Lump Solutions to a Jimbo-Miwa Like Equations

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1 Lump Soluons o Jmbo-Mw Lke Equons Hrun-Or-Roshd * M. Zulkr Al b Deprmen o Mhemcs Pbn Unvers o Scence nd Technolog Bngldesh b Deprmen o Mhemcs Rjshh Unvers Bngldesh * Eml: hrunorroshdmd@gml.com Absrc A Jmbo-Mw lke nonlner derenl equon n -dmensons s developed hrough generlzed blner equon wh he generlzed blner dervves D D D z nd D. Bsed on he generlzed blner orms wo clsses o lump soluons ronll loclzed n ll drecons n he spce nd clss o comple lump pe soluon re genered rom Mple serch o qudrc polnoml uncon soluon o he proposed Jmbo-Mw lke equon. The ppose condons o ssurnce nlc nd ronl loclzon o he soluons re oered. Ech o he resulng lump soluons hold s ree prmeers wo o whch re due o he rnslon nvrnce o he Jmbo-Mw lke equons nd our o whch onl requre o ss he presened ppose condons o beng lump soluons. We lso genere crcle O or ellpse shpe soluons rom he obned soluons usng deren condons on he lumps soluons. Moreover gures re gven o vsulze he properes o he eplc soluons. PACS numbers: 0.0.Jr 0.0.Hj. MSC codes: Q; Q; 7K0. Kewords: Lump soluon; he Jmbo-Mw lke equon; generlzed blner orm.. Inroducon Solon equons connec rch hsores o ecl solvble ssems consruced n mhemcs lud phscs mcrophscs cosmolog eld heor ec. To epln some phscl phenomenon urher becomes more nd more mporn o seek ec ronl

2 soluons especll rogue wve soluons. Recenl here hs been ncresng neres n ndng ronl soluons lump knd o rogue wve lke soluon o nonlner derenl equons. A specl knd o ronl soluons rogue wve soluons drws mjor wreness o reserchers worldwde whch llusre sgncn nonlner wve phenomen n ocenogrph [] nd nonlner opcs []. Ronl soluons o he negrble derenl equons hve been sedl consdered wh he help o he Wronskn ormulon or he Csorn ormulon see e.g. [ 7] bsed on Hro blner orms. One o recen neress o us s o dscuss bou ronl soluons o novel clss o nonlner derenl equons reled wh blner equons. Rel vlued soluons b mens o ronll dec or lumps hve been brodl delbered n recen reserches. Lumps n he Res.[] nd [] or he DS-II nd he Sne-Gordon Equons re esblshed. Nonrvl dnmcs o lumps o he KPI [0] re delbered n he lerure. Ths soluons ehbs neresng scerng properes h were rs noced n Re.[]. The Jmbo-Mw [-] equon s used o descrbe cern neresng -dmensonl wves n phscs nd s he second equon n he well known Pnlev`e herrch o negrble ssems. The smples non-lner -dmensonl Jmbo-Mw equon [-]: u u u u u u u 0 z Mn reserches epend consderble eors o solve he Eq.. Tng nd Lng [] suded he equon hrough he mul-lner vrble sepron scheme. Xu [] esed he negrbl hrough he Pnlev e es nd showed h Eq. s no negrble nd hrough he obned runced Pnlev e epnsons esblshed wo blner equons o nlze one solon wo solon nd dromn soluons. D e. l. [] used wo-solon mehod blner mehod nd rnsormng prmeers no comple ones mehod o cheve new clss o cross knk-wve nd perodc solr-wve soluon o he Eq.. Wzwz [] emploed Hro s blner mehod o develop mul-solon soluons or he Eq.. Asd nd M []

3 ppled blner Bäcklund rnsorms o nd eended grm-pe deermnn nd nd wve ronl soluons o he -dmensonl Jmbo-Mw equon. In hs rcle we would lke o ocus on he -dmensonl Jmbo-Mw lke equon rom s blner orm nd would lke o presen wo clsses o lump soluons nd clss o lump pe comple soluons reducng dmenson o he equon. In hs presen pper we show h ew sucen condons should be ssed or he soluons beng lump pe soluons. Oherwse he gve deren shped soluons even sngulr soluons. All he soluons re llusred wh D plo nd dens plo.. Formulon o he Jmbo-Mw lke Equon nd s Blner orms In hs secon we would lke o consruc Jmbo-Mw lke equon. To do h rs rnsormng he -dmensonl nonlner Jmbo-Mw equon Eq. no he blner orms hrough he dependen vrble rnsormons: u ln. The bove -dmensonl nonlner evoluon Eq. s mpped no he Hro blner equons: D D D D D D. 0 z where he blner derenl equon s dened s n M M n D. g g where L L M L L M nd n L nm re rbrr nonnegve negers. More precsel under he vrue o he Eq. he Eq. s reduced o 0. z z We would lke o consder noher blner derenl equon generlzed blner derenl equon smlr o he Jmbo-Mw equon s D D D D D D. 0. z

