lecture 3: short summary - portal ifsclaf/shnir/lecture_4.pdf · e[Á]= r for a simple modelfor a...
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Lecture 3: Short SummaryLecture 3: Short Summary
sinesine--Gordon equation:Gordon equation: uutttt ¡¡ uuxxxx ++ ssiinn uu == 00
Kink solution:Kink solution: Á = §4 arctan exp (x¡ x0)
QQ == 1122¼¼
11RR¡¡11ddxx @@ÁÁ@@xxTopological charge:Topological charge:
@¡@+Á = sinÁLightLight--cone coordinates:cone coordinates: xx§§ == 1122 ((xx §§ tt))
BBääcklundcklund transformation:transformation:
@+à = @+Á¡ 2¸ sinµÁ+ Ã2
¶; @¡Ã = ¡@¡Á+ 2¸ sin
µÁ¡ Ã2
¶
sinesine--Gordon model: 2Gordon model: 2--soliton interactionssoliton interactions
Á2 = 4arctan
"v sinh xp
1¡v2cosh vtp
1¡v2
#KKKK--collisioncollisionv=0.8
x
tKKKK--collisioncollisionv=0.8
¸2 =1
¸1; v =
1¡ ¸211 + ¸21
; ¸1 > 0
x
t
ÁÁ22 == 44 aarrccttaann
""ssiinnhh vvttpp
11¡¡vv22vv ccoosshh xxpp
11¡¡vv22
##
Breather:Breather:
v =i!p1¡ !2
Á2 = 4 arctan
"p1¡ !2!
sin!t
coshp1 ¡ !2x
#Á2
Á2
tx
sinesine--Gordon model: 2Gordon model: 2--soliton interactionssoliton interactions
KKKK--collisioncollision
xt
Asymptotic:
Á2
The phase shift:
ÁÁ22 == 44 aarrccttaann
··ssiinnhh °°vvtt
vv ccoosshh °°xx
¸;; °° ==
11pp11 ¡¡ vv22
== 44 ttaann¡¡11··ee°°vvtt¡¡llnn vv ¡¡ ee¡¡°°vvtt¡¡llnn vv
ee°°xx ++ ee¡¡°°xx
¸tt !! ¡¡11
ÁÁ22 ¼¼ ÁÁKK··((xx ++ vv
µµtt ++±±tt
22
¶¶°°
¸++ ÁÁ ¹¹KK
··((xx ¡¡ vv
µµtt ¡¡ ±±tt22
¶¶°°
¸Asymptotic: tt !! ++11
ÁÁ22 ¼¼ ÁÁKK··((xx ++ vv
µµtt ¡¡ ±±tt22
¶¶°°
¸++ ÁÁ ¹¹KK
··((xx ¡¡ vv
µµtt ++±±tt
22
¶¶°°
¸
±±tt == 22llnn vv
°°vv== 22ppvv22 ¡¡ 11 llnn vv
sinesine--Gordon model: Lax pair formulationGordon model: Lax pair formulationRecall: Lax pair is givenby two linear equations
Ãx = LÃ; Ãt = AÃ Ã =
µÃ11 Ã12Ã21 Ã22
¶½Ãxt = Ltà + LÃt;Ãtx = Axà +AÃx:
Ltà + LAà = Axà +ALÃ; Lt ¡Ax = [A;L]Zero curvature condition
sinesine--Gordon:Gordon:L = i¸
µ1 00 ¡1
¶+i
2
µ0 uxux 0
¶= i¸ ¢ ¾3 + i
2ux ¢ ¾1; ¸ 2 C
A =cosu
4i¸
µ1 00 ¡1
¶+1
4i¸
µ0 ¡ii 0
¶=cosu
4i¸¢ ¾3 + 1
4i¸¢ ¾2
Lt =iutx2¢ ¾1; Ax = ¡ 1
4i¸ux sinu ¢ ¾3 + 1
4i¸ux ¢ ¾2
[A;L] =i
4¸¢ ¾2 ¡ i
4¸¢ ¾3 + i
2sinu ¢ ¾1
in 0th order in λ
iutx2¢ ¾1 = i
2sinu ¢ ¾1
sine-Gordon equation is recovered!
