem nd hree-opled KdV eqons engendered he emnn ssem sng Le lssl mehod. The shllow wer eqons re se of hperol prl dfferenl eqons h desre he flow elow pressre srfe n fld. KdV s mheml model of wves on shllow wer srfe. The KdV eqon hs nfnel mn negrls of moon whh do no hnge wh me. we sded nll solons of wo-omponen shllow wer ssem nd hreeopled KdV eqons engendered he emnn ssem Le lssl mehod. The smmer grop redes he orgnl eqon o he smple one. B Le lssl mehod we v 5

6 hve nvesged he smmeres. Afer hvng done sndrd hnge of dependen vrles we sed he ss of he nfnesml operor wren n he new vrle n order o fnd some new lsses of nvrn solons. Chper s onerned wh nonlner Klen-Gordon eqon. The nonlner Klen- Gordon eqon ppers n mn pes of nonlneres. The Klen-Gordon eqon rses n relvs qnm mehns nd feld heor whh s of gre mporne for he hgh energ phsss nd s sed o model mn dfferen phenomen nldng he propgon of dsloons n rsls nd he ehvor of elemenr prles. A nmerl sheme hs een developed for he nmerl smlon of nonlner Klen-Gordon eqon. The essenl dffl n he nmerl solons for he Klen-Gordon eqon nvolves he nondedness of he phsl domn n he hper we onsdered he nonlner Klen- Gordon eqon on onded domn. Gss-Loo-Cheshev grd pons re sed for he nmerl smlon. Chper 5 dels wh Fsher pe s eqons. In hs hper nl nd nmerl solons of nonlner dffson eqon of Fsher pe s wh he help of lssl Le smmer mehod nd dfferenl qdrre mehod DQM re sded. The Fsher s eqon ors n heml knes nd neron poplon n nler reon. Moreover he sme eqon lso ors n logs poplon growh models flme propgon nerophsolog ol heml reons nd rnhng Brownn moon proesses. Le smmer mehod s lzed o nvesge he smmeres nd nvrn solons of he eqons. The nfnesml generors n he opml ssem re sed for redons nd e solons. Fnll polnoml dfferenl qdrre mehod s sed o fnd he nmerl solons of he Fsher s pe eqons wh he help of nl nd ondr ondons ken from he nl solons oned lssl Le smmer mehod. I s onlded h he nmerl solons re n good greemen wh he nll solons. Chper 6 s devoed o derve nmerl solons of wo dmensonl hperol prl dfferenl eqon wh vrle oeffens. Hperol PDEs desre propgon of dsrnes n spe nd me when he ol energ f he dsrnes remn onserved. v 6

7 I s he ondon of energ onservon h mkes he hperol eqon dfferen from prol ones. In hs hper osne epnson sed dfferenl qdrre mehod eended for nmerl smlon of he seond order wo dmensonl hperol eqons. The mn dvnge of he presen sheme s h he sheme gves ver re nd smlr resls o he e solons hoosng less nmer of grd pons nd he prolem n e solved p o g me. The omped nmerl resls re ompred wh he resls presened n some erler works nd s fond h he presen nmerl ehnqe gves eer resls hn he ohers. v 7

8 Ls of Reserh Ppers. Anl Verm Rm Jwr nd Ssh Kmr A nmerl sheme sed on dfferenl qdrre mehod for nmerl smlon of nonlner Klen-Gordon eqon Inernonl ornl of merl mehods for He nd Fld Flow 9-. Emerld Imp for.99 SCI. Anl Verm Rm Jwr nd Mehme Emr Koksl Anl nd merl solons of nonlner dffson eqons v smmer redon Advnes n Dfferene eqons -. Sprnger Imp for.6 SCI. Anl Verm nd Rm Jwr Cosne epnson sed Dfferenl Qdrre lgorhm for merl smlon of Two Dmensonl Hperol eqons wh vrle oeffens Aeped n Inernonl Jornl of merl mehods for He nd Fld Flow. Emerld Imp for.99 SCI. Anl Verm nd Rm Jwr Ssh Kmr nd sh Gol Le smmer mehod for fndng e solons of fml of nonlner eqons. Commned A Mhem Sen. 5. Anl Verm nd Rm Jwr Ssh Kmr nd Vks Kmr Smmer Redon nd Anll solons of opled eqons Commned v 8

9 Ls of Fgres. Grphl represenon of solon.. for Grphl represenon of solon.. for Grphl represenon of solon..5 for 9. Grphl represenon of solon..5 for Grphl represenon of solon..6 for Grphl represenon of solon..6 for 7..7 Grphl represenon of solon..7 for Grphl represenon of solon..8 for Grphl represenon of solon..5 for 6. Grphl represenon of solon..55 for 6 6. Grphl represenon of solon..5 for 5 m 55. Grphl represenon of solon..9 for Grphl represenon of solon.. for Grphl represenon of solon.5.5 for v 9

10 . merl Lef nd e Rgh solons of Emple p o s n spe-me grph form 8. merl Lef nd e Rgh solons of Emple p o 5s n spe-me grph form 8. merl Lef nd e Rgh solons of Emple p o 5s n spe-me grph form 8. merl Lef nd e Rgh solons of Emple p o s n spe-me grph form 8.5 merl Lef nd e Rgh solons of Emple 5 p o s n spe-me grph form 8 5. merl lef nd e rgh solon of Emple for p o me merl lef nd e rgh solon of Emple for p o me merl lef nd e rgh solon of Emple for. 5 p o me merl Lef nd E Rgh solons of Emple wh me sep lengh Conors plos of Emple wh me sep lengh... respevel

11 6. merl Lef nd E Rgh solons of Emple wh me sep lengh Conors plos of Emple wh me sep lengh... respevel 6.5 merl Lef nd E Rgh solons of Emple wh me sep lengh Conors plos of Emple wh me sep lengh.... respevel.

12 Ls of Tles. Commor Tle.. Adon Tle. Comprson of errors of Emple dfferen me. 79. Comprson of errors of Emple dfferen me. 79. Comprson of errors of Emple dfferen me 8. Comprson of errors of Emple dfferen me wh Comprson of errors of Emple 5 me. wh L RMS nd L errors of Emple dfferen me nd onsns L RMS nd L errors of Emple dfferen me nd onsns L RMS nd L errors of Emple dfferen me nd onsns M Asole RMS Relve error nd CPU me of emple dfferen mes for M 9 6. Comprson of RMS errors of emple dfferen nmer of grd pons 9 6. M Asole RMS Relve errors nd CPU me of emple dfferen mes for M. 6. Comprson of M Asole errors of emple dfferen nmer of grd pons T M Asole RMS Relve errors nd CPU me of emple dfferen mes for M 6.6 Comprson of M Asole error of emple dfferen nmer of grd pons T. 5.

