Eurasian International Center of Theoretical Physics, Eurasian National University, Astana , Kazakhstan

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1 Joul o Mhems d sem ee do: 765/59-59/8 D DAVID PUBLIHIG E Loled oluos o he Geeled +-Dmesol Ldu-Lsh Equo Gulgssl ugmov Ao Mul d Zh gdullev Eus Ieol Cee o Theoel Phss Eus ol Uves As 8 Khs As: I hs ppe he geeled +-dmesol Ldu-Lsh equo wh poel s vesged Is e loled d opologl soluos e osued geelg Ho s mehod Ke wods: p ssem solos Ldu-Lsh equo Ioduo ole models o -dmesol mge suses pl mpo ole he heoel despo o ole poesses el mges I pul ee es +-dmesol LLE Ldu-Lsh equo hs ee vel vesged [: whee wh legh The pupose o hs ppe s o d e opologl solo soluos o +-dmesol GLLE geeled Ldu-Lsh equo wh sl poel +- d +-dmesol GLLEs wh sel-osse poel wee vesged Res [-5 The Geeled +-Meso Ldu-Lsh Equo d Is Ble Fom The oe o ou sud s he ollowg +-dmesol GLLE gve he ompoes o he sp veo he om Coespodg uho: Gulgssl ugmov eseh elds: he heo o solos ole egle ssems u u u oe h he m hs om d e Pul mes Applg he seeogph somo 5 o he ssem o Eqs - we ewe Eqs - ems o s ollows 6 u whee deoes omple ougo The we osu le om o he Eqs - g osdeg somo he ollowg om:

2 88 E Loled oluos o he Geeled +-Dmesol Ldu-Lsh Equo D D D D g 7 [ D g [ D g [ D g 5 D [ D [ D D u g D D D g 8 D D D g g 9 g {[ D g g g [ D g g g [ D g} g g [ D g Usg he le oms Eqs -5 we osu e opologl soluos o Eqs - I se he le oms Eqs -5 eld he GLLE ems o g he om g g g ug 6 Heee g e omple The soluos o hese equos e lso soluos o Eqs -5 The ovese s o lws ue e uos d Ho s D-opeo s deed Re he Eqs -5 hve wde lss o soluos h [6 he ssem Eqs 6-7 The Eq 6 s dm l l DD g g equo d he Eq 7 s odo o soluo Fs we wll d e soluos o he Eq 6 To l g g do so we ove om Ces oodes o oodl oodes odg o he E Topologl oluos omuls g g g g 7 oluos desg he homogeeous se o mgeo whh o e edused o he s se e deomos o he mgeo eld e lled opologll spel The oespodg soluos lled opologl soluos I pe ole eos mges whh e desed e opologl soluos o sp ssems e o hgh ees mog sess To osu hs pe o soluos o ou GLLE Eqs - we ewe Eq 6 s ssem o wo equos: u I s le h Eq s +-dmesol le hödge equo wh he poel u d Eq s some ddol equo o he omple uo ω Fo Eqs - he le oms e we s [ D g D D D g [ D D D l u sh os q 8 sh s q 9 s q whee q osh os > < < The evese omuls e gve l l I oodl oodes he ssem o Eqs 6-7 es he om

3 E Loled oluos o he Geeled +-Dmesol Ldu-Lsh Equo 89 osh os g sh sh g [ osh os sh g g ug osh os sh osh os g g g sh 5 e see soluo o he ssem Eqs -5 he ollowg om: 6 whee e eges d g s some omple os usug Eq 6 o Eq gves A osh os sh sh osh os u R l sh osh os B 8 osh oss m ui 9 wh The he ssem Eqs -5 es he om Hee 7 l osh os sh sh osh os sh osh os u R osh os s m u I I hs se he Eqs - e ssed he poel u u R u s epessed he omuls I osh os sh u R l sh osh os sh osh os osh os s ui m The Eq hs he ollowg wo soluos:

4 9 E Loled oluos o he Geeled +-Dmesol Ldu-Lsh Equo 5 6 whee odo e g e d wh he A m ml l 7 The oespodg soluos o GLLE Eqs - e gve whee l l g os[ m l 8 g s[ m l 9 g g os[ m m g os m l The soluos Eqs 8-9 e e opologl solo soluos o GLLE Eqs - d ull lso loled To d he poso o lolo ell h ompoe o he sp o he veo es vlue he evl To d he poso o he lolo sues o e oe po om hs evl Fo oe ompoe we e he po Thus he solo lolo pos e deemed he ollowg equo: g g l os[ m m l g os m l whh ollows om Eq I os o Re [7 ou se soluos deped o me ow we gve he e l soluo o he se I hs se he soluos Eqs 8 d espevel e he om d whee m l g e e os gos m l g gos m l 5 s gs m l g gos m l g osh sh m osh 6 g os m l m sh m Also he Eq e we s g m sh g os m l The solos e loled odg o he omuls g 5