InteractionInteraction bertweenbertween thethe solitonssolitons
2-solitons solutions
ÁÁ == ÁÁKK ((xx ¡¡ dd)) ++ ÁÁkk((xx ++ dd)) ¡¡ 22¼¼ EEiinntt((dd)) == EEKKKK ((dd)) ¡¡ 22MM ¼¼ 3322ee¡¡22dd
Homework: Prove it!Linear perturbations of the kink:Linear perturbations of the kink:
ÁÁ == ÁÁKK ((xx)) ++ ´((xx;; tt))
ÁÁKK == 44 aarrccttaann eexx
@@22tttt ´((xx;; tt)) ¡¡ @@22xxxx ´((xx;; tt)) ++ ´((xx;; tt)) ccooss ÁÁKK ((xx)) == 00
µµ¡¡ dd
22
ddxx22++ 11 ¡¡ 22
ccoosshh22 xx
¶¶»»((xx)) == !!22»»((xx))´((xx;; tt)) == << »»((xx))eeii!!tt
LinearizedLinearized perturbations of theperturbations of the sGsG kinkkinkµµ¡¡ dd
22
ddxx22++ 11 ¡¡ 22
ccoosshh22 xx
¶¶»»((xx)) == !!22»»((xx))
^aayy^aa »»((xx)) == !!22»»((xx));; ^aa ==dd
ddxx++ttaannhhxx;; ^aayy ==¡¡ dd
ddxx++ttaannhhxx
Vacuum state: ^aa »»00 ´µµdd
ddxx++ ttaannhh xx
¶¶»»((xx)) == 00 »»00((xx)) ==
11
ccoosshh xxZero mode
»»kk((xx)) == ((ttaannhh xx ++ iikk))eeiikkxx;; !!kk ==
pp11 ++ kk22Continuum modes:
Note:Note: a eikx ´µd
dx+ tanhx
¶eikx = »k(x) Reflectionless potential!Reflectionless potential!
ÁÁ == ÁÁKK ((xx)) ++ CC»»00((xx)) == 44 aarrccttaann eexx ++
CC
ccoosshh xx== ÁÁKK ++
CC
22
ddÁÁkk
ddxx¼¼ ÁÁKK
µµxx ++CC
22
¶¶
»»((¡¡11)) == ((¡¡11 ++ iikk))eeiikkxx ;; »»((11)) == ((11 ++ iikk))eeii((kkxx++±±)) ;; eeii±± ==iikk ++ 11
iikk ¡¡ 11
sine-Gordon ↔ massive Thirring model
SS ==
ZZdd22xx
··11
22@@¹¹ÁÁ@@
¹¹ÁÁ ¡¡ ®®¯22((11 ¡¡ ccooss ¯ÁÁ))
¸sine-Gordon model
ThirringThirring modelmodel
MesonMeson statesstates →→ fermionfermion--antianti fermionfermion boundbound statesstates
SolitonSoliton →→ fundamentalfundamental fermionfermion
°°00 == ¾¾11 ;; °°11 == ¡¡ii¾¾22;; °°55 == °°00°°11 == ¾¾33
SS ==
ZZdd22xxhhii ¹¹Ãð°¹¹@@
¹¹Ãà ++mm ¹¹ÃÃÃà ¡¡ gg22(( ¹¹Ãð°¹¹ÃÃ))(( ¹¹Ãð°
¹¹ÃÃ))ii
BosonizationBosonization:: ¯22
44¼¼ ==11
11++gg==¼¼
InvarianciesInvariancies::
mm ¹¹ÃÃ11 ¨ °°5522Ãà == ¡¡ ®®
¯22ee§§iiÁÁ
ÁÁ !! ÁÁ00 == ÁÁ ++ 22¼¼nn¯ ;; ÃÃ !! ÃÃ00 == eeii®®VV ÃÃ;; ÃÃ !! ÃÃ00 == eeii°°55®®AAÃÃ
The topological current of the sineThe topological current of the sine--Gordon modelGordon modelcoincides with thecoincides with the NoetherNoether current of the massivecurrent of the massive ThirringThirring modelmodel
J¹ =12¼"¹º@
ºÁ
jj¹¹ == ii ¹¹Ãð°¹¹@@¹¹ÃÃ
((S.ColemanS.Coleman, 1975), 1975)
EquationEquation SolutionSolution
YESYES
NO!NO!