13 Conens Cerfe Delron Aknowledgemen Asr Ls of Reserh Ppers Ls of Fgres Ls of Tles v v v Inrodon. Inrodon. Prelmnres.. Defnon.. Defnon: One prmeer Le Grops.. Infnesml form of Le Grop 5.. Le seres Grop Operor nd Infnesml Invrne Condons for Fnon 5..5 Le Alger 6..6 Defnon 7..7 Clssfon of Sgrop nd Grop Invrn Solons 7..8 Invrne for Ssem of PDEs 8..9 Defnon 8.. Deermnng Eqons 9.. Defnon

14 .. Mehodolog... Le s Algorhm... The G / G - Epnson Mehod. merl Solons of Prl Dfferenl eqons.. Weghng Coeffens for he Frs Order Dervve 5.. Bellmn s Approh 5.. Qn nd Chng s Approh 7.. Sh s generl pproh 7..5 Weghng Coeffens for he seond order dervve 9..6 Qn nd Chng s Approh 9..7 Sh s Generl Approh..8 Sh s Rerrene Formlon for hgher order dervves..9 Weghng Coeffens for ml-dmensonl se..9. Cosne Epnson sed Dfferenl Qdrre Mehod E Solons of Fml of onlner Evolon Eqons. Inrodon 9. Le Smmer Anlss. Redons nd E Solons of Fml of onlner Eqons.. Slger V.. Slger V V.. Slger V. Conlson 7 Smmer Redon nd Solons of Shllow Wer nd KdV-Ssem of Eqons. Inrodon 9

15 . Le Smmer Anlss for Two-omponen Generlzon of Shllow Wer 5. E Solons of Shllow Wer Wve Eqon 5.. Slger V V 5.. Slger V V 55.. Slger V 58. Le Smmer Anlss Three-opled KdV Eqons Engendered he emnn Ssem 58.5 E Solons of Three-opled KdV Eqons 6.5. Slger V 6.5. Slger V V V 6.5. Slger V V Slger V 6.6 Conlson 65 merl Solon of Klen-Gordon Eqon. Inrodon 67. Klen-Gordon Eqon 69. Polnoml Dfferenl Qdrre Mehod 7. merl Sheme for Klen-Gordon Eqon 7.. Tme dsrezon nd Qslnerzon Proess 7.. Spl Dsrezon 7.. Implemenon of Bondr Condons 7.5 merl Epermens 7.6 Conlson 8 v 5

16 5 Comprve Solon Sd of Fsher s Tpe Eqons 5. Inrodon Le Clssl Anlss for onlner Dffson Fsher s Tpe Eqons Redons one-dmensonl S lgers 9 5. Comprve Sd of E nd merl Solons of Fsher s Tpe 9 Eqons 5.. merl Epermens nd Dssson 9 5. Conldng Remrks 98 6 Two Dmensonl Hperol Eqons wh Vrle Coeffens 6. Inrodon Cosne Epnson-Bsed Dfferenl Qdrre Mehod 6. merl Sheme for Two Dmensonl Hperol Eqons wh Vrle Coeffens 6.. Implemenon of Drhle Bondr Condons 6.. Implemenon of emnn Bondr Condons 6. merl Epermens nd Dssson Conlson Blogrph 5 v 6

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18 Chper Inrodon. Inrodon Prolems of phsl neres re ofen rnsled n erms of dfferenl eqons h m rn o o e lner or nonlner ordnr or prl. Prl Dfferenl eqons PDE pl n mporn role n mn rnhes of mheml phss where hese eqons provde mheml desrpon of mn nrl phenomen. PDEs re he s felds n he ppled nlss re. In ppled mhems he sdes of phss fld dnms e. nvolve mn fnons whh dono depend pon sngle ndependen vrle. To sd he hnges n hese fnons prl dervve wh respe o more hn one ndependen vrle re reqred nd hs son fnll leds o prl dfferenl eqon n he onsdered feld. Drng he ls hlf of nneeenh enr lrge nmer of mhemns eme ve n he nvesgon of nmeros phsl prolems modelled prl dfferenl eqons n order o see he phsl ehvors of he prolems. The solons of prl dfferenl eqons n e fond n nll [ ] nd nmerl [ ] form sng nll nd nmerl mehods. Anl nd nmerl solons of prl dfferenl eqons pl n mporn role n he proper ndersndng of vros feres of mn phenomen I s well known o s h mos of he Ordnr Dfferenl Eqons ODEs re eser o solve s ompred o Prl Dfferenl Eqons PDEs. Ths n mn pplons s desrle o rede PDEs o ODE. The reded ssem eomes es o nlse n erms of oh nlll nd nmerll. 8

19 In he hess we hve sed oh nll nd nmerl mehods n order o fnd e nd nmerl solons. Dfferenl Qdrre Mehod DQM s sed for fndng nmerl solons nd Le lssl mehod for e solons. In rren me mh enon s gven for onng e nd nmerl solons of nonlner PDEs. B sng Smmer grop ehnqes we n fnd on e solons of PDEs. In lol lngge smmer s rnsformon of n oe levng he oe nvrn. Le [-] developed he heor of Le smmer grops of dfferenl eqons n he perod of Le s lssl heor of smmeres of dfferenl eqon s n nsprng sore for lnerzng onservon lws for onng epl solons. The erl epressons of Le heor re fond n ooks whh re omposed Sophs Le wh hs ssoe from 888 o 896. Geome s he orgns of Le heor. Le s semnl de ws o look he on nfnesmll. The heor gve relevn resls n onree ppled prolems wh he work of I. Sedov [97] nd G. Brkhoff [7] on dmensonl nlss Frher ws he Rssn shool wh L. V. Ovsnnkov [99 ] h sne 96 egn o eplo ssemll he mehods of smmer nlss of dfferenl eqons n he epl onsron of solons of n sor of prolems even ompled of mheml phss. For fndng e solons of nonlner prolems Le lssl mehod s he mos mporn pplle mehod. Le s heor s powerfl ool o he developmen of ssem proedres nd o he deermnon of nvrn solons of nl nd ondr vle prolems. I s dffl o hndle hndreds of eqons o fnd sngle solon. Tod we hve ess o powerfl Comper Alger Ssems lke Mple nd Mhem. Some reen onrons re from [ 7 ] Bsws e l. [6 7 76] Gp e l. [ ] Hll [8 85] Olver [65 66] nd Clrkson e l. [8 5]. The s oe medng eween Le grops nd Le lgers s he one-prmeer grop. There es mn oher mehods whh re no dependen pon Le grop heor sh s Inverse serng rnsformon [] dre mehod [55 ] Bklnd rnsformon [] nd Pnleve nlss [89]. Presenl vre of powerfl mehods sh s Hro mehod [] Homogenos lne mehod [59] F-epnson mehod [87] Homoop 9

20 perron mehod [5] nh-seh mehod [] G modfed -epnson mehod [] e re developed. G G -epnson mehod [99] nd G. Prelmnres A ref smmr of Le grop hs een provded n hs seon... Defnon A se of elemens G s sd o e grop wh omposon f ssfng he followng oms: Closre Proper: If p nd q re elemens of G hen f s n elemen of G. Assove Proper: If p q nd r re elemens of G : f p f q r f f p q r... Iden elemen: For n elemen p of G nqe den elemen e of G ess: f p e f e p p... v Inverse elemen: If p s elemen of G nqe nverse elemen p n G lso ess sh h f p p f p p e..... Defnon: One prmeer Le Grops Le he veors... n le n some onnos open se D on he n -dmensonl Elden mnfold R n. Defne he rnsformon * T :{ Y }...

21 The fnon Y s dfferenl nfnel wh respe o nd n nll fnon of rel onnos prmeer whh le on open nervl S. The rnsformon T s Oneprmeer Le grop wh respe o he nr operon of omposon f nd onl f There s n den elemen sh h T { Y ; }...5 : For ever vle of here s n nverse nv sh h T nv :{ Y * ; nv }...6 The nr operon of omposon prodes rnsformon.e. memer of he grop. T T T.e. he grop s losed. Consder wo memers of grop nd ** * T { Y ; }..7 : * T { Y ; }..8 : If we ompose T T we ge ** T { Y ; }...9 : where f. The fnon f defnng he lw of omposon of S T s n nl fnon of eln. S nd S s ommve.e. f f ; hs le grop s vthe grop s ssove.e. T. T. T T.. T. T...

