NEGATIVE-ORDER FORMS FOR THE CALOGERO-BOGOYAVLENSKII-SCHIFF EQUATION AND THE MODIFIED CALOGERO-BOGOYAVLENSKII-SCHIFF EQUATION

Transcription

1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Ss A OF THE ROMANIAN ACADEMY Volum 8 Numb /07 pp 7 NEGATIVE-ORDER FORMS FOR THE CALOGERO-BOGOYAVLENSKII-SCHIFF EQUATION AND THE MODIFIED CALOGERO-BOGOYAVLENSKII-SCHIFF EQUATION Abdul-Majd WAZWAZ San Xa Uns Dpamn of Mahmacs Chcago IL USA E-mal: wazwaz@sudu Absac In hs wo w dlop h nga-od Calogo-Bogoalns-Schff (CBS) quaon and h nga-od modfd CBS quaon B mans of h cuson opaos of h Kowg-d Vs (KdV) quaon and h modfd KdV quaon w d nga-od foms fo h wo CBS quaons W fomall d mulpl solon soluons fo h nwl dlopd quaons W compa h suls of h sandad CBS quaon and h modfd CBS quaon wh h suls fo h nga-od sons of hs quaons K wods: Calogo-Bogoalns-Schff (CBS) quaon modfd CBS quaon ns cuson opao mulpl solon soluons INTRODUCTION Th cuson opao plas a sgnfcan ol n h fld of ngabl quaons n (+) dmnsons A cuson opao s dfnd as an ngo-dffnal opao ha maps a gnalzd smm of a nonlna paal dffnal quaon o a nw smm [ 0] Th cuson opao as dlopd b Ol [] and ohs fo an nonlna oluon quaon ndcas ha hs quaon has nfnl man hgh-od smms whch s a fau of s compl ngabl Ol [] pod ha h cuson opao maps a smm o a nw smm How Mag [6] ald ha som ssms admd wo dsnc bu compabl Hamlonan sucus now nown as b-hamlonan ssm Th hda smm Φ ( u( ) s a cuson opao of h followng hach of oluon quaons u + Φ u) u 0 () ( I s obous ha hs quaon gs s o a a of (+)-dmnsonal quaons dpndng on h sucu of Φ (u) Th cuson opao Φ () fo h KdV quaon s gn as Φ( ) + + () wh and dno h oal da and s ngaon opao wh spc o spcl Usng h cuson opao () gs h clbad KdV quaon as [ 5] () 6 whch ncluds h nonlna m and h dspson m How h cuson opao fo h modfd KdV (mkdv) quaon as h fom Φ ( ) + + ( ) whch gs h mkdv quaon (5) 6 ()

2 8 Abdul-Majd WAZWAZ Th las m n () s h opao ha as a polnomal P R{ } D and fnall mulpls h sul b mulpls b hn appls h [] Th concp of h cuson opao () was hooughl usd n h lau n pacula n [ 0] o dlop nw quaons n hgh dmnsons How h Calogo-Bogoalns-Schff (CBS) quaon was fs consucd b Bogoalns wh h modfd La fomalsm was usd I was also dd b Schff b ducng h slf-dual Yang Mlls quaon [ 9] W us h followng hach of oluon quaons + Φ ) 0 (6) ( wh Φ s h cuson opao of h KdV quaon () and n () s placd b Ths n un gs h CBS quaon o qualnl (7) u + u + u u + u u 0 (8) oband b usng h ponal u On h oh hand w us h followng hach of oluon quaons + Φ ( ) 0 (9) wh Φ s h cuson opao of h modfd KdV quaon () and n () s placd b Ths n un gs h modfd Calogo-Bogoalns-Schff quaon ( ) 0 Ol [] pod a gnal hom abou cuson opaos fo smms of an oluon quaon wh was shown ha such an opao cas a nw smm gnao whn appld o a nown smm gnao How Vos [8] ndd h wo of Ol n h nga dcon o oban a squnc of quaons of ncasngl nga ods Rcall ha () ndcas Φ( ) () B h nga od hach w f o Φ () h pows of φ go o h oppos dcon [7 9] In oh wods h nga od quaon can b dnod b Φ ) () ( and smlal fo h modfd son w us Φ ( ) () Th dmnaon of ac soluons spcall fo ngabl quaons offs a ch nowldg of h phscal bhao and dnamcal phnomna of h amnd nonlna quaons Eampls of h mhods usd a h Panlé analss [ 9] h ns scang mhod La pas and man ohs Th goals of hs wo a wo fold In pa I w am o sablsh nga-od quaon fo h CBS quaon (8) and o compa h solon soluons of h sandad CBS quaon wh s nga-od fom In pa II w dlop a nga-od fom fo h modfd CBS quaon whch wll b dd as wll W wll compa h suls of h modfd CBS quaon wh h suls of h nga-od fom of W plan o d mulpl solon soluons fo h nwl dlopd quaons and o compa h oband suls wh h suls of h sandad CBS and h modfd CBS quaons (0)

