Kinks in a Ginzburg-Landaucampus.usal.es/~mpg/General/TalkWMTFT4(2011).pdf · The Model Trial Orbits and Basic Kinks General solutions Kinks in a Ginzburg-Landau non-linear S2-sigma

The ModelTrial Orbits and Basic Kinks

General solutions

Kinks in a Ginzburg-Landaunon-linear S2-sigma model

A. Alonso Izquierdo1, M.A. Gonzalez Leon1, J. Mateos Guilarte2

and M. de la Torre Mayado2

1Departamento de Matemática Aplicada and IUFFyM (Universidad de Salamanca, Spain)2Departamento de Física Fundamental and IUFFyM (Universidad de Salamanca, Spain)

WMTFT4, Salamanca, 2011

September 16, 2011A. Alonso Izquierdo, M.A.G.L., J. Mateos Guilarte and M. de la Torre Mayado Kinks in a Ginzburg-Landau non-linear S2-sigma model


General solutions

Outline

1 The Model

2 Trial Orbits and Basic Kinks

3 General solutionsMore Basic KinksComposite Kinks

A. Alonso Izquierdo, M.A.G.L., J. Mateos Guilarte and M. de la Torre Mayado Kinks in a Ginzburg-Landau non-linear S2-sigma model


General solutions

Introduction

Continuing on the point where my colleague Alberto Alonso has finished histalk, i.e. the non-linear S2-sigma model given by the (1+1)-dimensionalfield-theory equivalent to the “Neumann potential” of Classical Mechanics:

S[~χ] =

∫d2y

{12∂µ~χ · ∂µ~χ − V(~χ)

}~χ = (χ1, χ2, χ3) ; y0 = ct, y1 = y ; ~χ · ~χ = χ2

1 + χ22 + χ2

3 = R2

characterized by the quadratic function:

V(χ1, χ2, χ3) =12(α2

1 χ21 + α2

2 χ22 + α2

3 χ23

)A. Alonso Izquierdo, M.A.G.L., J. Mateos Guilarte. Phys. Rev. Lett. 101, 131602 (2008).A. Alonso Izquierdo, M.A.G.L., J. Mateos Guilarte. Phys. Rev. D79 125003 (2009).A. Alonso Izquierdo, M.A.G.L., M. de la Torre Mayado. SIGMA 6 (2010), 017.



General solutions

I will speak now about a continuation of this work, replacing the quadraticterm by a quartic (Ginzburg-Landau type) one. Using non-dimensionalvariables, fields,

~Φ : R1,1 −→ S2 , ~Φ(xµ) = φ1(xµ)~e1+φ2(xµ)~e2+φ3(xµ)~e3 , ~ea·~eb = δab

~Φ · ~Φ = φ21 + φ2

2 + φ23 = R2

and coupling constants, the action reads:

S[~Φ] =m2

κ

∫dt dx

{12∂µ~Φ · ∂µ~Φ− V(φ1, φ2, φ3)

}

V(φ1, φ2, φ3) =12

(3∑

a=1

α2aφ

2a − 1

)2

+12

3∑b=1

η2bφ

2b (1)

We assume maximal anisotropy in the quartic terms, i.e.:

α21 > α2

2 > α23 > 0

A. Alonso Izquierdo, M.A.G.L., J. Mateos Guilarte, M. de la Torre Mayado, JHEP 8 (2010)1-29.



General solutions

Figure 1: Right) Graphics of V(φ1, φ2, φ3) as the radial coordinate over S2.



General solutions

The field equations of the system are:

∂2φa

∂t2 −∂2φa

∂x2 +∂V∂φa

= 0 ⇒

∂2φa

∂t2 −∂2φa

∂x2 = −2α2aφa

(3∑

b=1

α2bφ

2b − 1

)2

− η2aφa + λφa (2)

with a = 1, 2, 3, and λ being the Lagrange multiplier forcing the constraint

λ =1

R2

3∑a=1

(−(∂tφa)2 + (∂xφa)2 + ~Φ · ~gradV

)



General solutions

Kink solutions ≡ non-singular solutions of the field equations (2) suchthat their energy density has a space-time dependence of the form:ε(t, x) = ε(x− vt), where v is some velocity vector.