22 .. Infnesml form of Le Grop Consder one-prmeer Le grop of he form * T :{ Y ; }.. If we epnd.. o n erms of Tlor s seres we ge * Y ; Y ;..... Y ; o. The dervves of Y ; wh respe o grop prmeer evled re lled he Y ; nfnesmls of he grop nd re denoed where... Le seres Grop Operor nd Infnesml Invrne Condons for Fnon eessr nd sffen onddon for fnon s o e nvrn nder Le grop of rnsformon...if *.. or eqvlenl ssfes he ondon The operor n.. Y n Y...5

23 s lled he grop operor or nfnesml generor nd Y s lled Le dervve of. Proeedng o hgher order dervves we n wre * s Tlor seres rond : n * n n Y Y... Y o..6 n! If he seres onverges s ermed s Le seres nd * n n Y..7 n! n Ths n e wren s * ep Le Alger For he Le grop of rnsformon wh nfnesml generors VV he ommor Le rke ofv V s he frs order operor defned [ V VV V V..9 V ] The ommor hs followng properes:. Blner: pv qv V ] p[ V V ] q[ V ].. [ V [ V pv qv ] p[ V V ] q[ V V ]... where p q R.. Skew smmer: V V ] [ V ]... [ V

24 . Jo Iden: [ V [ V V ]] [ V [ V V ]] [ V [ V V ]]... An veor spe ssfng ove hree properes s lled Le Alger...6 Defnon Le G e Le grop wh Le lger L. If veory L he don veor Ad Y Z L s Ad Y / Z [ Z Y] [ Y Z].. The Le grop n e onsred smmng he Le seres Ad ep Y Z n n dy Z n n! Z [ Y Z ] [ Y[ Y Z ]] Clssfon of Sgrop nd Grop Invrn Solons Le G e Le grop. An opml ssem of s-prmeer sgrops s ls of eqvlen s- prmeer sgrops wh he proper h n oher sgrop s onge o presel one sgrop n he ls. Smlrl ls of s-prmeer s lgers forms n opml ssem f ever s-prmeer s lger of he Le lger L s eqvlen o nqe memer of he ls nder some elemen of he d on represenon. The prolem of fndng n opml ssem of sgrops s eqvlen o h of fndng n opml ssem of s lgers. For one dmensonl s lgers hs Clssfon prolem s essenll he sme s he prolem of lssfng he ors of he don represenon sne eh one-dmensonl s lger s deermned nonzero veor n L.

25 The lssfon done ssoed Le s lgers wh respe o he d on represenon [68 9] nd o ompe he d on represenon Le seres s sed Ad ep Y Z Z [ Y Z ] [ Y[ Y Z ]] Le V V... Vr e he ss of Le lger hen we sr wh generl nfnesmls V PV P V... P r V r...7 nd smplf s mh s possle mens of don ons. We wll fnd he ls of n eqvlen one-dmensonl s lgers. On sng he neqvlen one-dmensonl s lgers of he mml Le nvrne lger grop nvrn redons n e esl rred o...8 Invrne for Ssem of PDEs Le ssem of PDEs wh m dependen vrles v v... v m nd n ndependen vrles... gven F n l v v... v Defnon The one prmeer Le grop of pon rnsformons Y v; v V v; l If nd onl f s lh eenson leves nvrn he srfe n v v... v spe hen s pon smmer dmed..8 5

26 6 Theroem. Infnesml reron. Le v v v Y.. he eqon.. e nfnesml generor of he Le grop of pon rnsformons Consder l k v l l v v v v v v v v v v Y e nfnesml generor of.. lh-eended where v D D.. nd v D D.. v...m nd n.. for k.. n erms of... v v v v.... v v v v m Then he one-prmeer Le grop of pon rnsformons s dmed he ssem of PDEs ff... v v v v F Y k l.. when v ssfes..8 for eh Deermnng Eqons Le ssem..8 of PDEs v v v v f v l l..5 In erms of some spef l h-order prl dervve of v for some m... where.. v v v v f l s ndependen epll v k From heorem. he pon smmer dms ssem of PDEs..5 v v v Y..6 wh he lh eenson of..6 gven..6 f nd onl f v f v f f f l v k k l l

27 wh l v... k f v v v... v we n esl see h s s polnoml n he omponens of v v... v. The s... s oeffens re lner nd homogeneos n he omponens of v v. The order of dervve s s. Frs we elmne he omponens v... l nd her dfferenl onseqenes l from..7. Comopnen v nd he remnng omponens of v v... v h re n he smmer deermnng eqons..7 re ndependen vrles. Sne he reslng epresson for..7 holds for n vles of hese ndependen vrles one ons ssem of lner homogeneos PDEs for nd h onses se of deermnng eqons for he nfnesml generors X dmed he gven ssem of PDEs..8. In prlr f eh f v v l v... v... s Polnoml n he omponens of l v v... v hen he ssem of smmer deermnng eqons..7 elds polnoml l eqons n he ndependen omponens of v v... v. The oeffens of hese polnoml eqons ms vnsh seprel. Ths elds he se of lner deermnng eqons for nd. Tpll he nmers of deermnng eqons re fr greer hn n m... Defnon An nvrn solon reslng from dmed pon smmer wh nfnesml generor..7 s v wh omponens v... mf nd onl f: v s n nvrn srfe of.. for eh... m v solves..8. ssfes: v s n nvrn of he ssem of PDEs..8 f nd onl f v Y v when v... m;.e. n... m ;..9 F l v v... v whenv....e. F l

28 where denoes... n for... l... Eqon..9 s he nvrn srfe ondons for he nvrn solons of he ssem of PDEs... Invrn solons n e deermned he followng proedre:. Invrn form Mehod Solve epll orrespondng hrerss eqons for v gven d v m d dn dv dv dv..... m v v v v v n m If z v z v... zn v v... v re n m fnonll ndependen onsns h rse from solvng he ssem of n m frs order ODEs m..... wh he on m v v... v ssem of ODEs.. s gven mpll hen he generl solon v of he n v v z v z v... z.. where s n rrr dfferenle fnon of z z... zn for... m. m oe h... n v v... v re n m fnonll ndependen grop nvrns of... Le z n v e n m h nonl oordne ssfng Yz.. n If we rnsform he ssem of PDEs..8 whh onns ndependen vrles z... zn nd dependen vrles m... hen he rnsformed ssem of PDEs dms he oneprmeer Le grop of rnsformons. z z... n z n z n..... m. 8

29 The vrle z n does no pper epll n he rnsformed ssem of ODEs nd he solons of rnsformed ssem of ODEs n he form... The solons of PDEs..8 onn nvrn solons he nvrn form.. gven mpll. The solons re fond solvng reded ssem of dfferenl eqons wh n ndependen vrles... z z n nd m dependen vrles... m. The vrles z... z n re ommonl lled smlr vrles. We n fnd he reded ssem of dfferenl eqons ssng he nvrn form.. no he gven ssem of PDEs Mehodolog In presen work we re delng wh he mehod of grop nvrn ehnqes whh re sed on he heor of grop rnsformon known s Le grops whh s on he spe G of dependen nd ndependen vrles of he ssem. We lso se Dre mehods: - G epnson mehod on some nonlner prl dfferenl eqons n order o fnd e solons. For nmerl solons we mnl se Dfferenl Qdrre Mehod DQM. We now provde he ref olnes of he mehods menoned ove. Bese of smpl he grop heore ehnqes re sed o solve mn eqons. One rrves n over deermned lner ssem of dfferenl eqons for he grop nfnesmls when ssem of PDEs s se o nvrne nder one-prmeer Le grop of rnsformon. To on he redon of he ssem he nfnesmls of he rnsformon helps s.... Le s Algorhm The sepwse proedre of Le s mehod s s follows: Le ssem of PDEs wh m v v... v m nd n.. dependen vrles nd ndependen vrles respevel F v v n l v... v Consder he one-prmeer Le grop of pon rnsformons..6 s nvrn. Appl he prolonged operor..5 nd reqre h l Y gven.. o eh eqon of he ssem 9