3 Nga od foms fo h Calogo-Bogoalns-Schff quaon 9 PART I In hs pa w wll bfl w h oband suls fo h CBS quaon W hn wll mo o d h nga-od CBS quaon W wll compa h solon suls of ach quaon Th sandad CBS quaon As sad al w wll summaz h suls w oband bfo n [] whn w amnd h CBS quaon u + u + u u + u u 0 (5) In wha follows w onl ls h oband suls n []Th sngl solon soluons was found o b + u( (6) + + wh h dspson laon was dd as c and h soluon u ( was assumd as u ( ln f ( and h aula funcon was s as ( ) wh h dspson aabl s gn b f ( + (7) + (8) In a l mann w can oban h wo solon soluons b usng h aula funcon as + f ( a (9) wh h phas shf a was found as ( ) a (0) ( + ) Th h solon soluons can b oband b usng h aula funcon as f ( a + a + a + a a a () Fo mo nfomaon abou hs suls plas loo a Rf [] and som of h fncs hn Th nga-od CBS quaon In hs scon w am o dlop h nga-od CBS quaon W fs us h nga od hach Φ ) () ( wh h cuson opao Φ fo h CBS quaon s h sam cuson opao fo h KdV quaon gn n () In oh wods w us whch gs o qualnl ( + + )( ) + + ( ) () ()

4 0 Abdul-Majd WAZWAZ u N w wll sud h mulpl solon soluons fo h nga- oband upon usng h ponal od CBS quaon (5) u + u u + u u + u 0 (5) Mulpl solon soluons To dmn h dspson laon w s wh h aula funcon f ( s gn as ( ln f ( ) u ( R (6) + c f( fo h sngl solon soluon Subsung (6) no h nga-od CBS quaon (5) gs h dspson laon b c (8) and (7) R (9) and hfo w s h phas aabl as ( ) + (0) Usng (6) gs h sngl solon soluon u( wh h soluon of h nga-od CBS quaon () s oband b usng h ponal ( u ( Fo h wo solon soluons w s h aula funcon as ( ( ( + ( f ( a () wh a s h phas shf and ( ) s gn n (0) Subsung () and (6) no h nga-od CBS quaon (5) w oban h phas shf b ( ) a () ( + ) and hnc w s h phas shfs b ( j) aj < j () ( + j) Noc ha h nga-od CBS quaon dos no show sonanc bcaus a 0 o fo Combnng () and () and subsung h oucom no (6) w oban h wo solon soluons Fo h h solon soluons w s () f ( a + a + a + b (5) Pocdng as bfo w fnd

