Domain wall dynamics in expanding spaces
F ancisco J. Cao,1,2,*Elías Zamo a-Sille o,3,† and Niu ka R. Quin e o3,‡
1Depa amen o de Física A ómica, Molecula y Nuclea , Uni e sidad Complu ense de Mad id,
A enida Complu ense s/n, E-28040 Mad id, Spain
2LERMA, Obse a oi e de Pa is, Labo a oi e Associé au CNRS UMR 8112, 61 A enue de l’Obse a oi e, 75014 Pa is, F ance
3Depa amen o de Física Aplicada I, E.U.P., Uni e sidad de Se illa, Vi gen de Á ica 7, 41011 Se illa, Spain
共Recei ed 1 Decembe 2005; published 12 May 2006兲
We s udy he e ec s on he dynamics o kinks due o expansions and con ac ions o he space. We show ha
he p opaga ion eloci y o he kink can be adiaba ically uned h ough slow expansions and/o con ac ions,
while i s wid h is gi en as a unc ion o he eloci y. We also analyze he case o as expansions and/o
con ac ions, whe e we a e no longe on he adiaba ic egime. In his case he kink mo es mo e slowly a e an
expansion-con ac ion cycle as a consequence o he loss o ene gy h ough adia ion. All hese e ec s a e
nume ically s udied in he nonlinea Klein-Go don equa ions 共bo h o he sine-Go don and o he
␾
4po en-
ial兲, and hey a e also s udied wi hin he amewo k o he collec i e coo dina e e olu ion equa ions o he
wid h and he cen e o mass o he kink. These collec i e coo dina e e olu ion equa ions a e ob ained wi h a
p ocedu e ha allows us o conside e en he case o la ge expansions and/o con ac ions.
DOI: 10.1103/PhysRe E.73.056603 PACS numbe 共s兲: 05.45.Y , 45.20.Jj, 02.30.J
I. INTRODUCTION
Domain walls a e he in e ace be ween wo egions
whe e he sys em is close o wo di e en s able o me a-
s able s a es. Thei o ma ion and dynamics a e ele an o
many domains o physics as solid s a e physics 共magne ic
ma e ials兲关1,2兴, liquid c ys als science 关3兴, plasma physics
关4,5兴, and cosmology 共phase ansi ions in he ea ly uni e se兲
关6,7兴. Spa ially ex ended sys ems wi h mo e han one s able
o me as able s a e can p esen domain walls, as is he case o
sys ems ha ha e unde gone a i s o de phase ansi ion. An
expansion o con ac ion o hese sys ems can be ealized
h ough an ex e nal ac ion on he sys em 共magne ic ma e i-
als, liquid c ys als, plasmas兲, o hey a e al eady p esen in
he sys em dynamics 共phase ansi ions in he ea ly uni e se兲.
The kink solu ions o he one-dimensional nonlinea
Klein-Go don equa ion o po en ials wi h wo o mo e
s able s a es 共as sine-Go don o
␸
4兲p o ide models o he
p opaga ion o plane domain walls in he di ec ion pe pen-
dicula o he domain wall. These models ha e been success-
ully used in se e al con ex s as in he desc ip ion o he
dynamics o domain walls in e odis o i e ma e ials 关2兴,
and he magne ohyd odynamic mode ajec o y in e e sed-
ield pinch expe imen s 关4,5兴. On he o he hand, expansions
and con ac ions can be ealized in magne ic ma e ials
h ough he mechanical ac ion on he ma e ial 关8兴, and in
plasmas h ough con inemen 关9兴; while in he case o phase
ansi ions in he ea ly uni e se, he expansion is al eady
p esen in he sys em dynamics. These expansions and/o
con ac ions can be pa ame ized wi h a scale ac o a共 兲
when hey a e homogeneous in space 关10兴. The cha ac e is ic
imes o he ime a ia ion o a共 兲can be sho e o longe
han he cha ac e is ic imes o he kink dynamic.