How do we know if it isHow do we know if it is integrableintegrable or it is a nonor it is a non--integrableintegrable??
Historically, combination ofHistorically, combination of ““experimental mathematicsexperimental mathematics”” ((44) and) andknown analytic solutions (Sknown analytic solutions (S--G), then inverse scattering transform,G), then inverse scattering transform,group theoretic structure (Kacgroup theoretic structure (Kac--Moody Algebras), PainlevMoody Algebras), Painlevéé test.test.
λλ44 ::
SS--G:G:
Does any part ofDoes any part of ““hierarchyhierarchy”” ofof solitonssolitons inin integrableintegrabletheories (Stheories (S--G breather) exist in nonG breather) exist in non--intergrableintergrabletheories?theories?
Solitons vs. Solitary Waves
ÄÄÁÁ ¡¡ ÁÁ0000 ++ ssiinn ÁÁ == 00 ÁÁKK ¹¹KK == §§44 aarrccttaann ((ee¡¡xx++xx00 ))
ÄÄÁÁ ¡¡ ÁÁ0000 ¡¡ ÁÁ ++ ÁÁ33 == 00 ÁÁKK ¹¹KK == §§aa ttaannhh³³mm((xx¡¡xx00))pp
22
´
Topology primer: maps and windings
Kinks in 2d:Kinks in 2d:
Space: ++∞∞--∞∞
Vacuum:+1+1
-- 11Maps:
Circles: SCircles: S11 →→SS11
Space: Vacuum:
Maps:
ÁÁ®® == ((ssiinn '';; ccooss ''))
Topological charge: QQ == 1122
11RR¡¡11ddxx @@ÁÁ@@xx == ÁÁ((11)) ¡¡ ÁÁ((¡¡11))
Circles: SCircles: S11 →→SS11
Vacuum:
Topological charge: QQ == 1122¼¼
22¼¼ZZ00
dd'' ""®®¯ddÁÁ®®
dd''ÁÁ¯
Q=0:Q=0: ÁÁ®® == ((00;; 11))
ÁÁ®® == ((ssiinn '';; ccooss''))Q=1:Q=1:
Q=2:Q=2: ÁÁ®® == ((ssiinn 22'';; ccooss 22''))
Consider a model with scalar field in dConsider a model with scalar field in d--dimdim
Scaling agruments: Derrick’s theorem
EE [[ÁÁ]] ==RRddddxx [[@@¹¹ÁÁ@@
¹¹ÁÁ ++ UU ((ÁÁ))]] == EE22 ++ EE00
Scale transformation:Scale transformation: ~~xx !! ~~yy == ¸~~xx;; @@¹¹ÁÁ((~~xx)) == @@ÁÁ((~~xx))@@xx¹¹
!! ¸@@ÁÁ((¸~~xx))@@((¸xx¹¹))== ¸@
@ÁÁ((~~yy))@@yy¹¹
ddddxx !! dddd((¸xx))¸¡¡dd == ¸¡¡ddddddyyEE [[ÁÁ]] !! ¸22¡¡ddEE22 ++ ¸¡¡ddEE00
Each term is positive. If there is a stationary point of E(Each term is positive. If there is a stationary point of E(λλ)?)?ddEE[[¸ÁÁ]]dd¸ == ((22 ¡¡ dd))¸11¡¡ddEE22 ¡¡ dd¸¡¡dd¡¡11EE00
d=1d=1 d=2d=2 d=3d=3
EE [[ÁÁ]] ==RRddddxx [[@@¹¹ÁÁ@@
¹¹ÁÁ ++ UU ((ÁÁ))]] == EE22 ++ EE00For a simple modelFor a simple model
nontrivial solutions (Enontrivial solutions (E22 ≠≠ 0,0, EE00 ≠≠ 00 ) are possible only in d=1) are possible only in d=1
There are 3 possibilities to evade DerrickThere are 3 possibilities to evade Derrick’’s theorem:s theorem:
•• d=2:d=2: taketake EE00 = 0, then the model is scale= 0, then the model is scale--invariantinvariant•• Extend the model including higher derivatives inExtend the model including higher derivatives in ϕϕ ((SkyrmeSkyrme model inmodel ind=3, babyd=3, baby SkyrmeSkyrme model in d=2,model in d=2, FaddeevFaddeev--SkyrmeSkyrme model in d=3)model in d=3)