30 Y l F F The ondon..6 mens h l Y dsperse on he solon se of he orgnll..5.. As defned n seon.. ssem of lner PDEs h omprse he se of deermnng eqons for he nfnesml generor Y dmed he gven ssem of PDEs..5 s oned.. Solvng deermnng eqons he solons wll led o he epl form of nd. 5. A new ndependen vrle n erms of n s oned solvng he hrerss eqons To ge he reded form of he ssem rework he ssem The G / G - Epnson Mehod In 8 Wng e l. [] nroded mehod for fndng rvellng wve solons lled G / G - epnson mehod. Aordng o hs mehod frsl we sppose h nonlner eqon s no wo ndependen vrles nd s gven p v v v v v v where p s polnoml n v v nd v v re nknown fnon nd s prl dervves n whh hghes order dervves nd he nonlner erms re nvolved. The mn seps of hs mehod re gven elow Sep Le s sppose h v v. The rvellng wve vrle llows s redng o n ODE for v v p v v v he prme denoes he dervve wh respe o. Sep Aordng o hs mehod we sppose h solon of eqon s n polnoml n G / G s follows: G v d...5 G

31 where d re rel onsns wh d o e deermned s posve neger o e deermned. The fnon G s he solon of he lr lner ordnr dfferenl eqon G G G...5 where nd re rel onsns o e deermned. Sep In hs sep sse..5 no..9 nd sng seond order lner ordnr dfferenl eqon..5. All he erms wh sme order of G / G ogeher re sepred he eqon..9 s onvered no noher polnoml n G / G. ow eqe eh oeffen of polnoml o zero. Then we ge lger eqons for d... nd. Sep The generl solon of eqon..5 hs een well known for s hen ssng d d nd generl solon of eqon..5 no..5. We hve rvellng wve solons of nonlner prl dfferenl eqon..8.. merl Solons of Prl Dfferenl eqons There re mn PDEs n he lerre whh n e solved nlll. So hese solons n e ppromed nmerll. In he lerre here re mn nmerl ehnqes o fnd he nmerl solons of prl dfferenl eqons mong hem re fne dfferene mehod FD [ 9] fne volme mehod FV [ ] fne elemen mehod FE [] e. In he nneeen enr lmos ll he engneerng prolems were solved low order mehods sh s fne dfferene mehod fne volme mehod nd fne elemen mehod sng lrge nmer of grd pons. B presenl Dfferenl Qdrre mehod DQM s hgher order nmerl ehnqe sed for solvng prl dfferenl eqons. In DQ mehod he dervve of fnon s ppromed weghng sm of fnon vles ll dsree pons. Rhrds Bellmn nd hs ssoes [66] nroded he mehod of Dfferenl Qdrre mehod n he erl 97s. Imporn onrons o hs mehod nd s pplons hve een mde mn Sholrs. To smplf ompng effors of he evlon of DQ weghng oeffens for hgher order dervves Mngle [9] proposed lner rnsformons n he DQ mehod. Cvn e l [68] eended hs mehod o mldmensonl prolems nd negro-dfferenl eqons. The epl formls o ompe

32 rel nd effenl he DQ weghng oeffens for he frs nd seond order dervve ws gven Qn e l. [97]. Chng e l. [] emploed he proper fnons s s fnon nsed of polnoml fnon n he DQ mehod for delng wh prolems nvolvng seep grdens sessfll. Reenl he DQ mehod s mosl sed o nlze defleons vron nd klng of lner nd nonlner srrl omponens Ber e l. [] ng e l. [] srz e l. [7] Kkre e l. [] nd Mlk [7]. DQM hs een effenl emploed n vre of prolems sh s engneerng nd phsl senes. Some s defnons nd fndmenls for weghng oeffens of Dfferenl Qdrre mehod re gven n hs seon... Weghng Coeffens for he Frs Order Dervve Consder one dmensonl prolem over losed nervlp q nd here re grd pons wh oordnes s p... q. I s ssmed h fnon f s sffenl smooh over he nervl p q so h s frs order dervve f n grd pon n e ppromed he followng formlon. f p f..... where f represens he fnonl vle grd pon f ndes he frs order dervve of f nd p s he weghng oeffens of he frs order dervve. The ke proedre s deermnon of weghng oeffens... Bellmn s Approh Bellmn e l. [66] proposed wo pprohes o ompe he weghng oeffen p n eqon... Boh pprohes sed on wo dfferen es fnons. Bellmn s frs pproh: In hs pproh followng es fnon s hosen: l gl l......

33 eqon.. gves es fnons. In eqon.. nd re ken o. The ol nmer of weghng oeffens s. To on hese weghng oeffens he es fnons shold e ppled grd pons.... For p he followng lger eqons re oned: p p. p. k l. l l... for Ssem of eqons.. hs nqe solon ese s mr s of vndermonde form. Unfornel when s lrge he mr s ll-ondoned nd s nverson s dffl. Usll s hosen less hn. Bellmn s seond pproh: Bellmn s seond pproh s smlr o frs pproh ses dfferen es fnons L gl l L l where L s he Legendre Polnoml of degree nd L s he frs order dervve of L. B hoosng l o e he roo of he shfed Legendre Polnoml nd pplng eqon.. grd pons.... Bellmn e l. [99] oned smple lger formlon o ompe p. p L. L for..5 p for...5 Usng eqon..5 he ompon of weghng oeffen s smple. However hs pproh s no s flele s he frs pproh ese he oordnes of he grd pons n hs pproh nno e hosen rrr. The shold e hosen s he roo of he Legendre

34 Polnoml of degree. So eqon..5 s spel se. In prl pplons frs pproh s sll doped... Qn nd Chng s Approh In ompng he weghng oeffens mn emps hve een mde o mprove Bellmn s pproh. One of he mos sefl pproh ws nroded Qn nd Chng [9 97]. Qn nd Chng sed followng Lgrnge s nerpolon Polnoml s he es fnon. M gl l.... M..6 l l where M M..8 l l l Afer pplng eqons..6 o grd pons he oned he followng lger formlons o ompe he weghng oeffen p p l l l ll l l l for..9 p l l l There s no resron on he hoe of grd pons n he eqon Sh s generl pproh Sh s generl pproh ws nspred from Bellmn s pproh. I overs ll he weghng oeffens ll pprohes re sme de o polnoml ppromon nd lner veor spe. Sh ses he followng ses of he se polnomls: M gl l... M.. l l l gl l.....