The energy functional is:

E[~Φ] =

∫dx(

12∂t~Φ · ∂t~Φ +

12∂x~Φ · ∂x~Φ + V(~Φ)

)=

∫dx ε(t, x) (3)

Lorentz invariance of the model implies that it suffices to know thet-independent solutions ~Φ(x) in order to obtain the kinks of the model:~Φ(t, x) = ~Φ(x− vt). For static configurations (3) reduces to:

E[~Φ] =

∫dx

(12

d~Φdx· d~Φ

dx+ V(~Φ)

)=

∫dx ε(x)

and the PDE system (2) becomes the following system of three ordinarydifferential equations:

d2φa

dx2 = −2α2aφa

(3∑

b=1

α2bφ

2b − 1

)2

− η2aφa + λφa a = 1, 2, 3 (4)



General solutions

Equations (4) can be re-interpreted as the Newton equations of the so-called“Analogous Mechanical System", with the following identifications:

Field Model ⇔ Mechanical System

x (spatial coordinate) ⇔ Mechanical time τ

φa(x), field configurations ⇔ Paths in S2: Xa(τ)

Field theory: V(φ1, φ2, φ3)-function

⇔ Mechanical potential:−V(X1,X2,X3)

Energy functional ⇔ Mechanical action func-tional



General solutions

A better understanding of the characteristics of the V-function can beobtained writing the model directly on S2, using the substitution:

φ23 = R2 − φ2

1 − φ22

and re-defining the parameters and variables in a simpler way:

S[φ1, φ2] =m2

κ

∫dt dx {TS2 − VS2(φ1, φ2)}

TS2 =12

(∂µφ1∂

µφ1 + ∂µφ2∂µφ2 +

(φ1∂µφ1 + φ2∂µφ2)(φ1∂µφ1 + φ2∂

µφ2)

R2 − φ21 − φ2

2

)

VS2(φ1, φ2) =12(φ2

1 + σ2φ22 − α

)2+γ

2φ2

1 +β

2φ2

2 (5)

up to additive constants.



General solutions

where:

σ2 =α2

2 − α23

α21 − α2

3, α =

1− α23 R2

α21 − α2

3, γ =

η21 − η2

3

(α21 − α2

3)2 , β =η2

2 − η23

(α21 − α2

3)2

α21 > α2

2 > α23 > 0 ⇒ 0 < σ2 < 1

Thus the model apparently depends only in four significative parametersσ, α, γ and β.

Following our experience with the “Neumann” model, where the separabilityof the analogous mechanical system in sphero-conical coordinates over S2

was the responsible of the obtention of the complete kink solutionsmanifold, and taking into account the famous dictum of Jacobi:



General solutions

C.J.K. Jacobi, Vorlesungen über Dynamik, Reiner, Berlin (1884). Chap. 16, page 198.

“The main difficulty in integrating a given set of differential equations is to introducesuitable variables which cannot be found by a general rule. Therefore, we must go inthe opposite direction and, after finding some remarkable substitution, look forproblems to which it could be successfully applied . . . ”



General solutions

We select the points in the parameter-space such that separability insphero-conical coordinates over the S2-sphere is encountered.We introduce sphero-conical coordinates (λ0, λ1, λ2) in R3:

φ21 = λ0

(a1 − λ1)(a1 − λ2)

(a1 − a2)(a1 − a3)

φ22 = λ0

(a2 − λ1)(a2 − λ2)

(a2 − a1)(a2 − a3)(6)

φ23 = λ0

(a3 − λ1)(a3 − λ2)

(a3 − a1)(a3 − a2)

with associated separation constants:

a1 = 0 , a2 = 1− σ2 = σ̄2 , a3 = 1 ⇒ 0 < λ1 < σ̄2 < λ2 < 1

The constraint is simply:

λ0 = φ21 + φ2

2 + φ23 = R2



General solutions

Thus sphero-conical coordinates over S2, (λ1, λ2), verify:

φ21 =

R2

σ̄2 λ1 λ2 , φ22 =

R2

σ2σ̄2 (σ̄2−λ1)(λ2−σ̄2) , φ23 =

R2

σ2 (1−λ1)(1−λ2)

The map is eight-to-one, i.e. each octant of the S2 sphere is mappedone-to-one to the rectangle P2 in the (λ1, λ2) plane.