4 Ths s he sme blner orm s n he bsc Jmbo-Mw equon Eq.. The operors doped beond re knd o generlzed blner derenl operors lnked wh prme number p whch re known s n vrous rcle [7 ]: n M M n D. g g p α 7 where L n re rbrr nonnegve negers L M L L M n L M nd m h power o α s dened b m r m α m r m mod p 0 r m < p. I p n prculr we hve α α α α α α LL Thus when p we hve D D. D D. D D.. More precsel under he vrue o he Eq. he Eq. s reduced o z z z 0. z z Accordng o generl bell polnoml heores see e.g. [7 ] we dop dependen vrble rnsormon Eq. o rnsorm blner equons o nonlner equons. Then cn drecl be shown h he generlzed blner Eq. or Eq. s lnked o Jmbo-Mw lke nonlner derenl equon: u u v u u u v. uvu u v u u uu u u u u u u u z 0. 0 Acull under he Eq. we hve he ollowng equl: D D D D D D. z u u v u u uvu u v u u uu u u u u u u u 0 z

5 where u v. Snce u ln s rnsles PDEs o s blner orm. Thereore solves Eq. hen rnsormon u ln wll solve he Jmbo-Mw lke equon Eq.0. The Eq.0 hs more er erms wh hgher nonlner hn he usul Jmbo-Mw equon. Besde hs n conrs her blner counerprs we scrunze deren phenomenon h he generlzed blner Jmbo-Mw Eq. or Eq.0 s much smpler hn h o he usul Jmbo-Mw Eq. or Eq... Lump Soluons o he Jmbo-Mw lke Equons: Wh he d o he smbolc compuonl sowre Mple/mhemc we re gong o nqure or posve qudrc soluons o he blner equons reducng dmenson v z or z n Eq. In he wo-dmensonl spce soluon nvolved summng o one squre does no genere ec soluons whch re ronll loclzed n ll drecons n he spce under he rnsormons u ln or u ln. Thus we would lke o consder he rl soluon s he sum o wo lner polnomls s ollows: h g h where g 7 where re rel prmeers o be deermned.. The Jmbo-Mw Equon I s dcul o nd he soluon o he dmensonl equon. So we reduce he dmenson seng or or z. A rs we would lke o serch lump soluon o he Eq. or Eq.0 replcng z b where he blner equon Eq. s convered o 0. Consder he Eq. s he rl soluon smlr o he equon Eq.. Inserng Eq. no Eq. nd solvng or unknown prmeers ; L L we obn se o consrn resuls:

6 7 } { } { The qudrc polnoml uncon soluons n Eq. wh he bove se o resuls nvolved s ree prmeers o n urn elds clss o lump soluons under he deermnn condon: 0. nd 0 > gurnee he posveness o he correspondng qudrc uncon nd he condon 0 nd 0 gurnee he loclzon o u n ll drecons n he -plne. Now he prmeers n he Eq. wh he bove se o resuls produces clss o posve qudrc polnoml soluons o he -dmensonl Jmbo-Mw lke equon Eq.: } { } { 7 nd re rbrr consns. The resulng clss o soluons n urn elds clss o lump soluons o he dmensonl Jmbo-Mw lke Eq.0 under he rnsormon: u ln where s dened n Eq.7 ssng he bove condons Eqs. nd.

7 7 There re s ree prmeers nvolved n he soluon. The soluons re well dened nd posve.e. he condons n Eqs. nd re ssed. The deermnn condon Eq. precsel mens h wo drecons nd n he plne re no prllel whch s essenl n ormng lump soluons n - dmensonls b usng sum nvolvng wo squres. Secondl we would lke o dsclose lump soluon o sndrd Jmbo-Mw lke Eq. or Eq.0 replcng z b he blner equon Eq. cn be convered o 0. To nd lump soluon o he Eq. we hve o consder rl soluon o lke o he Eq.. Inserng Eq. no Eq. nd solvng or unknown prmeers ; L L elds wo se o consrnng equons or he prmeers: 7 nd ± ± 7 I I Thus he soluons re u ln where 0 nd re rbrr consns. The cheved resulng qudrc polnoml uncon soluons Eq. nvolvng s ree prmeers o n urn elds clss o lump soluons o he Jmbo-Mw lke equon under he condon 0. The

Ths s clss o comple soluon urned no lump soluon under he condons 0 bu hve no rel phscl menng.