35 where M nd M re he sme s n eqon..7 nd..8 he ompon of weghng oeffens s dfferen. For smpl we se M l. l l..... wh M. where s he kroneker operor. Usng eqon.. eqon.. n e smplfed o l gl l... M.. l Ssng eqon.. no.. gves p.. M In eqon.. M n e esl omped from he eqon..8. To evle we sessvel dfferene eqon.. wh respe o nd on he followng rerrene formlon M m m m. m... l l l for m... ; l... where M m m nd nde he mh order dervve of M nd. l From eqon.. we n on he epresson for s l M for..5 M..5 Ssng eqon..5 no eqon.. we on p M M for..6 M..6. M 5

36 If s gven s es o ompe M from eqon..6 nd hene p for. The llon of p s no es o lle. Ths dffl n e elmned sng he seond se of se polnoml..8. Aordng o lner veor spe proper f one se of se polnomls ssfes lner operor s eqon.. so does noher se of se polnomls. The ssem of eqons for deermnon of p derved from he Lgrnge nerpolon polnoml eqon.. shold e eqvlen o h derved from noher se of se polnomls l l... eqons... Ths p ssfes he followng eqons whh re oned se polnoml l where l p or p p..7 eqon..6 nd..7 re wo formlons o ompe he weghng oeffens...5 Weghng Coeffens for he seond order dervve For seond order dervve we nroded smlr ppromon form gven f q f where f ndes he seond order dervve of f nd q s he weghng oeffen of he seond order dervve. Ovosl eqon..8 s lner operor. The eqon..8 s sme s eqon... The dfferene eween hese eqons s he se of dfferen weghng oeffens. Two pprohes n e sed o deermne q h re followng:..6 Qn nd Chng s Approh In hs pproh Qn nd Chng [9 97] sed he Lgrnge Inerpolon polnomls s he es fnons nd hen derved 6

37 7 for q k k k l l l l..9 l l k l k k k q Sh s Generl Approh Sh s [6] generl pproh s lso sed on polnoml ppromon nd lner veor spe nlses. Two ses of se polnomls.. re sed ssng eqons.. no eqons..8 gves M q.. from eqon.. we on for M.. M.. Ssng eqons.. no eqon.. elds for M M q... M M q.. fnll ssng eqon..6 no eqon.. we on for p p q.. when q n e esl omped from eqon... However from eqon.. n e seen h ompon of q nvolves he hrd order dervve M whh nno e omped esl. Smlrl o he nlss for he se of frs order

38 dervve he eqon ssem for q derved from he Lgrnge Inerpolon polnomls.. shold e eqvlen o h derved from noher se of se polnomls l l Ths q ssfes he followng eqon whh s oned se polnoml l when l q or q q for he pplon of Sh s generl pproh..5 q s frsl omped from eqon.. when nd q s omped from eqon Sh s Rerrene Formlon for hgher order dervves For he dsrezon of hgher order dervves he followng wo lner operors re ppled. f m m w f..6 f m w m f..7 For... ; m... m m where f f ndes he h h m nd m order dervves of f wh respe o. m m m w w re he weghng oeffen reled o f nd f m. Two ses of se polnomls wll lso e sed o derve epl formlon for m w. The frs se of se polnomls s gven eqons... Ssng eqons.. no eqons..6 nd..7 gves w m m..8 M m m w..9 M B rewren eqon..8 we on 8

39 9. m m M w.. eqon.. s vld for n nd on he oher hnd from he rerrene formlon.. we hve m M m m.. for m M m m m.. m M m m.. Ssng eqon.. no eqon.. leds o for m m m m.. eqon.. n e frher smplfed sng eqon.. for M w M w m m m m..5 Ssng eqon..5 no eqon..9 sng eqon..6 rerrene formlon s oned s follows:... ;... m for w w p m w m m..6 where p s he weghng oeffen of he frs order dervve desred ove. The formlon for m w n e oned ssng eqon.. no eqon..8 whh gves

40 w m m M m. M for... ; m Is w o ompe he weghng oeffens m w for from..6 s ver dffl o ppl eqon..7 o ompe he weghng oeffens w m. Ths dffl n e overome he properes of lner veor spe. In erms of he nlss of he dmensonl lner veor spe he eqon ssem for m w derved from he Lgrnge s nerpolon polnomls shold e eqvlen o h derved from he se polnomls l l.... Ths m w shold ssf he followng eqon oned from se polnoml l when l m m w or w w m..8 From hs formlon w m..9 Weghng Coeffens for ml-dmensonl se In generl mos of he prolems re reled o wo or hree dmensonl ses. In hs seon we wll epln Cosne epnson sed dfferenl Qdrre mehod for wo dmensonl ses...9. Cosne Epnson sed Dfferenl Qdrre Mehod If fnon v z s sffenl smooh over he whole domn he frs nd seond order prl dervves of he fnon v z wh respe o nd z grd pon z re ppromed lner sm of ll fnonl vles n he whole domn h s v v z w v z..9 l l l l l z w v z..9 l

41 v v z zz M l l z w v z.. M l l l z w v z for M... l where v z v z nd v z vzz z re he frs nd seond order prl dervves of v z wh respe o nd z grd pon z nd w l w l w l w l re orrespondng weghng oeffens. For fndng he weghng oeffens Sh [7] proposed he mos generl pproh sng Lgrnge s nerpolon s se fnons. Reenl he mos freqenl sed dfferenl qdrre proedres o solve one nd wo dmensonl dfferenl eqons re Lgrnge nerpolon polnomls sed dfferenl qdrre mehod PDQM Cosne epnson sed dfferenl qdrre mehod CDQ nd oher mehods [ ]. The weghng oeffens n eqon n e deermned sng se se fnons. For CDQM he followng se fnons re sed o on weghng oeffens [7] d l os l l..... d dl l... os os.. l l where d el z l z os z os z l l d os os os os.. l k l k os os z os z os z..5 k l l l l l

42 In f he es fnon gven n eqon.. s Lgrnge nerpoled osne fnon nd oned sng he rnsformons w os nd w l os l nerpolon polnomls. o he Lgrnge L w rl w k... w w L w..6 where l l L w w w w w... w w..7 L w w w..8 k l l when he prolem domn s hosen s [ ] for ndependen vrle. Usng he se of se fnons gven n eqon.. nd.. he weghng oeffens of he frs order dervve re fond s w sn... os os for..9 w w w z sn z os z os z z... M for..5 w w... M..5 Smlrl he weghng oeffens for he seond order prl dervves re gven he forml [7] w sn w w o... os os for..5 w w.....5

43 w w w sn z o z... M os z os z for..55 w w... M...56 For he weghng oeffens of fnon defned on he nervl [ ] [ d] rnsformon n e presred s [7] p z d..57 q p e d To mp he nervl [ p q] [ d e] n nd z domn o he nervl [ ] [ ] n he nd domn. Bsll here re wo ws o solve he prolems defned on he nervl [ p q] [ d e]. One w s o rnsform he ll he dervves wh respe o nd. Then he dsrezon s mde n he rnsformed domn. For hs se he ove weghng oeffens..9 o..56 n e ppled drel. Anoher w s o mke dsrezon drel n he n he nd z domns for hs se he weghng oeffens..9 o..57 shold e modfed ordngl sng he rnsformon..58 we hve v v v q p v v v z z e d v q p v v z e d v..6 Then sng eqons..59 nd..6 he epl formlons gven for he weghng oeffens n he nd z domn n e modfed s w sn... q pos os for..6

44 w sn... M q pos os for..6 w w w sn o q pos os q p... for..6 w w w sn o q pos os e d... M for..6 In he llon of he weghng oeffens w nd w sng eqon he vles of he fnon o s ndefned he prolem ondres nd

