Noncommutative Solitons and Quasideterminants
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1 Nocommutatve Soltos ad Quasdetemats asas HNK Nagoya Uvesty ept. o at. Teoetcal Pyscs Sema Haove o eb.8t ased o H ``NC ad's cojectue ad tegable systems NP ep-t/69 H ``Notes o eact mult-solto solutos o NC tegable eaces HEP794ept/66 d otcomg papes
2 Successul pots NC teoes ppeaace o e pyscal objects escpto o eal pyscs gauge teoy Vaous successul applcatos to -bae dyamcs etc.
3 ad s cojectue: ay peaps all? tegable equatos ae eductos o te SY eqs. Soluto Geeatg Tecques Ite gauge goup S SY eq. s a maste eq.! KP SY Tsto Teoy Yag s om ad s cal Zakaov CS ae Toda NS KdV gauge equv. oussesq pkdv mkdv gauge equv. N-ave se-godo ouvlle Tteca
4 NC ad s cojectue: ay peaps all? NC tegable eqs ae eductos o te NC SY eqs. ecteeld-popov Soluto Geeatg Tecques Ne pyscal objects NC S NC SY eq. s a maste eq.? NC SY NC Tsto Teoy pplcato to -baes NC ad s cal NC KP Reductos NC Zakaov NC CS NC ae Toda NC KdV NC mkdv NC se-godo NC NS NC pkdv NC ouvlle NC oussesq NC N-ave NC Tteca
5 Pla o ts talk. Itoducto. acklud tasoms o NC SY eqs. Eact solutos o tya-ad asat 3. acklud tasoms o NC KdV eqs. Eact N-solto solutos 4. Cocluso ad scusso I tems o Quasdetemats
6 . acklud tasoms o NC SY eqs. I ts secto e deve NC SY eq. om te vepot o lea systems c s sutable o dscusso o tegable aspects. e dee NC Yag s equatos c s equvalet to NC SY eq. ad gve a acklud tasomato o t. Te geeated solutos ould cota ot oly teacto solutos NC statos but also te-acto solutos o-lea plae aves ad so o. Ts acklud tasomato ould be applcable o loe-dmesoal tegable eqs. va ad s cojectue.
7 Hee e dscuss GGN NC SY eq. om te vepot o lea systems t a spectal paamete. ea systems commutatve case: Compatblty codto o te lea system: ζ ζ ζ. ζ ζ 3 3 :SY equato e.g. : ν µ µ ν ν µ µν
8 Yag s om ad Yag s equato SY eq. ca be ette as ollos I e dee Yag s mat: te e obta om te td eq.: :.. etc etc :Yag s eq. Te soluto epoduce te gauge elds as s gauge vaat. Te decomposto to ad coespods to a gauge g
9 Q Ho e get NC veso o te teoes? e ave oly to eplace all poducts o elds oday commutatve gauge teoes t sta-poducts: g g Te sta poduct: NC ad assocatve µν s µν θ g : ep θ µ ν g g µ ν g O θ Note: coodates ad elds temselves ae usual c-umbe uctos. ut commutato o coodates becomes µ ν µ ν ν µ : θ µν
10 Hee e dscuss GGN NC SY eq. om te vepot o lea systems t a spectal paamete. ζ ll poducts ae sta-poducts. ea systems NC case: Compatblty codto o te lea system: ζ ζ : ν µ µ ν ν µ µν. ζ ζ 3 3 :NC SY equato e.g. θ θ θ θ θ µν O O
11 Yag s om ad NC Yag s equato NC SY eq. ca be ette as ollos I e dee Yag s mat: te e obta om te td eq.: : :NC Yag s eq. Te soluto epoduces te gauge elds as s gauge vaat. Te decomposto to ad coespods to a gauge g
12 acklud tasomato o NC Yag s eq. Yag s mat ca be decomposed as ollos Te NC Yag s eq. becomes Te ollog t. leaves NC Yag s eq. as t s:. : e e e e e e β H NP ep-t/69 ad collaboato t Glso-sa ad Nmmo-sa et. al.
13 e could geeate vaous o-tval solutos o NC Yag s eq. om a tval seed soluto by usg te pevous acklud t. togete t a smple t. γ : e C C C : cost. Ts combed t. ould geeate a goup o dde symmety o NC Yag s eq. c ould be also appled to loe-dmeso. o GG e ca peset te tasoms moe eplctly ad gve a eplct om o a class o solutos tya-ad asat.