Wi h hese ac s in mind, in his pape we conside kinks
o he nonlinea Klein-Go don equa ions o he sine-Go don
and he
␾
4po en ials 共as a pa adigma ic examples o in e-
g able and nonin eg able nonlinea Klein-Go don equa ions,
espec i ely兲, and we s udy hei dynamics in slow and as
expanding and/o con ac ing spaces.
In o de o achie e his pu pose, ou pape is o ganized as
ollows: In Sec. II we in oduce he nonlinea Klein-Go don
equa ions in expanding and/o con ac ing spaces, and de i e
he con inui y equa ions o he ene gy and momen um den-
si ies. In he nex sec ion, Sec. III, we use he Rice Ansa z
关11兴in hese con inui y equa ions, and we educe he p ob-
lem wi h an in ini e numbe o deg ees o eedom o an
app oxima e desc ip ion in e ms o wo collec i e coo di-
na es 共CC兲, he cen e and he wid h o he kink. In Sec. IV,
we s udy he kink dynamic in bo h slow and as expanding
and/o con ac ing spaces using he collec i e coo dina e
e olu ion equa ions, and in eg a ing nume ically he com-
ple e nonlinea Klein-Go don equa ions. We i s conside
he case o he slow expansion and/o con ac ion wi hin he
adiaba ic app oxima ion, and la e we discuss he as expan-
sion and/o con ac ion case and he depa u es om he
adiaba ici y ha eme ge. Finally, in he conclusions, we sum-
ma ize and discuss he esul s.
The p e ious sec ions a e complemen ed wi h wo Appen-
dixes whe e some echnical de ails ela ed o expanding
and/o con ac ing spaces a e e iewed, and he equi alence
be ween se e al collec i e coo dina es app oaches 关12–14兴 o
his p oblem is shown.
II. KLEIN-GORDON EQUATION
IN EXPANDING SPACES
The Klein-Go don equa ion in nonexpanding spaces can
be ob ained by ex emizing he ac ion de ined by he La-
g angian densi y,
S=
冕
d dx L=
冕
d dx
冋
1
2
␾
2−1
2
␾
x
2−U共
␾
兲
册
.共1兲
*Elec onic add ess: [email p o ec ed]
†Elec onic add ess: elias@eule .us.es
‡Elec onic add ess: niu ka@eule .us.es
PHYSICAL REVIEW E 73, 056603 共2006兲
1539-3755/2006/73共5兲/056603共8兲©2006 The Ame ican Physical Socie y056603-1
In an expanding 共o con ac ing兲space wi h one spa ial
dimension he space- ime dis ance be ween e en s is gi en
by
ds2=d 2−a2共 兲dx2,共2兲
whe e a共 兲is he scale ac o ha gi es he dila ion o he
physical spa ial dis ances 共see Appendix A兲. Thus, he physi-
cal spa ial dis ance be ween wo poin s wi h coo dina es x1
and x2is
d12共 兲=a共 兲兩x2−x1兩.共3兲
The Klein-Go don equa ion in his expanding space can
be ob ained no ing ha a e he spa ial dila ion dx→a共 兲dx
he ac ion o a Klein-Go don ield is
S=
冕
d dxa共 兲L=
冕
d dxa共 兲
冋
1
2
␾
2−1
2
␾
x
2
a2共 兲−U共
␾
兲
册
,
共4兲
whe e he subsc ip s and xindica e he ime and space
pa ial de i a i es, espec i ely, and U共
␾
兲is he nonlinea
Klein-Go don po en ial.
The Eule -Lag ange equa ion o he ield
␾
,
⳵
⳵
␦
关a共 兲L兴
␦
␾
+
⳵
⳵
x
␦
关a共 兲L兴
␦
␾
x
−
␦
关a共 兲L兴
␦
␾
=0, 共5兲
gi es he equa ion o mo ion
␾
+H共 兲
␾
−
␾
xx
a2共 兲+dU
d
␾
=0, 共6兲
wi h H=a /a.
We ha e o keep in mind ha he physical dis ance is
a共 兲兩x2−x1兩and no 兩x2−x1兩共 he la e is usually called he
como ing dis ance兲. Thus, when a共 兲g ows he space expe-
iences an elonga ion and when a共 兲dec eases he space ex-
pe iences a con ac ion.