45 5

46 Chper E Solons of Fml of onlner Evolon Eqons. Inrodon In mheml phss he fml of eqons.. hs een se of eensve sd. I represens lss of nonlner evolon eqons. The eqon.. represens verson of Koreweg-de Vres KdV eqon [7] Cmss-Holm eqon [6] he Hner-Son eqon [8] nd Invsd Brgers eqon []. In mhems he KdV s mheml model of wves on shllow wer wve s srfe. More hn hlf enr elpsed sne So Ressell's dsover efore wo Dh senss Koreweg nd de Vres [7] derved n 895 he eqon h governs he propgon of solr wve n shllow hnnel. Sh e l. [7] proved h he modfed KdV eqon dms nor fml of smll mplde qso-perod solon. Dkh e l. [5] emne he energ rnsfer mehnsm drng he nonlner sge of he modlon nsl n he modfed KdV eqon. The prlr of hs sd onsss n onsderng he prolem essenll n he Forer spe. Trk e l. [77] sd wo fmles of ffh order KdV eqon wh me-dependen oeffens nd lner dmpng erm. These models ppl o he hrerzon of envelope wve dnms n nhomogeneos ssems modelled KdV-pe eqon. In order o sd shllow wer wves Cmss nd Holm [6] derved prl dfferenl eqon whh s now lled he Cmss- Holm eqon nd s onsdered one of he mos fsnng PDEs n mheml phss. In fld dnm he Cmss-Holm eqon s negrle dmensonless nd nonlner prl dfferenl eqon. Rehmn e 6

47 l. [8] ppl wo reen nll pprohes o nvesge he possle lsses of rvellng wve solons of some memers of reenl n negrle fml of generlzed Cmss-Holm eqon. L e l. [7] onsder he nmerl ppromon of he Cmss-Holm eqon whh sppors peken solon. The develop nd es hgh order enrl dsonnos Glerkn-fne elemen mehod for solvng he eqon. J e l. [9] sed smpo denses o sd n smpo proper of he dsperson Cmss- Holm eqon. The Hner-Son eqon desres he propgon of wves n mssve dreor feld of nem lqd rsl wh orenon of moleles desred he feld of n veors. Hner nd Son nvesged he se when vsos dmpng s gnored nd kne energ erm s nlded n he model. As s well known he Brgers eqon n e oned s lmng se of ver-sokes eqon nd serves rnspor prolems. For he nvsd se he Brgers eqon redes o where s prle velo. Smmer grop ehnqes provde one mehod for onng e solons of PDEs. The heor of Le smmer grops of dfferenl eqons ws developed Le. Severl pplons of he Le grops n he heor of dfferenl eqons were dsssed n lerre; mos mporn ones re he orgnl eqon n e reded o n eqon wh fewer ndependen vrles redon of order of ordnr dfferenl eqons onsron of nvrn solons solvng he hrers eqon mppng solons o oher solons nd deeon of lner zng rnsformons for mn oher pplons of Le Smmeres see [ ]. In hs hper sng he Le pon smmer mehod we hve nvesged eqon... We fond he prelmnr lssfon of grop-nvrn solons lookng he represenon of he oned smmer grop on s Le lger nd hen we reded he eqon. o ordnr dfferenl eqon ODEs. Then we ppl G / G - epnson mehod on one ODE nd ge rvellng wve solons. The work s orgnzed s follows. In Seon. we derved he generl form of n Infnesml generor dmed.. nd n opml ssem of one-dmensonl smmer grop of.. s onsred. Seon. s devoed o he orrespondng one- 7

48 prmeer redons nd grop-nvrn solons nd G / G -epnson mehod hs een ppled on one reded ODE. Fnll some onlsons nd dsssons re gven n Seon... Le Smmer Anlss In rel-lfe pplons models re epressed n erms of PDEs s opposed o ODEs whh re relvel eser o solve. Fornel smmeres n e sed o onsr nvrn solons.e. solons h re nhnged nder he on of sgrop of smmer grop whh n rn llows he redon of PDEs o ODEs. An eqvlene rnsformon of ssem s hnge of oh dependen nd ndependen vrles no new dependen nd ndependen vrles kng he orgnl ssem no ssem of he sme form. In hs seon we wll ppl Le's mehod of nfnesml rnsformon grops on fml of eqons. We le nd re nfnesmls orrespondng o nd respevel nd mpose he ondon of nvrne on... The nvrne nder nfnesml rnsformons mens h f s solon of eqon.. nd hen * s lso solon of. The followng relon from he oeffens of he frs order s derved on nvokng he nvrne reron: O O.. O. where re nd eended prolonged nfnesmls orrespondng o nd respevel. For he nfnesmls nd he deermnng eqon s se whh s oned from nvrne ondon fer eqng he oeffens of vros dervve erms o zero s s follows 8

49 The se of eqons.. helps s o orrespondes he nfnesmls.. nd s follows:... where nd re rrr onsns. The Le lger ssoed wh eqon.. onsss of he followng hree veor felds V V V... The smlr vrle nd form n e oned solvng he hrers eqons d d d..5 The generl solon of hese eqons nvolves wo onsns; one eomes he new ndependen vrle nd he oher ss F pls he role of new dependen vrle. On ssng hese solons of..5 n eqon.. one ges he reded ordnr dfferenl eqon. As menoned n Olver [65] he ommor Tle. nd he d on Tle. for ove Le lger n e esl onsred s follows 9

50 . COMMUTATOR TABLE V V V V V V V V V V. ADJOIT TABLE V V V V V V V V V V V - V V V V V V V e In generl o eh s -prmeer sgrop of he fll smmer grop here wll orrespond fml of grop-nvrn solons. Sne here re lmos lws n nfne nmer of sh sgrops s sll no fesle o ls ll possle grop-nvrn solons o he ssem. Th needs n effeve ssem mens of lssfng hese solons ledng o n opml ssem of grop-nvrn solons from whh ever oher solon n e derved. Ao he opml ssem lo of eellen work hs een done epers [57 99 nd 7] nd some emples of opml ssem n lso e fond n []. Up o now severl mehods hve een developed o onsr he opml ssem. The don represenon of he Le grop on s Le lgers ws lso known o Le. Is se n lssfng grop-nvrn solons ppered n [] nd [9] whh re wren Ovsnnkov [99] nd Olver [57] respevel. The ler referene onns more dels on how o perform he lssfon of sgrop nder he don on. Here we hve sed Olver s mehod whh onl depends pon frgmens of he heor of Le lgers o onsr he opml ssem of eqon... An opml ssem omprses of hree veor felds vz. V V V V. ow or prmr fos on he redons ssoed wh hese veors felds nd emps o fnd some e solons. 5

51 . Redons nd E Solons of Fml of onlner Eqons In hs seon we wll derve some e solons of fml of eqons. One w o on e solons of.. s redng o ordnr dfferenl eqons. Ths n e heved wh he se of Le pon smmeres dmed... I s well known h he redon of prl dfferenl eqon wh respe o r-dmensonl solvle s lger of s Le smmer lger leds o redng he nmer of ndependen vrles r... Slger V The reded ODEs n hs se re F..6 The ovos solon of.. s.. Slger V V onsn..7 On sng he hrers eqons he orrespondng vrle nd he form of he orrespondng solon re s follows..8 F..9 On sng hese n eqon.. he reded ODE s gven F FF F FF FF F.. In hs seon we seek solon of eqon.. G/ G -epnson mehod s desred n seon... The reded ODE s F FF F FF FF F.. The solons of eqon.. n e epressed polnoml n G/ G s follows: 5

52 5 G G F.. where s posve neger o e deermned. The fnon G s he solon of lr lner ordnr dfferenl eqon G G G.. where nd re rel onsns o e deermned. ow lnng F wh F F gves. Therefore we n wre solons of eqon.. n he followng form: G G G G F.. Ssng eqon.. no eqon.. nd sng.. olleng ll erms wh he sme order of ogeher he lef hnd sde of eqon.. s onvered no polnoml n...7 / G G seng eh oeffen of hs polnoml o zero we derve se over deermned dfferenl eqon for nd. 7 8 : / G G : / G G : / G G : / G G : / G G..9

53 : / G G : / G G : / G G.. Solvng hese ssems of lger eqons we hve followng resls Cse 8.. Ssng he generl solons of eqon.. no eqon.. we hve hree pes of e solons of eqon... Then we ge he solons of PDE hngng he fnon when hen we ge hperol fnon solons snh osh osh snh snh osh osh snh..