14 acklud t. o NC Yag s eq. GG et s cosde te ollog acklud t. I e take a seed sol. te geeated solutos ould be : a a b b b b β Collaboato t Glso-sa ad Nmmo-sa et. al. Vey Hot o α α β γ α 3 O. NC tya-ad asat : a b b a γ
15 acklud t. o NC Yag s eq. GG et s cosde te ollog acklud t. I e take a seed sol. te geeated solutos ould be Collaboato t Glso-sa ad Nmmo-sa et. al. Vey Hot o α α β γ α 3 ll gedets asat ca be detemed om oly Vaous coce o Vaous solutos NC tya-ad asat. O
16 acklud t. o NC Yag s eq. GG et s cosde te ollog acklud t. I e take a seed sol. te geeated solutos ould be ecteeld-popov HEPep-t/99 o α α β γ α 3 NC tya-ad asat ll gedets asat ca be detemed om oly e.g.. ep ' ' `` o c lea. O
17 acklud t. o NC Yag s eq. GG et s cosde te ollog acklud t. I e take a seed sol. te geeated solutos ould be o α α β γ α 3 NC tya-ad asat ll gedets asat ca be detemed om oly e.g.. ep ' ' `` o c lea Quasdetemats! O
18 Quas-detemats Quas-detemats ae ot just a NC geealato o commutatve detemats but ate elated to vese matces. o a by mat X j ad te vese Y y j o X quas-detemat o X s dectly deed by X Recall tat X Y C j y θ j j det j det X some acto X C C C X C C C e ca also dee quas-detemats ecusvely
19 Quas-detemats eed ductvely as ollos j j j j j j j j j j j j j X X X : : : X X X X X X j j j X : te mat obtaed om X deletg -t o ad j-t colum
20 I ts ay e could geeate vaous complcated solutos o NC Yag s eq. om a smple seed soluto by usg te pevous acklud t. α γ o β NC CYG t. seed soluto: `` '' ep lea o NC statos NC No-ea plae-aves NC CYG t. ould elate to a abou tasom o NC SY Glso&Nmmo&Ota et. al ad `eakly o-assocatve algebas c. Quasdetemats sols. o NC KP ae atually deved om a abou t. ad te `eakly o-assocatve algebas. GNO maks&uelle-hosse
21 3. acklud tasoms o NC KdV eq. I ts secto e gve a eact solto solutos o NC KdV eq. by a abou tasomato. Glso-Nmmo Pto appea l.s/77 e see tat gedets o quas-detemats ae atually geeated by te abou tasomato. a og o quas-detemats e also make a commet o asymptotc beavo o solto scatteg pocess H HEP 7 94 ep-t/66.
22 ea systems: a pa o NC KdV eq. t u λ 3 3/ u 3/ 4 u. Compatblty codto o te lea system: 3 u& u uu u u 4 4 :NC KdV equato abou tasom o NC KdV et us take a ege ucto o ad dee Φ Te te ollog t. leaves te lea systems as t s: Φ Φ Φ Φ Φ ad θ u u u log
23 Te abou tasomato ca be teated et us take ege cs. o ad dee Iteated abou tasom o NC KdV Te ollog t. leaves te lea systems as t s ad N Φ Φ Φ Φ Φ log N N N u u u θ Φ Φ I act ae quas-detemats o osk matces! Φ Φ Φ Φ 3
24 Eact N-solto solutos o te NC KdV eq. logdet N N u θ... : ep ep a λ ξ λ ξ 3 3 t t t λ λ λ ξ m m m m m m m O
25 Quas-det solutos ca be eteded to NC tegable eacy Eact N-solto solutos o te NC KP eacy Φ Φ u... : Φ N N N ep ep a β ξ α ξ 3 3 α α α α ξ logdet N N u θ... : osk mat: oe geealato s possble. H ep-t/66 m m m m m m m O
26 Itepetato o te eact N-solto solutos e ave oud eact N-solto solutos o te de class o NC eaces. Pyscal tepetatos ae o-tval because e g ae eal * g s ot geeal. Hoeve te solutos could be eal some cases. -solto solutos ae all te same as commutatve oes because o vt* g vt vt g vt I asymptotc ego coguatos o multsolto solutos could be eal solto scattegs ad te same as commutatve oes. H HEPep-t/66 maks-uelle-hosse ep-t/75
27 -solto soluto o KdV eac packet as te coguato: 3 u k cos k 4k t v 4k k velocty egt
28 -solto soluto o NC KdV eac packet as te coguato: 3 u k cos k 4k t v 4k k velocty egt H HEP 7 94 ep-t/66 c Paak ep-t/585
29 NC S 4. Cocluso ad scusso OK! Soluto Geeatg Tecques Ite gauge goup Hep-t/57 NC SY eq. s a maste eq.! NC KP NC SY NC Tsto Teoy Summaed H Yag s om ep-t/69 OK! OK! NC ad s cal NC Zakaov NC CS NC ae Toda NC NS NC KdV gauge equv. NC pkdv NC mkdv gauge equv. NC se-godo NC ouvlle NC oussesq NC N-ave NC Tteca
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