We de ine he ene gy densi y
␳
E共 ,x兲, he momen um den-
si y
␳
P共 ,x兲, and he momen um cu en jP共 ,x兲, as he ime-
ime, space- ime, and space-space con a a ian componen s,
espec i ely, o he ene gy-momen um enso . We ob ain
␳
E=1
2
␾
2+1
2
␾
x
2
a2+U共
␾
兲,共7兲
␳
P=− 1
a2
␾
␾
x,共8兲
jP=1
a2
冉
1
2
␾
2+1
2
␾
x
2
a2−U共
␾
兲
冊
.共9兲
The ene gy-momen um conse a ion 共in he co a ian sense兲
gi es he ollowing ela ions:
⳵␳
E
⳵
+H共
␳
E+a2jP兲+
⳵␳
P
⳵
x=0, 共10兲
⳵␳
P
⳵
+3H
␳
P+
⳵
jP
⳵
x=0, 共11兲
ha also can be ob ained using he equa ion o mo ion 共6兲.
This also implies ela ions be ween he o al ene gy E
=兰adx
␳
E, he o al momen um P=兰adx
␳
P, and he o al
momen um cu en JP=兰adxj
P. They a e gi en by
dE
d +Ha2JP=0 共12兲
关p o ided
␳
P共 ,⬁兲=
␳
P共 ,−⬁兲兴, and
dP
d +2HP =0 共13兲
关p o ided jP共 ,⬁兲=jP共 ,−⬁兲兴. No e ha o a space ha is
nei he expanding no con ac ing 关i.e., a共 兲=1兴Eand Pa e
conse ed quan i ies 关1兴.
III. COLLECTIVE COORDINATES ANALYSIS
FOR THE PERTURBED KLEIN-GORDON EQUATION
In he p e ious sec ion we ha e shown ha he momen-
um and he ene gy a e no longe conse ed quan i ies in
expanding and/o con ac ing spaces, ins ead hey sa is y he
con inui y equa ions 共12兲and 共13兲, espec i ely. We a e in-
e es ed in he e olu ion o he kink which we desc ibe
h ough wo collec i e coo dina es, i s wid h and i s cen e o
mass posi ion. The e o e, we neglec he possible ene gy
ans e o o he deg ees o eedom.
In o de o ob ain he equa ions o mo ion o hese wo
CCs we use he Rice Ansa z 关11兴in he con inui y equa ions
共12兲and 共13兲关wi h he ene gy and momen um gi en by Eqs.
共7兲–共9兲兴, and we s udy wo pa icula nonlinea po en ials,
he sine-Go don 共SG兲po en ial U共
␾
兲=1−cos共
␾
兲, and he
␾
4
po en ial U共
␾
兲=1
4共1−
␾
2兲2. The domain walls in hese sys-
ems a e ep esen ed by kinklike solu ions ha desc ibe he
ansi ion be ween wo egions in di e en minima o he
po en ial. Fo he sine-Go don po en ial he Rice Ansa z as-
sumes he unc ion
␾
共x, 兲= 4 a c an
再
exp
冋
x−X共 兲
l共 兲
册
冎
,共14兲
whe eas o he
␾
4, he Rice Ansa z eads
␾
共x, 兲= anh
冋
x−X共 兲
l共 兲
册
,共15兲
whe e X共 兲and l共 兲 ep esen in bo h cases he cen e and he
wid h o he kink, espec i ely, in he co-mo ing ame. No e
ha o he unpe u bed sys em 关 ake, in Eq. 共6兲,a共 兲=1兴 he
kinklike solu ions a e ep esen ed by Eqs. 共14兲and 共15兲wi h
X共 兲= and l共 兲=ls=l0冑1− 2, being and ls he cons an
eloci y and wid h o he kink, espec i ely, and l0=1 共l0
=冑2兲 he wid h o he soli on a es o SG 共
␸
4兲. The ime
dependence o he scale ac o a共 兲共expansions and/o con-
ac ions兲implies changes in he wid h o he kink, as we
show la e , jus i ying he choice o he Rice Ansa z.