5 5 Nga od foms fo h Calogo-Bogoalns-Schff quaon b (6) aaa Ths shows ha h h solon soluons a obanabl Th snc of h solon soluons ofn ndcas h compl ngabl of h quaon und amnaon How oh ca such as La pa should b usd o confm ngabl W conclud fom h suls oband abo ha h CBS quaon and h nga-od CBS quaon ha dsnc solons Ths s du o h dsnc dspson laons Th dspson laons fo h CBS quaon and h nga-od CBS quaon w dd as and / spcl How h wo quaons g mulpl solon soluons wh h phas shfs of h nacon of solons a dncal and boh do no show sonanc PART II In a paalll mann o h analss psnd n pa I w plan o conduc a compaa sud of h modfd CBS quaon and h nga-od modfd CBS quaon whch w wll d la Th modfd CBS quaon Th modfd CBS quaon was alad dd n (0) and s gn as ( ) 0 W fs us + + ( ) ( ) ψ (8) whch cas (7) o (7) ψ ( 0 (9) Dffnang (9) wh spc o and usng (8) w oban (0) ψ To dmn h dspson laon w s wh h aula funcon f ( s gn as Mulpl n soluons ( acan f ( ) ( R () + c f( fo h sngl solon soluon Subsung () no h modfd CBS quaon (0) gs h dspson laons b and and hfo w s h phas aabl as () c () R ()

6 Abdul-Majd WAZWAZ 6 Usng () gs h sngl n soluon + (5) ( ) + ( (6) ( + + Fo h wo solon soluons w s h aula funcons as f ( + (7) + g ( ) a wh a s h phas shf and ( ) s gn n (5) Subsung (7) and () no h modfd CBS quaon (0) w oban h phas shf b ( ) a (8) ( + ) and hnc w s h phas shfs b ( j) aj < j (9) ( + j) Noc ha h modfd CBS quaon dos no show sonanc bcaus a 0 o fo Combnng (7) and (8) and subsung h oucom no (0) w oban h wo solon soluons Fo h h solon soluons w s ( ) f b ( ) g a a a Pocdng as bfo w fnd aaa (50) b (5) Ths shows ha h h n soluons a obanabl Th snc of h solon soluons ofn ndcas h compl ngabl of h quaon und amnaon bu oh ca such as La pa a ndd o confm ngabl Th nga-od modfd CBS quaon W n dlop h nga-od modfd CBS quaon Pocdng as bfo w us h nga od hach Φ ( ) (5) wh h cuson opao Φ fo h modfd CBS s h sam cuson opao fo h modfd KdV quaon gn n () In oh wods w us whch gs Usng cas (5) o ( + u + u ( ) ) u + + ( ) (5) (5) + + ( ) ψ ( ) (55)

7 7 Nga od foms fo h Calogo-Bogoalns-Schff quaon + + ψ ( (56) Dffnang (56) wh spc o and usng (55) w oban ψ ( + 0 (57) To dmn h dspson laon w s wh h aula funcon f ( s gn as Mulpl solon soluons ( acan f ( ) ( R (58) + c f( (59) fo h sngl solon soluon Subsung (58) no h nga-od modfd CBS quaon (57) gs h dspson laons b c (60) and R Thfo w s h phas aabl as ( ) + (6) Usng (58) gs h sngl solon soluon ( + ( + + Fo h wo solon soluons w s h aula funcons as f ( + (6) + g( a wh a s h phas shf and ( s gn n (6) Subsung (6) and (58) no h nga-od CBS quaon (56) w oban h phas shf b ( ) a (6) ( + ) and hnc w s h phas shfs b ( j) aj < j (65) ( + j) Noc ha h nga-od CBS quaon dos no show sonanc bcaus a 0 o fo Combnng (6) and (6) and subsung h oucom no (58) w oban h wo solon soluons Fo h h solon soluons w s ( ) aaa f b g ( ) a a a (6) Pocdng as bfo w fnd b Ths shows ha h h n soluons a obanabl W conclud fom h suls oband abo ha h modfd CBS quaon and h nga-od modfd CBS quaon ha dsnc n soluons Ths s du o h dsnc dspson laons Th (66)

