NC Solitons and Integrable Systems

Size: px

Start display at page:

Download "NC Solitons and Integrable Systems"

ゆきひらつくとの
5 years ago
Views:

1 t-sl-ual Ya-lls Basd maly o C.Glso Glaso H.Nmmo Glaso ``Backlud trasorms ad t tya-ward asat or o-commutatv NC at-sl-dual S Ya-lls Y s. Proc. Roy. Soc arxv:8. p-t. µ ν µν θ µν θ

2 NC

3 Ward : SY H&Todap-t/48 NC It au roup NC S Hp-t/57 NC KP NC SY 4 N Summard H Ya s orm p-t/69 NC Ward s cral 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

4 4-dm. SY. GGN µ ν 3 µν µν : µν µ ν ν µ µ ν 3 3

5 Rducto to NC KdV rom NC SY :NC SY. GG 4 t Rducto codtos uu u u u u & :NC KdV.! u

6 T NC KdV. as tral-lk proprts: posssss t cosrvd dsts: σ 3 rs rs u rs u θ 4 r rsr : : s as act N-solto solutos: u N W W W : W... s s r j θ s! j 3 pξ α a p ξ α ξ α α tα s t θ

7 Rducto to NC NS rom NC SY t :NC SY. GG Rducto codtos & :NC NS.!

8 NC SY NC SY. 3 4

9 .Backlud trasorms or NC SY s. NC SY. NC Ya s uatos NC Ya s uatos Sd soluto Quasdtrmats NC statos o-lar pla avs ad so o.

10 NC SY ad Ya s uato GG NC SY : ν µ µ ν ν µ µν θ θ θ θ θ µν. p : θ θ θ ν µ µν ν µ µν r s µν µ ν ν µ ν µ θ :

11 NC SY ad Ya s uato GG NC SY. ca rrtt as ollos I d Ya s matr: t ota rom t trd.: :.. * * * * tc tc :Ya s. T soluto rproduc t au lds as K

12 Backlud tr. or NC Ya s. GG Ya s matr ca rparamtrd as ollos T t ollo to kd o trs. lav 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 tya- Ward solutos d d γ γ o o o α α γ α : : γ γ o

14 Gratd solutos NC tya-ward sols. t s cosdr t comd Backlud tr. T t ratd solutos ar : o α α γ α Quasdtrmats! NC Glad-Rtak t a sd soluto:

15 Quas-dtrmats Quas-dtrmats X j Y y j X uasdtrmat X j y j dt X j j dt X X j : X j j j X j j X j j j j X j j j j j j covt otato

16 Quas-dtrmats : : : X X X X X X j j B C C B C B C B C B C B X Y C B X C.

17 Eplct tya-ward asat solutos o NC Ya s. GG 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!

18 W could rat varous solutos o NC SY. rom a smpl sd soluto y us t prvous Backlud tr. α γ sd soluto: `` p lar o '' Proo s mad smply y us spcal dtts o uasdtrmats NC aco s dtts ad a omolocal rlato tc. otr ords ``NC Backlud trs ar dtts o uasdtrmats. o NC statos NC No-ar pla-avs &

19 SY : NC SY. Ya s orm Quas-dtrmat : : γ o α α γ α

20 3. Itrprtato rom NC tstor tory NC NC : Kapust-Kutsov-rlov Takasak Haauss ctld-popov Bra-ajd NC Pros-Ward 4 NC SY NC QT???

21 5 l m ± λ µ ralty 4 :

22 NC Pros-Ward NC SY ``NC.. / Patc matr Takasak : or : NC SY. Ya s orm P : ; ;

23 NC tya-ward W asat T -t W asat or t Patc matr T Brko actorato lads to: Udr a au t a smpl uasdtrmats sols! P ; ;! P

24 NC tya-ward W asat -t W asat or t Patc matr T rcurso rlato s drvd rom: P ; ; ; ; m l

25 o α α γ α 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. : : C B PC C P P B B P γ : P B C B C P γ α a o γ : s B s a C C k k k - compot o : γ

26 : Quas-dtrmat P ; ; : : C B PC C P P B B P γ

27 4. Cocluso ad scusso NC SY Backlud sd sol. Quas-dtrmats N Pla av? QT BH

28 SY SY K K va Ward KdV K K N K Quas-dtrmats Quas-dtrmats SW

t Z

t Z t Z 1 11 1 1 1 1 1 1 1 1 1 1 1 1 UU 1 as a s 1 1 1 1 1 1 G q w e r q q w e r a s 1 1 1 1 1 1 1 a s a gg g q w e r 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1