54 5 when hen we ge rgonomer fnon solons sn os os sn sn os os sn..5 when we on ronl fnon solons..6 Grphll represenon of solons hs gven n Fgre.-.6.

55 Fgre.: The dsron of some snglres of he solon for 9 6. Fgre.: The knk wve solon for

56 Fgre.: The dsron of some snglres of he solon for. Fgre.: The dsron of some snglres of he solon for

57 Fgre.5: The Snglr = of solon for Fgre.6: The perod solon for 7. 57

58 58.. Slger V Usng hrers eqons..5 he smlr vrle nd he form of he smlr solon re s follows: F...7 On sng hese n eqon.. he reded ODE s gven 8 F F F F F FF F..8 Solons orrespondng o hs ODE re: sn 5 F sn 6 F sn sn 6 F.. os F v os 6 F v os os 6 F v.. osh F v osh 6 F v osh osh 6 F..7 e F e F e F.. sn F..

59 sn 6 F v sn sn 6 F v.. snh F v snh 6 F v snh snh 6 F v..6 Afer ssng nd sng F we n ge solons of eqon.. s followng Solon : sn 5..7 Solon : 6 6 sn 6..8 Solon : 7 7 sn sn 6..9 Solon : os v..5 Solon 5: 5 5 os 6 v..5 Solon 6: 6 6 os os 6 v..5

60 6 Solon 7: osh v..5 Solon 8: 5 5 osh 6 v..5 Solon 9: 6 6 osh osh Solon : e..56 Solon : 6 6 e..57 Solon : 5 6 e..58 Solon : sn..59 Solon 5: 5 5 sn 6 v..6 Solon 6: 6 6 sn sn 6 v..6

61 Solon 7: v snh..6 Solon 8: v 6 5 5snh..6 Solon 9: v 6 6 snh 6 snh...6 On hoosng rrr onsns eql o one grphl represenon of some solons of eqon.. re shown n Fgre

62 Fgre.7: The perod solon for. 5 Fgre.8: The perod solon for. 6 6

63 Fgre.9: The knk wve solon v for. Fgre.: The perod solon for. 6 6

64 . Conlson In smmr we nvesge he smmer of he eqon.. mens of lssl Le smmer mehod. The smmer lgers nd grops of.. re oned. Spell he mos generl one-prmeer grop of smmeres s gven nd mos generl solons re gned. e we hve lssfed he one-dmensonl s lgers of he Le lger of... Then he redons nd some solons of eqon.. sng he ssoed veor felds of he oned smmer re gven o. B one-dmensonl s lgers.. s reded o ODEs. We sed G / G -epnson mehod o fnd rvellng wve solons of one ODE. ew epl solons of eqon.. re derved. We hve lso shown grphl represenon of solons n fgres. 6

65 65

66 Chper Smmer Redon nd Solons of Shllow Wer nd KdV-Ssem of Eqons. Inrodon In hs hper we hve sded wo-omponen shllow wer ssem nd hree-opled Koreweg-de Vres KdV eqons engendered he emnn ssem sng Le lssl mehod. The wo-omponen shllow wer ssem s n he form ww w k w k k w.. The shllow wer eqons re se of hperol prl dfferenl eqons h desre he flow elow pressre srfe n fld. The eqons re derved from deph negrng he ver-sokes eqons n he se where he horzonl lengh sle s mh greer hn he verl lengh sle. L e l. [8] onsder he Ch prolem for D vsos shllow wer ssem n Besov spe. Csrodz e l. [] desres he developmen of hgh-order mehods for he shllow wer ssem h preserve ever sonr solon of he ssem no onl he wer--res ones. L e l. [5] sded +-dmensonl dsplemen shllow wer wve ssem for perfe fld. Ln e l. [9] ompre some frs order well-lned nmerl shemes for shllow wer ssem wh spel neres n pplons where here re rp vrons of he opogrph. Bov e l. [] nrodng he ol energ of he wer olmn o move hnge of vrles whh smmerzes he shllow wer onservon ssem nd sremlne pwnd Perov-Glerkn sheme s developed sed on he proposed smmer form. Imn e l. [] sed reonsron of vronl eron mehod for ompng he opled shllow wer wve eqon. Hgen e l. [86] sed Commd-pe grphs rd o smle nonlner wer wves desred ssem lne lws lled he shllow wer ssem nd o solve hs hperol ssem eplo hgh resolon enrl-p wnd shemes s sed. Asnon e l. [6] kle he eleron of he shllow wer smlon n rnglr meshes 66

67 eplong he omned power of severl CUDA-enled GPUs n GPU lser. Lsr e l. [] oned nmerl solons of he shllow wer wve ssem n D domns sng modern grphs proessng ns. Levn e l. [9] presen he dervon of he dsree Eler-Lgrnge eqons for n nverse sperl elemen oen model sed on he shllow wer eqons. The hree-opled KdV eqons engendered he emnn ssem s w vv v v wv v w.. w w ww. 8 KdV s mheml model of wves on shllow wer srfe. The KdV eqon hs nfnel mn negrls of moon whh do no hnge wh me. L e l. [9] sed mehod of dnml ssem o sd he opled KdV ssem nd some epl prmer represenon of he solr wve nd perod wve solons re oned n gven prmeer regon. Y e l. [67] presen he generlzed Kpershmd deformon of he ffh order opled KdV eqon herrh wh self-onssen sores nd s l represenon. Zho e l. [5] nvesged hree-opled KdV eqons orrespondng o he emnn ssem of he forh order egen vle prolem. Adem e l. [9] sed Le smmer nlss on he opled KdV ssem nd onservon lws of he opled KdV ssem re derved sng he mlpler pproh nd onservon heorem. H e l. [79] onsred new poson negon nd ompleon solons of he opled KdV-mKdV ssem from zero seed solon mens of Dro rnsformon. Wzwz [8] sd wo ompleel negrle opled KdV nd KP ssem nd Hro s lner mehod s emploed o formll derve mlple solon solons nd mlple snglr solons for eh ssem. Fn e l. [6] oned severl knds of e solons for ssem of opled KdV eqon sng n mproved Homogenos lne mehod. Zheng e l. [5] sed eended mppng mehod nd lner vrle sepron mehod nd new pes of vrle sepron solons wh 67

68 wo rrr fnons for +-dmensonl Korweg-de Vres ssem s derved. Lee [9] presens he dfferenl operors for he ffh order KdV eqon. The work p n hs hper hs een srred s follows: In seon. Le smmeres re sed on wo-omponen generlzon of shllow wer. In seon. he e solons of wo-omponen generlzon of shllow wer re fond. In seon. nd.5 Le lssl mehod s ppled on hree-opled KdV eqons engendered he emnn ssem nd e solons re oned. In seon.6 onldng remrks re gven.. Le Smmer Anlss for Two-omponen Generlzon of Shllow Wer The pplon of he lssl Le grop mehod reqres onsderng one-prmeer Le grop of nfnesml rnsformons n he vrles w s follows * * * w O X w O T w O U w * w O w W.. whh leves he ssem.. nvrn. The mehod for deermnng he smmer grop of.. onsss of fndng he nfnesmls X T U nd W whh re fnons of nd w. Assmng h ssem.. s nvrn nder he rnsformons.. he nfnesmls X T U nd W ms ssf he smmer ondons U U U U U U U U w w W k wu W k k W U w.. 68