•• Extend the model including gauge fields (monopoles in d=3,Extend the model including gauge fields (monopoles in d=3,instantonsinstantons in Euclidean space d=4)in Euclidean space d=4)
~~xx !! ¸~~xx == ~~yy;; AA¹¹((~~xx)) !! ¸AA¹¹((~~yy));; DD¹¹ÁÁ((~~xx)) !! ¸DD¹¹ÁÁ((~~yy));; FF¹¹ºº ((~~xx)) !! ¸22FF¹¹ºº ((~~yy))
EE [[ÁÁ]] ==RRddddxx££jjFF¹¹ºº jj22 ++ jjDD¹¹ÁÁjj22 ++ UU ((ÁÁ))¤¤ == EE44 ++ EE22 ++ EE00EE [[ÁÁ]] !! ¸44¡¡ddEE44 ++ ¸22¡¡ddEE22 ++ ¸¡¡ddEE00
•• d=1:d=1: there arethere are solitonsoliton solutions in thesolutions in the models with gaugemodels with gaugeand scalar fields or in pure scalar models with a potentialand scalar fields or in pure scalar models with a potentialU(U(ϕϕ)) ((KinksKinks).).
•• d=2:d=2: there arethere are solitonsoliton solutions in thesolutions in the models with gaugemodels with gaugeand scalar fieldsand scalar fields ((vorticesvortices)) or in pure scalar modelsor in pure scalar models withoutwithoutpotential U(potential U(ϕϕ)) ((LumpsLumps))..•• d=3:d=3: there arethere are solitonsoliton solutions in thesolutions in the models with gaugemodels with gaugeand scalar fieldsand scalar fields ((monopolesmonopoles))•• d=4:d=4: there arethere are solitonsoliton solutions in thesolutions in the models with gaugemodels with gaugefield onlyfield only ((instantonsinstantons))•• d>4:d>4: there are nothere are no solitonsoliton solutions, higher derivatives aresolutions, higher derivatives arenecessary.necessary.
If we restrict ourselves to the models with quadraticIf we restrict ourselves to the models with quadraticterms in derivatives, there are possibilities:terms in derivatives, there are possibilities:
Alternative:Alternative: one can consider time-dependentstationary configurations!
44 model
LL == 1122@@¹¹ÁÁ@@
¹¹ÁÁ ¡¡ UU ((ÁÁ));; UU ((ÁÁ)) == ¸44
¡¡ÁÁ22 ¡¡ aa22¢¢22
Field equation:Field equation: @@¹¹@@¹¹ÁÁ ++ ddUU
ddÁÁ == 00
Potential energy:Potential energy:
Kinetic energy:Kinetic energy:
VV ==11RR¡¡11ddxx
··1122
³³@@ÁÁ@@xx
´22++UU((ÁÁ))
¸TT == 1122
11RR¡¡11ddxx³³@@ÁÁ@@tt
´22Vacuum:Vacuum: ÁÁ00 == §§aa Static configuration:Static configuration: T=0T=0
EE==VV ==11RR¡¡11ddxxhh11pp22ÁÁ00 §§ppUU ((ÁÁ))ii22 ¨ 11RR
¡¡11ddxxpp22UU ((ÁÁ)) ÁÁ00 ¸ 00Energy bound:Energy bound:
44 model: Applications
Phenomenological theory of second orderPhenomenological theory of second orderphase transitionsphase transitionsA model of theA model of the displacivedisplacive phase transitionsphase transitionsA model ofA model of uniaxialuniaxial ferroelectricsferroelectricsA phenomenological theory of the nonA phenomenological theory of the non--perturbativeperturbative transition intransition inpolyacetylenepolyacetylene chainchainCondensed matter physics: solitary waves in shapeCondensed matter physics: solitary waves in shape--memorymemory