General solutions

The sphero-conical coordinates distinguish four special points in S2, the fociF1, F2, F3 and F4, mapped in P2 to the point (λ1, λ2) = (σ̄2, σ̄2).Sphero-conical coordinates are related in a direct way with spherical-ellipticcoordinates in S2. Choosing two not-antipodal foci (for instance F1 and F2),(λ1, λ2) can be written as:

λ1 = sin2(

r1 − r2

2R

), λ2 = sin2

(r1 + r2

2R

)being r1 and r2 are the geodesic distances from a given point to F1 and F2,respectively.v = r1−r2

2 and u = r1+r22 are the spherical-elliptic coordinates in S2.



General solutions

Thus the iso-curves λ1 = constant and λ2 = constant, are “ellipses" and“hyperbolas" on S2.



General solutions

The kink equations is separable in sphero-conical coordinates if and only ifthe parameters verify:

β = σ2 (γ + σ̄2R2) (7)

Rearranging the additive constants in such a way that the function VS2 takesthe value zero in its minima, and defining a new constant:

δ2 =2α− γ

2R2 , δ̄2 = 1− δ2

it is possible to write the potential in terms of two single free parameters, σand δ:

VS2(φ1, φ2) =12(φ2

1 + σ2φ22 − δ2R2)2

+12

R2σ2σ̄2φ22 (8)

VS2(λ1, λ2) =R4

2(λ2 − λ1)

((σ̄2 − λ1)(δ2 − λ1)2 + (λ2 − σ̄2)(λ2 − δ2)2)



General solutions

VS2 has 18 critical points but, if 0 < δ < 1, (the more generic situation), onlyfour of them are minima: (φ2

1, φ22, φ

23) = (R2δ2, 0,R2δ̄2).

The finite energy requirement (finite action of the analogous mechanicalsystem) selects between the static and homogeneous solutions the setM ofzeroes (and absolute minima) of VS2 :

M ={

v1 ≡ (R δ , 0,Rδ̄), v2 ≡ (−R δ , 0,Rδ̄), v3 ≡ (R δ , 0,−Rδ̄), v4 ≡ (−R δ , 0,−Rδ̄)}



General solutions

Finally, we will choose the regime:

σ̄2 < δ2

The other possibility, σ̄2 > δ2, is equivalent to this one modulo a π2 rotation

around the φ2-axis.

All the four points v1, v2, v3 and v4 are mapped into the point v ≡ (σ̄2, δ2) inP2.



General solutions

Finite energy forces also the verification of the asymptotic conditions:

limx→±∞

d~Φdx

= 0 , limx→±∞

~Φ ∈M .

The space of finite energy configurations:

C ={

Maps(R, S2)/E < +∞}

=

4⋃i,j=1

Cij

is the union of the disconnected sectors: Cij, where the different sectors arelabeled by the element ofM reached by each configuration at x→ −∞ andx→∞.

If i 6= j Cij the finite energy solutions will be termed as topological kinks,whereas non-topological kinks belong to Cii, i = 1, . . . , 4.The eight-to-one correspondence between S2 and P2 maps all the Cij sectorsinto only one. The asymptotic conditions, in sphero-conical coordinates,read:

limx→±∞

dλ1(x)

dx= lim

x→±∞

dλ2(x)

dx= 0 , lim

x→±∞(λ1(x), λ2(x)) = (σ̄2, δ2) .



