MULTIPLE SOLITON SOLUTIONS FOR TWO INTEGRABLE COUPLINGS OF THE MODIFIED KORTEWEG-DE VRIES EQUATION
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1 THE PUBLISHING HOUSE POCEEDINGS OF THE OMANIAN ACADEMY Srs A OF THE OMANIAN ACADEMY Volm Nmbr /0 pp 9 5 MULTIPLE SOLITON SOLUTIONS FO TWO INTEGABLE COUPLINGS OF THE MODIFIED KOTEWEG-DE VIES EQUATION Abdl-Md WAZWAZ S Xvr Uvrsy Dprm of Mhmcs Chcgo IL USA E-ml: wzwz@sd I hs wor w s h lgbr of copld sclrs o dvlop wo ds of olr grbl coplgs of h modfd Korwg-d Vrs (mkdv qo O of h grbl coplgs of h mkdv qo gvs mlpl solo solos of dsc mplds whrs h scod d gvs mlpl sglr solo solos of dsc mplds s wll Th Bäcld rsformo d h smplfd Hro s mhod wll b sd for hs sdy W show h hs coplgs possss mlpl solo solos h sm s h mlpl solo solos of h mkdv qo b dffr oly h coffcs of h Bäcld rsformo Ths dffrc hbs solo solos wh dsc mplds Ky words: grbl coplgs bäcld rsformos modfd Korwg-d Vrs qo mlpl solo solos INTODUCTION Th bqos Korwg-d Vrs (KdV qo [ ] dmsolss vrbls rds ± + 0 ( 6 Ths qo modls vry of olr wv phom sch s shllow wr wvs cosc wvs hrmoc crysl d o-cosc wvs plsms Th KdV qo s complly grbl d gvs rs o mlpl-solo solos Ths qo hs b sdd by vry of mhods sch s h vrs scrg mhod d h Bäcld rsformo mhod Th KdV qo dms mlplsolo solos d hbs f mbr of cosrvo lws of rgy Th modfd KdV (mkdv qo + + ( 6 0 s mpor my rs of olr scc Th mkdv qo pprs cosc wvs cr hrmoc lcs modls of rffc cogso rsmsso ls Schoy brrr Alfvé wvs collso lss plsm o cosc solos lsc md d ohr pplcos I posssss my rmrbl proprs sch s cosrvo lws vrs scrg rsformo blr rsformo mlpl solo solos brhr solos Plvé grbly d Drbo rsformo Th hory of olr grbl coplgs of ordry solo sysms ws prsd [] d frhr sdd [] d ohrs I [ ] M l proposd h prrbo mhod for sblshg grbl coplgs Zhg l [] prsd h lrgd L lgbr mhod o ob grbl coplgs Prclrly oworhy r h cosrcos of grbl coplgs bsd o h o-sm smpl L lgbrs [] I fc hr r svrl mhods dopd o cosrc grbl coplgs sch s prrbos lrgg h spcrl problm crg w loop lgbrs d sm drc sms of L lgbr Lo of wor hs b do hs fld d my grbl coplgs wr cosrcd [56] I s ow ow h for grbl sysm w c cosrc w grbl dffrl qo sysm clld grbl coplgs whch clds h gv grbl qo s sb-sysm
2 0 Abdl-Md Wzwz I [] vry rl rglr olr coplgs of grbl sysms wr dvlopd Th cosrco [] ws md o h lvl of volo qos by modfco of h lgbr of dymcl flds Th lgbr of copld sclrs ws sd o dvlop -copld KdV (c-kdv gv h form Th lgbr of copld sclrs ws rodcd [] d ws show o b l commv d ssocv Morovr h grbly of h coplgs ( ws md [] My rlbl mhods r sd h solry wvs hory o vsg solos d prclr mlpl solo solos of complly grbl qos Th lgbrc-gomrc mhod h vrs scrg mhod h Bäcld rsformo mhod h Drbo rsformo mhod h Hro blr mhod d