8 Abdul-Majd WAZWAZ 8 dspson laons fo h modfd CBS quaon and h nga-od modfd CBS quaon w dd as and / spcl How h wo quaons ha mulpl n soluons wh h phas shfs of h nacon of solons a dncal and boh do no show sonanc DISCUSSION W usd h ns of h cuson opaos fo h KdV quaon and h mkdv quaon gn n () and () spcl o fomall d h nga-od CBS quaon and h nga-od modfd CBS quaon W ha shown ha h nwl dd quaons possss mulpl solon and mulpl n soluons W dmnd h dspson laon fo ach quaon and showd ha h phas shfs a of h Hoa s p Moo w showd ha h CBS quaons and h nga-od sons ha dsnc soluons du o h occunc of dsnc dspson laons REFERENCES D BALDWIN W HEREMAN A smbolc algohm fo compung cuson opaos of nonlna paal dffnal quaons Innaonal Jounal of Compu Mahmacs 87 pp A FOKAS Smms and ngabl Suds n Appld Mahmacs 77 pp AM WAZWAZ Mulpl-solon soluons fo h Calogo-Bogoalns-Schff Jmbo-Mwa and YTSF quaons Appl Mah Compu 0 pp PJ OLVER Eoluon quaons possssng nfnl man smms J Mah Phs 8 (6) pp S LOU Hgh dmnsonal ngabl modls wh a common cuson opao Commun Tho Phs 8 pp F MAGRI Lcus Nos n Phscs Spng Bln D ZHANG J JI S ZHAO Solon scang wh amplud changs of a nga od AKNS quaon Phsca D 8 pp JM VEROSKY Nga pows of Ol cuson opaos J Mah Phs pp Z QIAO E FAN Nga-od Kowg-d Vs quaon Phs R E R HIROTA Th Dc Mhod n Solon Tho Cambdg Uns Pss Cambdg 00 SA KHOURY Solon and podc soluons fo hgh od wa quaons of KdV p (I) Chaos Solons & Facals 6 pp SA KHOURY Eac soluons fo a class of nonlna oluon quaons: A unfd ansäz appoach Chaos Solons & Facals 6 pp SA KHOURY Nw ansäz fo obanng wa soluons of h gnalzd Camassa-Holm quaon Chaos Solons & Facals 5 pp H LEBLOND D MIHALACHE Modls of fw opcal ccl solons bond h slowl ang nlop appomaon Phs Rp 5 pp H LEBLOND D MIHALACHE Fw opcal ccl solons: Modfd Kowg-d Vs sn-godon quaon sus oh non slowl ang nlop appomaon modls Phs R A D MIHALACHE Muldmnsonal localzd sucus n opcal and ma-wa mda: a opcal su of cn lau Rom Rp Phs AM WAZWAZ Paal Dffnal Equaons and Sola Was Thom Spng and HEP Bln AM WAZWAZ Ingabl couplngs of h Bugs quaon and h Shama-Tasso-Ol quaon: Mulpl n soluons Rom Rp Phs 65 pp AM WAZWAZ Mulpl solon soluons fo wo ngabl couplngs of h modfd Kowg-d Vs quaon Poc Romanan Acad A pp N GOYAL AM WAZWAZ RK GUPTA Applcaons of Mapl sofwa o d ac soluons of gnalzd ffh-od Kowg-d Vs quaon wh m dpndn coffcns Rom Rp Phs 68 pp T COLLINS al Dnamcs of shallow wa was wh logahmc nonlna Rom Rp Phs 68 pp AM WAZWAZ Mulpl n soluons fo h (+)-dmnsonal ngabl Gadn quaon Poc Romanan Acad A 5 pp 6 0 AM WAZWAZ Gaussan sola was fo h logahmc-kdv and h logahmc-kp quaons Phsca Scpa AM WAZWAZ Nw (+)-dmnsonal oluon quaons wh Bugs and Shama-Tasso-Ol quaons consung h man pas Poc Romanan Acad A 6 pp AM WAZWAZ Mulpl n soluons fo h scond hanl quaon and h asmmc hanl quaon Poc Romanan Acad A 7 pp 0 06 Rcd Ma 5 07