5 E Loled oluos o he Geeled +-Dmesol Ldu-Lsh Equo 9 L l m 5 Hee L s he ume equl o he gees ommo dvso o m d K I wh ollows we eed he omuls: e e e s os osh sh os s e osde vous spel ses o he soluo Eqs 5-5 seg o oveee Emple : m Eqs 5-5 e we he om sh g 5 L 5 Eq 5 Ces oodes s we s sh 5 o 55 whh s equvle o he ollowg ous equo: 56 Ths equo s ssed wh he pmeo os 57 s 58 As o he Eq 5 we ewe he om L 59 o equvlel os 6 Usg omuls 8- hs equo Ces oodes e we he om [ [ 6 Hee we o V 6 Tg o ou he equo o he ous 56 om Eq 6 we ll hve 6 o 6 usug Eq 58 o Eq 6 we hve os s 65

6 9 E Loled oluos o he Geeled +-Dmesol Ldu-Lsh Equo e pss om Ces oodes o ldl oodes usg he omuls os s whee Tg o ou Eqs d 65 Ces oodes ems o ldl oodes e pmeed s ollows: os os s os 66 Fg Moo o he sp veo Cos pmees: m l ψ α os s s os 67 s 68 I he pul se whe he Eqs e he om os os os s os os s The gphs o oed soluos Eqs 5-7 e show Fgs d e lel see he ehvo o he sp veo he me evl [; d η Fg Moo o he sp veo Cos pmees: m l ψ α π Emple : m Fo hs se he Eqs 5-5 e he om g 69 sh L 7 Eq 69 Ces oodes s we s

7 E Loled oluos o he Geeled +-Dmesol Ldu-Lsh Equo 9 sh 7 o 7 The ls omul s equvle o he ollowg ous equo: 7 Ths equo s ssed wh he pmeo os 7 s 75 As o he Eq 7 we ewe he om os os os s s 76 so h os 77 s os s 78 The om Eqs we o [ 8 [ [ 79 Hee 8 Tg o ou he equo o he ous Eq 7 om Eq 8 we ll hve 8 o 8 usug Eqs 7-75 o Eq 8 elds s os 8 The somo om Ces oodes o ldl oodes s doe odg o he omuls 8- Fom Eqs 8 d 8 we o os s os s os s s os s os Hee we hve os os 8 h he osdeo o Eqs 7-75 d 8 Ces oodes ems o ldl oodes e pmeed he ollowg w: os s s os 85

8 9 E Loled oluos o he Geeled +-Dmesol Ldu-Lsh Equo os os s os 86 s 87 I he pul se whe he Eqs e he om os 9 s Ule he pevous wo emples hs se we do o hve vl o os s 88 Vsulo o oed soluos Eqs 5-7 o he d emple s peseed o Fgs d e see he moo o he sp veo he me evl [; d η Emple : m I hs se he Eqs 5-5 e we s osh g sh osh 9 sh 9 sh 89 L 9 A he sme me he Eq 9 e we s Fg Moo o he sp veo emple m l os V V V 9 I sml w o he pevous emples we o 9 Hee we hve Fg Moo o he sp veo emple m l

9 E Loled oluos o he Geeled +-Dmesol Ldu-Lsh Equo 95 heo s used o he heoel despo o he mosop ehvo o mges I he mewo o hs heo o us pple ws mpo o show h GLLE +-dmesos dms soluo eled o he o heo I hs sese he soluos +-dmesos hve o logue Thus we heoell poved h mges hee e moe omple solo eos e ole oes oeed wh odes I hs egd eseh o epemel sehes o odes d poo o he esee mges e doe Posve esuls hese es o eseh m hve pomsg ehl pplos Fg 5 Moo o he sp veo emple m h pmees: Fg 6 Moo o he sp veo emple m A opologl soluo o he Eqs - s show o Fgs 5 d 6 The moo o he sp veo e see he me evl [; d η Colusos The lssl heo o mges Ldu-Lsh Reeees [ Kosevh A Ivov B d Kovlev A 98 ole Mgeo ves: Dml d Topologl olos Russ Kev: uow Dum [ Ishmo Y 98 Mul-Voe oluos o Two-Dmesol ole ve Equo Pog Theo Phs 7: -7 [ Mulov R Vlshm ugmov G d Lshm M 997 A +-Dmesol Iegle p Model: Geomel d Guge Equvle Couep olos d Loled Cohee uues Phs Le A : 9-6 [ ugmov G gdullev Zh d Mulov R 7 Ho s Mehod o p Model wh el-cosse Poel J o Phs Co ees 8: 5 [5 Yesulov Z Zhssev M Yesmhov K ugmov G d Mulov R 6 Dou Tsomo d E oluos o he Iegle Heseeg Feomge Equo wh el-cosse Poels I J o Geom Meh Mod Phs : 55 [6 Ho R 97 E oluo o he Koeweg-de Ves Equo o Mulple Collsos o olos Phs Rev Le 7: 9 [7 Adm C Hop Mps s oluos o he Comple Eol Equo J Mh Phs 5: -7

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