ohr mhods r sd o m progrss d w dvlopms hs fld I hs wor w m o pply h Bäcld rsformos d h smplfd Hro s mhod [ 0] for rlbl sdy Or m from hs wor s wo fold Th frs gol s o mploy h dvlopd lgbr of copld sclrs [] o drv wo forms of olr grbl coplgs of h modfd KdV qo W m scod o sdy hs coplgs d show h possss mlpl solo solos d mlpl sglr solo solos h sm s h mkdv qo b dffr oly h coffcs of h Bäcld rsformos h rsl dsc mplds for ch qo of h sysm 0 ( FIST COUPLINGS OF THE MKDV EQUATION: MULTIPLE SOLITON SOLUTIONS I l mr o h pproch prsd [] whr h lgbr of copld sclrs ws dvlopd w s o fld solo sysm K[ ] K[ ] ( h c b dd o h sysm of copld PDEs of h form [] K[ ] K[ ] (5 whr Accordgly h sysm (5 s h form [] [ ] K K K ( (7
3 Mlpl solo solos for wo grbl coplgs of h modfd Korwg-d Vrs qo Usg ( w c s 6 Isrg (8 o (7 w dvlop h -copld modfd KdV (c-mkdv gv h form ( + 6( + 6 6( + + 6( (8 (9 Mlpl solo solos Sbsg ω ( o h lr rms of ch qo (9 gvs h dsprso rlo by d s rsl w ob h followg phs vrbls Th mlpl solo solos of h coplgs (9 r ssmd o b (0 ω ( ( F( FG GF ( rc G( F + G whr h lry fcos F ( d G ( for h sgl solo solo r gv by ( F( G ( ( Sbsg ( o (9 d solvg for d w ob wo dsc ss gv by (5 + ( for ( + for Combg h obd rsls gvs wo ss of sgl solo solos (6 + ( ( + (7
4 Abdl-Md Wzwz d for + ( + ( for + I ohr words w ob wo ss of sgl solo solos wh dsc mplds bw h wo ss for r r For h wo solo solos w s h lry fcos by (8 F( + G + ( Usg (9 ( d sbsg h rsl (9 w ob h followg phs shf coffc d hc w s ( ( + ( < ( + (9 (0 ( Usg (5 d (6 d h prvos rsls w lso ob wo dsc ss of wo solo solos I s wll ow h wo solo solo [] c dgr o rso rd dr h codos 0 or( 0 for ( Accordgly h rsoc phomo dos o s for hs coplg bcs 0 d ( 0 for For h hr solo solos w s F( G( Procdg s bfor w fd b ( b ( Two ss of hr solo solos wh dsc mplds r obd by sbsg ( o ( s prsd rlr Ths shows h ch qo of h coplg (9 possss h sm proprs s h mkdv qo: h sm phs vrbl h sm phs shf d h o rsoc phom Howvr h oly dffrc s h h mplds r dsc for dsc SECOND COUPLINGS OF THE MKDV EQUATION: MULTIPLE SINGULA SOLITON SOLUTIONS Usg h scod form of h mkdv qo 6 (5
5 5 Mlpl solo solos for wo grbl coplgs of h modfd Korwg-d Vrs qo d sg h lyss prsd bfor for h drvo of h coplgs of h mkdv qo w ob scod coplgs of h mkdv qo gv by I hs sco w wll m mlpl sglr solo solos of h scod d of h coplgs of h mkdv qo ( + + 6( ( ( (6 Mlpl sglr solo solos Sbsg ( ω (7 o h lr rms of ch qo (6 gvs h dsprso rlo by d s rsl w ob h followg phs vrbls Th sglr solo solos of h coplgs (6 r ssmd o b ω (8 (9 f ( ( l g( gf fg gf whr h lry fcos f ( d g ( for h sgl sglr solo solo r gv by (0 f ( g( ( d Sbsg (0 o (6 d solvg for w ob wo ss of solos ( + ( for ( for Sbsg ( o (0 gvs wo ss of sgl sglr solo solos gv by ( d + ( ( (