General solutions

Trial Orbits and Basic Kinks

Before searching for general kink solutions of the ODE system (4), we shallfind two ones by using to Rajaraman’s trial orbit method. We choose as trialorbit the meridian φ2 = 0

φ21 + φ2

2 + φ23 = R2

∣∣φ2=0 ⇒ φ2

1 + φ23 = R2

Let us introduce polar coordinates in S1:

φ1(x) = R sin θ(x) , φ3(x) = R cos θ(x)

VS2 |φ2=0 = VS1(θ) =R4

2(sin2θ(x)− δ2)2



General solutions

And the ODE system (4) reduces to the single second-order OD equation:

d2θ(x)

dx2 = R2 sin 2θ(x)(sin2 θ(x)− δ2) (9)

The mechanical energy I provides a first-integral for (9):

I =12

(dθdx

)2

+1

R2 U(θ) =12

(dθdx

)2

− R2

2(sin2θ(x)− δ2)2

And the asymptotic conditions derived from the finite energy requirementforce that I = 0. Thus the kink solutions correspond to the quadratures of

dθdx

= ±R(sin2θ − δ2)

which produce two types of analytical outcomes:



General solutions

1.- Polar Meridian Kinks (PMK): θPMK12 (x), θPMK

21 (x), θPMK34 (x), θPMK

43 (x).The indexes stand for the asymptotically connected vacua via the kink.

θPMK12 (x) = arcsin

δ sinh[Rδδ̄(x− x0)]√cosh2[Rδδ̄(x− x0)]− δ2

(10)

θPMK21 (x) = θPMK

12 (−x) , θPMK34 (x) = θPMK

12 + π , θPMK43 (x) = θPMK

34 (−x)

φPMK1 (x; ε1) =

(−1)ε1 Rδ sinh[Rδδ̄(x− x0)

]√cosh2 [Rδδ̄(x− x0)

]− δ2

, φPMK2 (x) = 0

φPMK3 (x; ε3) =

(−1)ε3 Rδ̄ cosh[Rδδ̄(x− x0)


]− δ2

, ε1, ε3 = 0, 1

ε3 = 0 ⇒ C12 if ε1 = 1, and C21 if ε1 = 0.ε3 = 1 ⇒ C34 if ε1 = 1, and C43 if ε1 = 0.



General solutions

Field profiles for ε1 = 0 = ε3, x0 = 0 (left). Orbits (red curves) in C21, C12 and C43, C34 (right).



General solutions

2.- Tropical Meridian Kinks (TMK): θTMK14 (x), θTMK

41 (x), θTMK23 (x),

θTMK32 (x).

θTMK41 (x) = arccos

δ̄ sinh[Rδδ̄(x− x0)]√cosh2[Rδδ̄(x− x0)]− δ̄2

(11)

θTMK14 (x) = θTMK

41 (−x) , θTMK32 (x) = θTMK

41 (x) + π , θTMK23 (x) = θTMK

32 (−x)

φTMK1 (x;κ1) =

(−1)κ1 Rδ cosh[Rδδ̄(x− x0)


]− δ̄2

, φTMK2 (x) = 0

φTMK3 (x;κ3) =

(−1)κ3 Rδ̄ sinh[Rδδ̄(x− x0)


]− δ̄2

, κ1, κ3 = 0, 1

κ3 = 0 ⇒ C41 if κ1 = 0, and C14 if κ1 = 1.κ3 = 1 ⇒ C32 if κ1 = 0, and C23 if κ1 = 1.



General solutions

Field profiles for κ1 = 0 = κ3, x0 = 0 (left). Orbits (red curves) in C41, C14 and C23, C32

(right).