8 deermnn condon under he condon 0 whch gurnees h wo drecons nd n he plne re no prllel nd s equvlen o 0. Thereore he condons on he prmeers > wll gurnee nlc nd loclzon o he soluons n Eq. nd hus presen lump soluons o he dmensonl Eq.0. And u ln where ± I ± I nd re rbrr consns. Ths s clss o comple soluon urned no lump soluon under he condons 0 bu hve no rel phscl menng. 0 > Some D plos nd correspondng dens plos o he bove cheve soluons re gven wh he prculr choce o he ree prmeers. Energ dsrbuon depends upon wve hegh. So we provde D plo o show he energ dsrbuon o he wve.e. wve hegh or deren vlues o. In he men me we see h when he squres re rel vlued nd < 0 gves resuls wh shpe ellpse or crcle pe solonc soluons or sngulr solon soluon.

9 Fg-:Prole o he Eq.7 or 0 0 : D plo le nd D plo rgh b or 00. b c d

10 Fg-: Prole o he Eq. or 0 0 : D plo b dens plo c D plo nd d Prole o he Eq. or.. Concludng remrks nd uure sks Bsed on generlzed blner dervves wh prme number p we successull ormuled Jmbo-Mw lke equon nvolvng lrge een hgher order nonlner erms. We presened posve qudrc polnoml uncons soluons o he Jmbo-Mw lke equon. Enough condons on he es s ree prmeers nvolved n he obned soluons re proposed o be lump soluons nlc nd loclzed n ll drecon o spce. Though he proposed Jmbo-Mw lke equon conns more nonlner erms her correspondng blner orm s comprvel smller nd eser o hndle hn n pe o soluons nd clculon o hese resuls cn be hndled ver esl wh less eor hn h o he sndrd Jmbo-Mw equon. A comple pe lump soluon lso presened n hs rcle m be useul or eplorng new phenomenon n cse o comple suons. Crcle or ellpse pe even smgulr solon soluons re lso epressed he cheve soluon when he presened condon re no ull ssed generll when < 0. The D plos dens plos o he presened soluons wh some prculr choces o he nvolved prmeers cn be ound n Fgs. nd whch show energ dsrbuon. Hgher order rogue wve soluons could be genered s well n erms o posve polnoml soluons beng mosl movng clss o ec soluons wh ronl uncon mpludes. Such wve soluons re used o llusre essenl nonlner wve phenomen n boh ocenogrph [] nd nonlner opcs[] whch s lkel gre del o curren wreness n he mhemcl phscs soce. To surve more solon phenomen would be ver remrkble sk o consder mul-solon mul-componen nd hgher order eensons o lump soluons more sgncnl n -dmensonl cses. 0

11 Reerences [] P. Müller C. Grre A. Osborne Rogue wves Ocenogrph [] C. Khr E. Pelnovsk A. Slunev Rogue Wves n he Ocen Advnces n Geophscl nd Envronmenl Mechncs nd Mhemcs Sprnger-Verlg Berln 00 [] D.R. Soll C. Ropers P. Koonh B. Jll Opcl rogue wves Nure [] N. Akhmedev J.M. Dudle D.R. Soll S.K. Tursn Recen progress n nvesgng opcl rogue wves J. Op pp.. [] W.X. M Y. You Solvng he Koreweg de Vres equon b s blner orm: Wronskn soluons Trns. Am. Mh. Soc [] W.X. M C.X. L J.S. He A second Wronskn ormulon o he Boussnesq equon Nonlner Anl. TMA [7] W.X. M Y. You Ronl soluons o he Tod lce equon n Csorn orm Chos Solons Frc [] V. A. Arkdev A. K. Progrebkov M. C. Polvnov Inverse scerng rnsorm mehod nd solon soluons or Dve-Sewrson II equon Phsc D 7. [] H. E. Nszks D. J. Frzeskks B. A. Mlomed Phs. Rev. E [0] M. J. Ablowz J. Vllrroel Soluons o he me dependen Schrodnger nd he Kdomsev-Pevshvl Equons Phs. Rev. Le [] V. M. Glkn D. E. Pelnovsk Y. A. Sepnns The srucure o he ronl soluons o he Boussnesq equon Phsc D 0. [] X.Y. Tng Z.F. Lng Vrble sepron soluons or he -dmensonl Jmbo- Mw equon Phs. Le. A [] G. Q. Xu The solon soluons dromons o he Kdomsev-Pevshvl nd Jmbo- Mw equons n -dmensons Chos Solons nd Frcls

12 [] Z. D. D Z. L Z. J. Lu D. L. L Ec cross knk-wve soluons nd resonnce or he Jmbo-Mw equon Phsc A [] A. M. Wzwz Mulple-solon soluons or he Clogero-Bogovlensk-Sch Jmbo-Mw nd YTSF equons Appl. Mh. Compu [] M. Asd nd W. X. M Eended Grm-pe deermnn wve nd ronl soluons o wo -dmensonl nonlner evoluon Appl. Mh. Compu [7] C. Glson F. Lmber J. Nmmo R. Wllo On he combnorcs o he Hro D- operors Proc. R. Soc. Lond. A. [] W. X. M Blner equons Bell polnomls nd lner superposon prncple J. Phs.: Con. Ser

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