Inse ing he Ansa z, Eq. 共14兲o 共15兲, in he exp essions
CAO, ZAMORA-SILLERO, AND QUINTERO PHYSICAL REVIEW E 73, 056603 共2006兲
056603-2
o
␳
P,
␳
E, and jPand in eg a ing o e x, we ob ain
P共 兲=M0l0X
˙
a共 兲l,共16兲
E共 兲=M0l0
2
aX
˙2
l+
␣
M0l0
2
al
˙2
l+M0
2
冉
l0
al +al
l0
冊
,共17兲
JP共 兲=M0l0
2
X
˙2
al +
␣
M0l0
2
l
˙2
al +M0
2a2
冉
l0
al −al
l0
冊
,共18兲
espec i ely; whe e
␣
=
␲
2/12 and M0=8 o he sine-
Go don and
␣
=共
␲
2−6兲/12 and M0=2冑2/3 o
␸
4. These
exp essions a e eplaced in he con inui y equa ions 共12兲and
共13兲, gi ing he ollowing sys ems o o dina y di e en ial
equa ions 共ODE兲 o X共 兲and l共 兲:
X
˙=a共 兲P共 兲l共 兲
M0l0
,共19兲
dP
d =−2HP,共20兲
␣
关l
˙2−2Hll
˙−2ll
¨兴=l2
l0
2
冉
1+a2共 兲P2
M0
2
冊
−1
a2,共21兲
whe e he do s deno e he de i a i e wi h espec o .I
he ini ial condi ions a e hose o a kink wi h an ini ial posi-
ion X共0兲and an ini ial eloci y X
˙共0兲, his implies
l共0兲=l0冑1−X
˙2共0兲and l
˙共0兲=0, and P共0兲is de e mined
by Eq. 共19兲a =0. The same collec i e coo dina es e olu-
ion equa ions a e ob ained using he Lag angian me hod
关14,15兴o he gene alized a elling wa e Ansa z 共GTWA兲
关12,16兴共based on p ojec ion echniques兲, as i is shown in
Appendix B.
This se o e olu ion equa ions, Eqs. 共19兲–共21兲, in ol es
he a iables X共 兲and l共 兲 h ough he momen um P共 兲de-
ined in Sec. II. No e ha he equa ion o he momen um is
linea , and he e o e i can be sol ed exac ly yielding
P共 兲=P共0兲
a2共 兲,共22兲
wi h P共0兲=X
˙共0兲M0l0/冑1−X
˙2共0兲. The o he wo equa ions
a e nonlinea and coupled, and hey a e analyzed in de ail in
he nex sec ion.
IV. KINK DYNAMICS IN EXPANDING
AND/OR CONTRACTING SPACES
Fi s o all, le us ema k ha he main physical a iables
associa ed wi h he kink p opaga ion a e he physical cen e
o he kink, Xphys共 兲=a共 兲X共 兲, i s physical wid h lphys共 兲
=a共 兲l共 兲, and i s physical eloci y
Vphys共 兲=HXphys共 兲+a共 兲dX共 兲
d .共23兲
The i s e m on he hs is he con ibu ion o he kink e-
loci y due o he expansion o con ac ion o he space 共i.e., i
u ns o ze o when he space inishes i s elonga ion o con-
ac ion H=0兲. On he o he hand, he second e m is he
peculia eloci y o he kink wi h espec o he p opaga ing
space,
Vpec共 兲=a共 兲dX共 兲
d .共24兲
Fo he sake o cla i y, we analyze sepa a ely he cases o
slow and as expansion and/o con ac ion.