Energy densities: (left) PMK in C12 and C21. (right) TMK kinks in C23 and C14.A. Alonso Izquierdo, M.A.G.L., J. Mateos Guilarte and M. de la Torre Mayado Kinks in a Ginzburg-Landau non-linear S2-sigma model


General solutions

More Basic KinksComposite Kinks

General solutions

The Energy functional for the static solutions (action functional of theanalogous mechanical system) in sphero-conical coordinates reads:

E(λ1, λ2) =

∫dx

[12

g11

(dλ1

dx

)2

+12

g22

(dλ2

dx

)2

+ VS2(λ1, λ2)

]

g11 = g11(λ1, λ2) =R2(λ2 − λ1)

4λ1(σ̄2 − λ1)(1− λ1)

g22 = g22(λ1, λ2) =R2(λ2 − λ1)

4λ2(λ2 − σ̄2)(1− λ2)



General solutions


The equations of the kink solutions can be reduced to a first-order system ofODE if a solution, W(λ1, λ2), of the PDE:

VS2(λ1, λ2) =12

(g11(∂W∂λ1

)2

+ g22(∂W∂λ2

)2)

(12)

is found. (12) is no more than the time-independent Hamilton-Jacobiequation of the analogous mechanical problem (for zero mechanical energy),such that W(λ1, λ2) is the Hamilton’s characteristic function. Separabilityimplies the existence of solutions of the form:

W(λ1, λ2) = W1(λ1) + W2(λ2)

General integration of (12) equation involves hyper-elliptical integrals, butthe quadratures reduce to simple irrational integrals taking into account theasymptotic conditions:

limx→±∞

dλ1(x)

dx= lim

x→±∞

dλ2(x)

dx= 0 , lim

x→±∞(λ1(x), λ2(x)) = (σ̄2, δ2) .

(13)forced by the finite energy requirement.



General solutions


Separability reduces the PDE to the system:

dW1

dλ1= (−1)ε1

R3

2(δ2 − λ1)√λ1(1− λ1)

,dW2

dλ2= (−1)ε2

R3

2(δ2 − λ2)√λ2(1− λ2)

.

And the solution is given by:

W(λ1, λ2) =(−1)ε1 R3

2

(√λ1(1− λ1) + (1− 2δ2) arctan

√1− λ1

λ1

)

+(−1)ε2 R3

2

(√λ2(1− λ2) + (1− 2δ2) arctan

√1− λ2

λ2

)ε1, ε2 = 0, 1



General solutions


The Bogomolnyi arrangement of the energy functional:

E(λ1, λ2) =

∫dx

12

2∑i=1

gii

(dλi

dx− gii ∂W

∂λi

)2

+

∫dx

2∑i=1

∂W∂λi

dλi

dx

shows that the absolute minima of the energy functional are the solutions ofthe first-order ODE system

dλ1

dx= g11(λ1, λ2)

∂W∂λ1

,dλ2

dx= g22(λ1, λ2)

∂W∂λ2

Explicitly:

dλ1

dx= −R(−1)ε1

2(σ̄2 − λ1)(δ2 − λ1)√λ1(1− λ1)

λ1 − λ2(14)

dλ2

dx= −R(−1)ε2

2(σ̄2 − λ2)(δ2 − λ2)√λ2(1− λ2)

λ2 − λ1(15)



General solutions


Outline

1 The Model





General solutions


More Basic Kinks. The orbit λ2 = δ2 solves the equation (15) and theequation (14) on this orbit can readily be integrated. We find kinks that liveon the tropical “ellipses”.

λTK1 (x) =

σ̄2 sinh2 [Rσσ̄(x− x0)]

cosh2 [Rσσ̄(x− x0)]− σ̄2, λTK

2 = δ2 (16)

TK orbit in the P2 rectangle displayed as a solid red line (left). TK energy density (right)



General solutions


The inverse image of (16) in S2 (κ1, κ2, κ3 = 0, 1)

φTK1 (x;κ1) =

(−1)κ1 Rδ sinh [Rσσ̄(x− x0)]√cosh2 [Rσσ̄(x− x0)]− σ̄2

φTK2 (x;κ2) =

(−1)κ2 R√δ2 − σ̄2√

cosh2 [Rσσ̄(x− x0)]− σ̄2(17)

φTK3 (x;κ3) =

(−1)κ3 Rδ̄ cosh [Rσσ̄(x− x0)]√cosh2 [Rσσ̄(x− x0)]− σ̄2

Topological kink profiles (17) (left) The orbits (right).