6 Abdl-Md Wzwz 6 for ( 8 ( for I ohr words w ob wo ss of sgl sglr solo solos wh dsc mplds for r r For h wo solo solos w s h lry fcos by + + f( + + g ( Usg (6 (0 d sbsg h rsl (6 w ob h followg phs shf coffc d hc w s ( ( + (5 (6 (7 ( < (8 ( + Combg h obd rsls gvs wo ss of wo sglr solo solos For h hr solo solos w s f ( g( + Procdg s bfor w fd b b (9 b (0 Ths r gvs wo ss of hr sglr solo d h hr sglr -solo solos r obd by sbsg (9 o (0 DISCUSSION I hs wor wo ds of coplgs of h mkdv qo wr dvlopd by sg h lgbr of copld sclrs Mlpl solo d mlpl sglr solos r drvd for h boh coplgs of h mkdv qo I fc wo ss of solo solos wr drvd for ch coplg W showd h ch qo of h wo coplgs posssss h sm proprs s h mkdv qo: h sm phs vrbl h sm phs shf d h o rsoc phom Howvr h oly dffrc s h h mplds r dsc for ch qo of h coplgs EFEENCES M BASZAK B SZABLIKOWSKI B SILINDI Cosrco d sprbly of olr solo grbl coplgs Appl Mh Comp 9 pp WX MA B FUCHSSTEINE Igrbl hory of h prrbo qos Chos Solos d Frcls 7 pp WX MA Z-N ZHU Cosrcg olr dscr grbl Hmlo coplgs Comp Mh Applcos 60 pp
7 7 Mlpl solo solos for wo grbl coplgs of h modfd Korwg-d Vrs qo 5 Y ZHANG H TAM Thr ds of coplg grbl coplgs of h Korwg d Vrs hrrchy of volo qos J Mh Phys Y FA-JUN L LI Igrbl coplgs of C-KdV qos hrrchy wh slf-coss sorcs ssocd wh sl( Phys L A 7 pp Y FA-JUN Prologo srcr for olr grbl coplgs of KdV solo hrrchy Ch Phys B CM KHALIQUE Ec solos d cosrvo lws of copld grbl dsprsolss sysm Flom 6 pp CM KHALIQUE A BISWAS Solos Plsms: A L Symmry Approch Irol Jorl of Thorcl Physcs 8 pp CM KHALIQUE A BISWAS Opcl solos wh powr lw olry sg L grop lyss Phys L A 7 pp BB KADOMTSEV VI PETVIASHVILI O h sbly of solry wvs wly dsprsv md Sov Phys Dol 5 pp HIOTA M ITO soc of solos o dmso J Phys Soc Jp 5 pp HIOTA A w form of Bäcld rsformos d s rlo o h vrs scrg problm Progrss of Thorcl Physcs 5 pp H LEBLOND H TIKI D MIHALACHE Drvo of grlzd dobl-s-gordo qo dscrbg lrshor solo propgo opcl md composd of mllvl oms Phys v A H LEBLOND D MIHALACHE Modls of fw opcl cycl solos byod h slowly vryg vlop ppromo Phys pors 5 pp AM WAZWAZ (+-dmsol KdV (N qos drvd by sg h KdV rcrso opror Physc Scrp AM WAZWAZ Two forms of (+-dmsol B-yp Kdomsv-Pvshvl qo: mlpl solo solos Physc Scrp AM WAZWAZ O d wo solo solos for svh-ordr Cdry-Dodd-Gbbo d Cdry-Dodd-Gbbo-KP qos Crl Erop Jorl of Physcs 0 pp AM WAZWAZ A rlbl sdy for sos of h Br problm wh bodry codos Mhmcl Mhods h Appld Sccs 5 pp AM WAZWAZ Solo solos for wo (+-dmsol o-grbl KdV-yp qos Mhmcl d Compr Modllg 55 pp AM WAZWAZ Igrbl coplgs of h Brgrs qo d h Shrm-Tsso-Olvr qo om pors Physcs 65 0 cvd Mrch 0
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