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 Kink-oscillon collisions

vviinn == 00::11

44 Kink-oscillon collisions

vviinn == 00::22

Sine-Gordon kink-breather collision

vviinn == 00::1155

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

Bounce:Bounce:

KK ¹¹KK !! KK ¹¹KKvviinn == 00::2277

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

H=-0.5 elastic recoil

H=0 2 bounces + excitations

BoundaryBoundary 44 model: Static solutions

BoundaryBoundary 44 model: phase diagrammodel: phase diagram

vcr =

q1 ¡ 4[(1 +H)3=2 + (1 ¡H)3=2]¡2