General solutions


Outline

1 The Model





General solutions


Degenerate families of polar zone non-topological kinks We first considerthe polar zone: λ2 ∈ (σ̄2, δ2). The first-order equations (14)-(15) written indifferential form:

(−1)ε1 dλ1

2R(σ̄2 − λ1)(δ2 − λ1)√λ1(1− λ1)

+(−1)ε2 dλ2

2R(σ̄2 − λ2)(δ2 − λ2)√λ2(1− λ2)

= 0

(−1)ε1λ1dλ1

2R(σ̄2 − λ1)(δ2 − λ1)√λ1(1− λ1)

+(−1)ε2λ2dλ2

2R(σ̄2 − λ2)(δ2 − λ2)√λ2(1− λ2)

= −dx∫(−1)ε1 dλ1

(σ̄2 − λ1)(δ2 − λ1)√λ1(1− λ1)

+

∫(−1)ε2 dλ2

(σ̄2 − λ2)(δ2 − λ2)√λ2(1− λ2)

= 2R C1∫(−1)ε1λ1dλ1

(σ̄2 − λ1)(δ2 − λ1)√λ1(1− λ1)

+

∫(−1)ε2λ2dλ2

(σ̄2 − λ2)(δ2 − λ2)√λ2(1− λ2)

= −2R(x−C0)

It is possible to integrate these equations and even to obtain explicitexpressions for the solutions using typical “Calculus I - rules”. First weperform classical Euler changes of variables and we re-define some of theconstants:

s1 =

√1− λ1

λ1, s2 =

√1− λ2

λ2; 0 < σ2

2 =δ̄2

δ2 < s22 < σ2

1 =σ2

σ̄2 < s21 < +∞



General solutions


The quadratures (18)-(18) are thus rationalized:

1δ2σ̄2R

2∑i=1

∫(−1)εi (1 + s2

i ) dsi

(σ21 − s2

i )(σ22 − s2

i )= C1 ,

1δ2σ̄2R

2∑i=1

∫(−1)εi dsi

(σ21 − s2

i )(σ22 − s2

i )= x− C0

(18)Plugging the simple fraction decompositions we obtain:

2∑i=1

∫(−1)εi dsi

σ22 − s2

i= δ2 R (x−C0−σ̄2 C1) ,

2∑i=1

∫(−1)εi dsi

σ21 − s2

i= σ̄2 R (x−C0−δ2 C1) (19)

Finally, direct integration provides a two-parametric family of kinksolutions, in implicit form:

arccoths1

σ2+ arccoth

s2

σ2= R δ δ̄ (x− x0) (20)

arccoths1

σ1+ arctanh

s2

σ1= Rσ σ̄ (x− x0 + ζ) (21)

in terms of the parameters: x0 = C0 + σ̄2 C1 and ζ = (σ̄2 − δ2)C1,x0, ζ ∈ R. The value of the integration constant x0 fixes the “center of mass"of the kink, whereas different values of ζ determine the different kink orbitsin S2 uniquely.



General solutions


In order to calculate explicit expressions for the solutions we remember theaddition formulas for the hyperbolic functions:

tanh (arccoth p + arccoth q) =p + q

1 + pq=

1tanh (arccoth p + arctanh q)

that allow to invert equations (20-21)

σ22(s1 + s2)

σ2(σ22 + s1s2)

= tanh[R δ δ̄(x− x0)

]≡ t1 (22)

σ1(σ21 + s1s2)

σ21(s1 + s2)

= tanh [Rσ σ̄(x− x0 + ζ)] ≡ t2 (23)

that can be written as a algebraic linear system in terms of the “Vietavariables”: A = s1 + s2 , B = s1s2.

