Backlund Transformations for Non-Commutative (NC) Integrable Eqs.

Transcription

1 Backlud Trasormatos or No-Commutatv NC Itral Es. asas HAANAKA Uvrsty o Naoya vst Glaso utl. 3 ad IHES 3 - ar3 9 Basd o Clar R.Glso Glaso H ad oata.c.nmmo Glaso ``Backlud trs or NC at-sl-dual AS Ya-lls Y s. arxv:79.69 to appar G & 8mm.. H NPB HEP 7 94 PB P

2 NC tso o tral systms atr ralato Quartro-valud systm oyal dormato tso to NC spacs prsc o matc lu 4-dm. At-Sl-ual Ya-lls E. plays mportat rols QT a mastr. o lor-dm tral s. Ward s cojctur All varals lo to NC r c mpls assocatvty.

3 4-dm. NC ASY. GGN µ ν 3 µν µν : A A A A µν µ ν ν µ µ ν 3 3

4 Rducto to NC KdV rom NC ASY :NC ASY. GG A A 4 A A t Rducto codtos uu u u u u & :NC KdV.! u

5 Rducto to NC NS rom NC ASY t A A A A :NC ASY. GG Rducto codtos & :NC NS.!

6 NC Ward s cojctur: ay praps all? NC tral s ar rductos o t NC ASY s. NC S N pyscal ojcts Soluto Grat Tcus It au roup Hp-t/57 NC KP Applcato to str tory NC ASY NC Tstor Tory Ya s orm NC Ward s cral Summard H NPB I au tory NC matc lds NC Zakarov NC CBS NC a Toda NC NS NC KdV au uv. NC pkdv NC mkdv au uv. NC s-gordo NC ouvll NC Bousss NC N-av NC Ttca

7 Pla o ts talk. Itroducto. Backlud Trasorms or t NC ASY s. ad NC Atya-Ward asat solutos trms o uasdtrmats 3. Or o t Backlud trs rom NC tstor tory 4. Cocluso ad scusso

8 . Backlud trasorm or NC ASY s. I ts scto drv NC ASY. rom t vpot o lar systms c s sutal or dscusso o tralty. W d NC Ya s uatos c s uvalt to NC ASY. ad v a Backlud trasormato or t. T ratd solutos ar NC Atya-Ward asat solutos trms o uasdtrmats c cota ot oly t-acto solutos NC statos ut also t-acto solutos o-lar pla avs ad so o.

9 A drvato o NC ASY uatos W dscuss GGN NC ASY. rom t vpot o NC lar systms t a commutatv spctral paramtr. ar systms: Compatlty codto o t lar systm: :NC ASY..

10 Ya s orm ad NC Ya s uato NC ASY. ca rrtt as ollos I d Ya s -matr: t ota rom t trd.: :.. tc A tc A :NC Ya s. A A A A T soluto rproduc t au lds as s au varat. T dcomposto to ad corrspods to a au

11 Backlud tr. or NC Ya s. GG Ya s matr ca rparamtrd as ollos T NC Ya s. coms T ollo tr. lavs NC Ya s. as t s:. : β

12 Backlud trasormato or NC Ya s. Ya s matr ca rparamtrd as ollos T NC Ya s. coms Aotr tr. also lavs NC Ya s. as t s: : γ.

13 Bot trs. ar volutv ut t comd tr. s o-trval. T could rat varous otrval solutos o NC Ya s. rom a trval sd soluto so calld NC Atya- Ward solutos d d γ γ β β o o o α α β γ α : β : γ β γ o

14 Gratd solutos NC Atya-Ward sols. t s cosdr t comd Backlud tr. T t ratd solutos ar : o α α β γ α Quasdtrmats! a kd o NC dtrmats O Glad-Rtak t a sd soluto:

15 Quas-dtrmats Quas-dtrmats ar ot just a NC ralato o commutatv dtrmats ut ratr rlatd to vrs matrcs. or a y matr X j ad t vrs Y y j o X uas-dtrmat o X s drctly dd y X y j Rcall tat X Y j A C B commutatv lmt dt X j j som actor dt X j X : t matr otad rom X dlt -t ro ad j-t colum A A B CA B CA A B CA B X CA B CA CA B W ca also d uas-dtrmats rcursvly

16 Quas-dtrmats d ductvly as ollos j j j j j j j j j j j j j j X X X : : : X X X X X X j j covt otato y y

17 Eplct Atya-Ward asat solutos o NC Ya s. GG O O O O Glso-H-Nmmo. arxv:79.69 Ya s matr au varat Glso-Gu GHN T Backlud tr. s ot just a au tr. ut a o-trval o! O

18 W could rat varous solutos o NC ASY. rom a smpl sd soluto y us t prvous Backlud tr. α γ A sd soluto: `` p lar o '' Proo s mad smply y us spcal dtts o uasdtrmats NC aco s dtts ad a omolocal rlato Glso-Nmmo s drvatv ormula tc. otr ords ``NC Backlud trs ar dtts o uasdtrmats. o β NC statos NC No-ar pla-avs commo atur commutatv Backlud lor-dm.!

19 3. Itrprtato rom NC tstor tory I ts scto v a or o t Backlud trs. rom t vpot o NC tstor tory. NC tstor tory as dvlopd y svral autors Kapust-Kutsov-Orlov Takasak Haauss ctld-popov Bra-ajd Wat d r s NC Pros-Ward corrspodc t sol. sp. o ASY ad ``NC olomorpc vctor udl o a NC tstor spac.

20 NC Pros-Ward corrspodc ar systms o ASY ``NC ol. Vc. dl.. / P Patc matr : ; O ; O Takasak W av oly to actor a v patc matr to ad to t ASY lds. Brko actorato or Rma-Hlrt prolm ASY au lds ar rproducd

21 Or o NC Atya-Ward AW asat sols. -t AW asat or t Patc matr T cas rlato s drvd rom: P ; ; ; ; m l OK!

22 Or o NC Atya-Ward AW asat sols. T -t AW asat or t Patc matr T Brko actorato lads to: Udr a au ts soluto cocds t t uasdtrmats sols! P ; ;! P O O O OK!

23 Or o t Backlud trs T Backlud trs ca udrstood as t adjot actos or t Patc matr: actually: T -tr. lads to T -tr. s drvd t a sular au tr. β : : C B PC C P P B B P γ β : P B C B C P β γ α a o γ : s B s β a C C k OK! k O k - compot o T prvous -tr! β T prvous -tr! γ

24 4.Cocluso ad scusso NC tral s ASY r-dm. AH OK Tstor OK Backlud tr OK Symmtry Nt Prooud rlato?? va Ward cojctur NC tral s KdV lor-dms. HrarcyOK Quasdtrmats It cosrvd uatts OK mt a ky Eact N-solto solutos OK Symmtry NC Sato s tory Nt Quas-dtrmats ar mportat! c NC ary arou tr. Salm-Hassa-Sdd Quas-dtrmats ar mportat! EtoGladRtak GlsoNmmoOtaSoomaTammaacarla maksullr-hoss H..