alloysalloysCosmology: model dynamics of the domain walls.Cosmology: model dynamics of the domain walls.Biophysics:Biophysics: solitonsoliton excitations in DNA double helices.excitations in DNA double helices.Quantum field theory: a model example to investigate transitionQuantum field theory: a model example to investigate transitionbetweenbetween perturbativeperturbative and nonand non--perturbativeperturbative sectors of the theory.sectors of the theory.A model of quantum mechanicalA model of quantum mechanical instantoninstanton transitions in doubletransitions in double--well potentialwell potential
44 model: Kink solutionsmodel: Kink solutions
Minimum of the energy:Minimum of the energy:
UU ((ÁÁ)) ==11
22((ÁÁ22 ¡¡ 11))22;; VV ==
11
22
11ZZ¡¡11ddxx
""µµ@@ÁÁ
@@xx
¶¶22++ ((ÁÁ22 ¡¡ 11))22))
##@@ÁÁ
@@xx== §§((11 ¡¡ ÁÁ22)) xx¡¡ xx00 == §§
ZZddÁÁ
11¡¡ ÁÁ22 == §§ ttaannhh¡¡11 ÁÁ
ÁK = tanh(x¡ x0); Á ¹K = ¡ tanh(x¡ x0)kink solution:kink solution:
Energy density: Mass of the kink: MM ==ZZEEddxx == 44
33EE == 11
ccoosshh44((xx ¡¡ xx00))
Topological current:
Topological charge:
J¹ =1
2¼"¹º@
ºÁ; @¹J¹ ´ 0
Q = 12
1R¡1dx@Á@x =
12 [Á(1)¡ Á(¡1)]
Interaction between the kinks
KinkKink--antikinkantikink pair (a=1, m =pair (a=1, m = √√2)2)::
Far away from the pair (somewhere at xFar away from the pair (somewhere at x ≈≈ 0)0)
Interaction energy:Interaction energy:
ÁÁ((xx)) == 11++ttaannhh((xx¡¡RR))¡¡ ttaannhh((xx++RR))
ttaannhh((xx ¡¡ RR)) ¼¼ ¡¡11 ++ 22ee22((xx¡¡RR));;ttaannhh((xx ++ RR)) ¼¼ 11 ¡¡ 22ee¡¡22((xx++RR))
Linear oscillations on the static kink background:Linear oscillations on the static kink background: ÁÁ == ÁÁKK ++ ±±ÁÁ
→→
KinksKinks attractsattracts eacheach otherother withwith thethe forceforce
EEiinntt ¼¼ ¡¡1166ee¡¡22LL;; LL == 22RR
FF == ddEEiinnttddLL ¼¼ 3322ee¡¡22LL
ÄÄÁÁ ¡¡ ÁÁ0000 ¡¡ ¸((aa22 ¡¡ ÁÁ22))ÁÁ == 00 ±± ÄÄÁÁ ¡¡ ±±ÁÁ0000 ++hh44 ¡¡ 66
ccoosshh22 xx))
ii±±ÁÁ == 00
LinearizedLinearized perturbations of theperturbations of the 44 kinkkinkµµ¡¡ dd
22
ddxx22++ 44 ¡¡ 66
ccoosshh22 xx
¶¶»» == !!22»»
Reflectionless potential, again!Reflectionless potential, again!
^aayy ^aa »»((xx)) == !!22»»((xx));; ^aa ==dd
ddxx++ nn ttaannhhxx;; ^aayy == ¡¡ dd
ddxx++ nn ttaannhhxx
[[^aayy ;; ^aa]] ==22nn
ccoosshh22 xx
n=2
Vacuum state:
Zero mode»»((nn))00 ((xx)) ==
11
ccoosshhnn xx^aa »»00 ´
µµdd
ddxx++ nn ttaannhh xx
¶¶»»((nn))00 ((xx)) == 00
Internal mode: ^aayy»»00 == »»11 ==ssiinnhh xx
ccoosshh22 xx!!11 ==
pp33
Continuum:
Homework: Prove it!