σ2A− t1B = t1σ22 , σ1t2A− B = σ2

1



General solutions


Solving by using Cramer’s rule we find the kink families in Vieta variables:

A(x; x0, ζ) =(δ2 − σ̄2)t1

δσ̄(σδt1t2 − σ̄δ̄

) , B(x; x0, ζ) =σδ̄(σδ − δ̄σ̄t1t2

)σ̄δ(σδt1t2 − σ̄δ̄

)Finally, inverting the changes of variables we find explicit expressions forthe family of Polar Zones kink solutions:

λPZK1 (x; x0, ζ) =

11 + s2

1(x; x0, ζ)

λPZK2 (x; x0, ζ) =

11 + s2

2(x; x0, ζ)



General solutions


φPZK1 (x; x0, ζ) =

(−1)κ1 Rδ(δ̄σ̄ − δσt1t2

)√(δ2 − σ̄2)2 t2

1 + σ2σ̄2t21t2

2 − 2δδ̄σσ̄t1t2 + δ2δ̄2(24)

φPZK2 (x; x0, ζ) =

(−1)κ2 R(δ2 − σ̄2)t1√

1− t22√

(δ2 − σ̄2)2 t21 + σ2σ̄2t2

1t22 − 2δδ̄σσ̄t1t2 + δ2δ̄2

(25)

φPZK3 (x; x0, ζ) =

(−1)κ3 Rδ̄(δσ − δ̄σ̄t1t2)√(δ2 − σ̄2)2 t2

1 + σ2σ̄2t21t2

2 − 2δδ̄σσ̄t1t2 + δ2δ̄2(26)



General solutions


Orbits in S2 for three PZK solutions: (1) (x0 = 0, ζ = 0), red. (2) (x0 = −1, ζ = 1), brown.(3) (x0 = 1, ζ = 6), blue. (right).



General solutions


Non-topological kink energy densities for:(1) (x0 = 0, ζ = 0), red. (2) (x0 = 1, ζ = 2),brown. (3) (x0 = −1, ζ = 5), blue.



General solutions


Degenerate families of tropical zone topological kinks. A similarcalculations, in the case δ2 < λ2 < 1, leads to the Tropical-zone kinkfamilies:

φTZK1 (x; x0, ζ) =

(−1)κ1 Rδ(δσ t1 − σ̄δ̄ t2

)√(σσ̄t1 − δδ̄t2

)2+ (δ2 − σ̄2)2

(27)

φTZK2 (x; x0, ζ) =

(−1)κ2 R(δ2 − σ̄2)√

1− t21√(

σσ̄t1 − δδ̄t2)2

+ (δ2 − σ̄2)2(28)

φTZK3 (x; x0, ζ) =

(−1)κ3 Rδ̄(δσ t2 − σ̄δ̄ t1

)√(σσ̄t1 − δδ̄t2

)2+ (δ2 − σ̄2)2

(29)



General solutions




General solutions


Orbits in S2 for three TZK solutions: (1) (x0 = 0, ζ = 0), red. (2) (x0 = −1, ζ = −2), brown.(3) (x0 = −1, ζ = −7), blue. (right).



General solutions


Topological kink energy densities for:(1) (x0 = 0, ζ = 0), red. (2) (x0 = −1, ζ = −2), brown.(3) (x0 = −1, ζ = −7), blue.



General solutions


Two final remarks:

1. Kink energy sum rules:

E (PZK) = E(PMK) + E(TK)

E (TZK) = E(TMK) + E(TK)

2. Kink Stability:The topological basic kinks TMK and TK (and their anti’s) are stable.The topological basic kinks PMK (and their anti’s) are unstable.The composite non-topological PZK kinks are unstable.The composite topological kinks TZK are stable.



General solutions


Thank you very much


Documents

Kinks in a Ginzburg-Landaucampus.usal.es/~mpg/General/TalkWMTFT4(2011).pdf · The Model Trial Orbits and Basic Kinks General solutions Kinks in a Ginzburg-Landau non-linear S2-sigma