69 where U U U W W W nd re eended prolonged nfnesmls ng on n enlrged spe e spe h nldes ll dervves of he dependen vrles for more dels see [99 7 9]. Ssng vles of U U W W W nd U no smmer ondons.. hen eqng he oeffens of he vros monomls n he frs seond nd he oher order prl dervves of nd w nd her powers we n fnd he deermnng eqons for he smmer grop of he eqons... Solvng hese eqons we ge he followng forms of nfnesmls: X T U.. W w. The Le lger ssoed wh ssem.. onsss of followng hree veor felds V w w V.. V. I s es o verf h V 5 s losed nder he Le rke. The lssfon of one-dmensonl slgers of he whole smmer lger.. s done n ndve pproh [7]. We wll work o frs n opml ssem nd hen emrk pon he vros redons ssoed wh generors n he opml ssem. We egn onsderng generl elemen V V V V of smmer lger nd se o vros don rnsformons o smplf s mh s possle. The don on s gven he Le seres V V V V V V V V Ad ep..5 69

70 where V V V V V V s he ommor for he Le lger nd s prmeer. The pes of one-dmensonl slgers of ssem.. re s follows: V V V V v V...6 where nd re rrr onsns. One he slger of he eqon.. hve een deermned he smmer vrles for he ssoed redon n e fond solvng he hrers eqon X d d d dw...7 w T w U w W w On solvng he eqon..7 for he vros operors n he opml ssem we on se of non-eqvlen Le nsäze smlr vrle smlr solons nd forms of he oeffen fnons for he fnon nd w. Usng hese nsäze we n rede he nonlner ssem.. o he ssem of ODEs.. E Solons of Shllow Wer Wve Eqon In hs seon he prmr fos s on he redons ssoed wh he slgers n he opml ssem. Correspondng o ever essenl slger n he opml ssem he reded ODEs re oned. Eh reded ssem of ODEs n e frher sded for e nd nmerl solons.. Slger V V In hs slger we on he followng orrespondng vrles nd solons for he redon s: log F 7

71 w G... B sng he smlr vrle nd solon.. he ssem of PDEs.. redes o he followng ssem of ODEs: F F F F F F FF FF G G.. G G k F G k k G F Heren we seek spel solon for.. n he followng form: F ep m G ep m.. B ssng hese epressons for F nd G n.. we ge he followng relons mong he vros prmeers: m m k k... m where m nd k re rrr onsns. Ths on sng he k rnsformon we ge he solon of he ssem.. s follows ep log m w ep m log...5 7

72 Fgre.: Grphl represenon of solon..5 for 5 m... Slger V V Under hs slger followng s he opml ssem desres he orrespondng vrle nd orrespondng solons s follows: F G w...6 Agn sng he orrespondng vrle nd solon he ssem of PDEs.. redes o he followng ssem of ODEs: F F F F FF FF G G G k FG k k GF..7 when k Eq...7 negres one esl o eld G K F 7

73 7 where K s n rrr onsn. On ssng G n Eq...7 we ge F F F F FF K F F..8 Ths solvng eqon..8 nd reverng k o he orgnl vrles we on he grop-nvrn solons of he eqon.. s follows Solon : nh K K..9 Solon : nh K.. Solon : nh I K K.. Solon : nh I K..

74 Fgre.: Grphl represenon of solon..9 for Fgre.: Grphl represenon of solon.. for

75 .. Slger V The reded ODEs n hs se re F G. eqon..5 he ovos solon of.. s onsn w onsn...6. Le Smmer Anlss Three-opled KdV Eqons Engendered he emnn Ssem The pplon of he lssl Le grop mehod reqres onsderng one-prmeer Le grop of nfnesml rnsformons n he vrles w s follows * X v w O * T v w O * U v w O * v v V v w O * w w W v w O... whh leves he ssem.. nvrn. The mehod for deermnng he smmer grop of.. onsss of fndng he nfnesmls X T U V nd W whh re fnons of v nd w. Assmng h ssem.. s nvrn nder he rnsformons.. he nfnesmls X T U V nd W ms ssf he smmer ondons 75

76 U U wu W Vv vv V V wv vw vw wv.. W W ww ww U. 8 where U U U W W W nd re eended prolonged nfnesmls ng on n enlrged spe e spe h nldes ll dervves of he dependen vrles for more dels see [99 7 9]. Ssng vles of U U W W W nd U no smmer ondons.. hen eqng he oeffens of he vros monomls n he frs seond nd he oher order prl dervves of nd w nd her powers we n fnd he deermnng eqons for he smmer grop of he eqons... Solvng hese eqons we ge he followng forms of nfnesmls: X T.. U V v W w. where nd re rrr onsns. The Le lger ssoed wh ssem.. onsss of followng for veor felds V v w v w V 76

77 V V.. I s es o verf h V 5 s losed nder he Le rke. The lssfon of one-dmensonl slgers of he whole smmer lger.. s done n ndve pproh [7]. We wll work o frs n opml ssem nd hen emrk pon he vros redons ssoed wh generors n he opml ssem. We egn onsderng generl elemen V V V V V of smmer lger nd se o vros don rnsformons o smplf s mh s possle. The don on s gven he Le seres V V V V V V V V Ad ep..5 where V V V V V V s he ommor for he Le lger nd s prmeer. The pes of one-dmensonl slgers of ssem.. re s follows: V V V V V V v V...6 where nd re rrr onsns. One he slger of he eqon.. hve een deermned he smmer vrles for he ssoed redon n e fond solvng he hrers eqon d d d dv dw X v w T v w U v w V v w W w..7 77

78 On solvng he eqon.8 for he vros operors n he opml ssem we on se of non-eqvlen Le nsäze smlr vrle smlr solons nd forms of he oeffen fnons for he fnon nd w. Usng hese nsäze we n rede he nonlner ssem.. o he ssem of ODEs..5 E Solons of Three-opled KdV Eqons In hs seon we wll drve some e solons of Three-opled KdV Eqons. One w o on e solons of.. s redng o ordnr dfferenl eqons. Ths n e heved wh he se of Le pon smmeres dmed Slger V In hs slger we on he smlr vrles nd smlr solons for he redon s follows: F.5. v G w H. Usng he smlr vrle nd solon.5. he ssem of PDEs.. redes o he followng ssem of ODEs: F F 9 F F HF GG G G 9 G G HG GH H H 9 H H HH F we re nle o fnd he solons of ODE.5. 78

79 .5. Slger V V V Under hs slger he followng s he opml ssem dere he orrespondng vrle nd solons s follows: F G v H w..5. Usng he orrespondng vrle nd solon he ssem of PDEs.. redes o he followng ssem of ODEs: F F HF GG G G HG GH.5. H H HH F. 8 Ths solvng eqon.5. nd reverng k o he orgnl vrles we on he grop-nvrn solons of he eqon.. s follows Solon : nh v nh 5 5 w nh where

80 8 Solon : I w I I v 8 nh nh nh 6 nh nh.5.6 where. Fgre.: Grphl represenon of solon.5.5 for Slger V V For hs slger we on he followng orrespondng vrles nd solons for he redon