»»kk == eeiikkxx¡¡33 ttaannhh22 xx ¡¡ 33iikk ttaannhh xx ¡¡ 11 ¡¡ kk22¢¢
LinearizedLinearized perturbations of theperturbations of the 44 kinkkinkµµ¡¡ dd
22
ddxx22++ 44 ¡¡ 66
ccoosshh22 xx
¶¶»» == !!22»»
Internal mode:
»(n)0 (x) =
1
coshn x; !0 = 0 Zero mode:
^aayy»»00 == »»11 ==ssiinnhh xx
ccoosshh22 xx;; !!11 ==
pp33
Coupling (negative radiation pressure)RRddxx´kk ´00
The 4 kink accelerates towardsthe source of the radiation
44 model: continuum modes:model: continuum modes:
!!22kk ==¡¡44 ++ kk22
¢¢;; »»kk((xx)) == <<
££eeiikkxx¡¡33 ttaannhh22 xx ¡¡ 33iikk ttaannhh xx ¡¡ 11 ¡¡ kk22¢¢¤¤
(I. L. Bogolubsky and V. G. Makhankov (JETP Lett. 24, 12 (1976))
In theIn the 44 model there is a long livedmodel there is a long livednonradiativenonradiative spatially localizedspatially localizedsolution (solution (at least 10at least 10 millionmillion oscillationsoscillations!!)
Oscillon state: 44 model
Gaussian initial data:
ÁÁ((xx;; 00)) == 11 ¡¡ 00::77ee¡¡00::220055xx22
Collective coordinate model:Collective coordinate model:
ÁÁ((xx;; tt)) == 11 ¡¡ AA((tt))ee¡¡³³xxxx00
´22
LL==xx00 == (( __AA))22 ¡¡ 22
33AA44 ¡¡ ¼¼AA33 ¡¡
³³44 ++ 11
33xx2200
´AA22
Anharmonic oscillator with
frequency 0=qq44 ++ 11
33xx00
44 KK collisions: fractal dynamics
M. J. Ablowitz, M. D. Kruskal and J. F. Ladik (SIAM J.Appl. Math. 36 (1979) 421)D. Campbell, J. Schonfeld and C Wingate (Physica 9D (1983) 1)P. Anninos, S. Oliveira and R. A. Matzner (Phys. Rev. D 44 (1991) 1147) etc
KK ¹¹KK !! oosscciilllloonnvviinn == 00::1177
Annihilation:Annihilation:
44 KK collisions: fractal dynamics
Three bounceThree bounceresonance:resonance:
KK ¹¹KK !! KK ¹¹KKvviinn == 00::2244338855
The resonanceThe resonance mechnismmechnism
Resonant energy exchange between the translational and the internal modes: thefirst collision excites the internal mode which takes the kinetic energy of the kinks,the second collision unbinds the pair taking the stored energy back: T = + 2n(D. Campbell, J. Schonfeld and C Wingate Physica 9D (1983) 1)
KinkKink--antikinkantikink collisions on :collisions on :xx 22 [[¡¡11;;11]]
TheThe energy can be stored not only in the internalenergy can be stored not only in the internalmode of the kink, but also in the collective modesmode of the kink, but also in the collective modes
BoundaryBoundary 44 model
Kink solution on :Kink solution on :
xx
00
--11
11
LL == 1122@@¹¹ÁÁ@@
¹¹ÁÁ ¡¡ 1122
¡¡ÁÁ22 ¡¡ 11¢¢22
ÁÁKK ¹¹KK == §§ ttaannhh ((xx ¡¡ xx00)) MM == 4433
Boundary energy:Boundary energy:
Neumann boundary condition
@@xxÁÁ((00;; tt)) == HH
BoundaryBoundarymagnetic fieldmagnetic field
¡¡HHÁÁ((00;; tt))xx 22 [[¡¡11;;11]]
BoundaryBoundary 44 model: Energy functional
EE [[ÁÁ]] ==11
22
00ZZ¡¡11ddxx[[ÁÁxx §§ ((ÁÁ22 ¡¡ 11))]]22 ¨ [[ 11
33ÁÁ33 ¡¡ ÁÁ]]
¯00¡¡11¡¡ HHÁÁ((00;; tt))
EE[[ÁÁ11]] ==22
33¡¡ 2233((11 ¡¡HH))33==22;; EE[[ÁÁ22]] == 22
33++22
33((11 ¡¡HH))33==22;; EE[[ÁÁ33]] == 22
33¡¡ 2233((11 ++HH))33==22
KinkKink--boundary forces:boundary forces: F = 32
µH
4+ e2x0
¶e2x0
repulsion far from the boundaryand attraction near it
repulsion far from the boundaryand attraction near it
x0 < 0
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