A. Slow expansion and/o con ac ion (adiaba ic egime)
No e om Eqs. 共19兲–共21兲 ha he adiaba ic app oxima ion
implies
␣
关l
˙2−2Hll
˙−2ll
¨兴Ⰶ1
a2.共25兲
Hence, om Eq. 共21兲we ob ain he ela ion
lphys共 兲=l0冑1−Vpec
2共 兲,共26兲
ha links he physical wid h and he peculia eloci y o he
kink. Fu he mo e, om Eqs. 共19兲and 共22兲–共26兲we ob ain
Vpec共 兲=P共0兲
冑M0
2a2共 兲+P2共0兲.共27兲
F om he p e ious Eqs. 共26兲and 共27兲we see ha an expan-
sion decele a es he peculia mo ion o he kink and makes i
wide , asymp o ically
lim
a共 兲→⬁
Vpec共 兲= 0, lim
a共 兲→⬁
lphys共 兲=l0.共28兲
On he o he hand, a con ac ion accele a es he peculia mo-
ion o he kink and makes i sha pe , asymp o ically
lim
a共 兲→0
Vpec共 兲= 1, lim
a共 兲→0
lphys共 兲=0. 共29兲
The adiaba ici y condi ions can be ob ained om Eq. 共25兲
and equi e only slow expansions and/o con ac ions
兩H兩Ⰶ⍀R
1−Vpec
2,共30兲
兩H
˙兩Ⰶ⍀R
2
共1−Vpec
2兲2,共31兲
whe e ⍀R=1/共冑
␣
l0兲is he so-called Rice equency o ze o
eloci y, ⍀R=冑12/
␲
=1.10..., o SG and ⍀R=冑6/共
␲
2−6兲
=1.24..., o
␸
4.
We ha e s udied he dynamics o di e en expansion and
con ac ion a es, bo h o he sine-Go don and he
␾
4po en-
ials. In pa icula , Fig. 1 共and also Fig. 2兲shows he esul s
DOMAIN WALL DYNAMICS IN EXPANDING SPACES PHYSICAL REVIEW E 73, 056603 共2006兲
056603-3
o an expansion ollowed by a con ac ion pa ame ized by
he unc ion
a共 兲=1+⌬a
2 anh
冉
− 0
⌬
冊
−⌬a
2 anh
冉
− 1
⌬
冊
,共32兲
wi h ⌬a, 0, 1, and ⌬ cons an s. 共⌬a ep esen s he changes
in a; 0and 1, he imes when he expansion and con ac ion
ake place, espec i ely, and ⌬ is he cha ac e is ic ime in-
e al whe e he changes in a ake place.兲
In Fig. 1 we show he e ec s on a kink o a slow expan-
sion ollowed by a slow con ac ion ha e i y he adiaba ic
condi ions 关H⬃⌬a/⌬ =0.1 and H
˙⬃⌬a/共⌬ 兲2=0.01兴. In his
case, when he adiaba ic condi ions a e e i ied, bo h he
CCs equa ions and he adiaba ic app oxima ions a e in good
ag eemen wi h he exac esul s, especially o he cen e o
he kink Xphys共 兲and i s peculia eloci y Vpec共 兲共see Fig. 1兲.
I is impo an o no e ha he ag eemen is good e en when
he a ia ions o aa e la ge.
This implies ha he main e ec s o a slow expansion
and/o con ac ion a e he change o he wid h and he speed
o he p opaga ion o he kink, ollowing he adiaba ic ela-
ions 共26兲and 共27兲. The e o e, hese esul s show ha he
speed o a kink can be uned by slowly expanding and/o
con ac ing he space.
No e ha he CCs e olu ion equa ions go u he and a e
able o p edic he small oscilla ions o he kink wid h 关23兴
p oduced by he slow expansion and/o con ac ion 共see Fig.
1兲. The e o e, hey can compu e de ia ions om adiaba ici y
excluding he adia i e e ec s 共because hey in ol e he
ans e o ene gy o o he deg ees o eedom兲.
I is impo an o s ess ha he collec i e coo dina e e o-
lu ion equa ions ob ained wi h an Ansa z wi h ixed wid h
and a a iable cen e o he kink does no p edic any a ia-
ion o he kink speed due o expansions and/o con ac ions.
The e o e, an Ansa z ha allows he wid h o he kink o
e ol e, like he Rice Ansa z, is an essen ial ing edien in
o de o ob ain he co ec a ia ion o he speed o he kink
unde expansions and/o con ac ions. I is in e es ing o e-
ma k ha , in he
␾
4model, he oscilla ions o he wid h a e
ela ed o exci a ions o he in e nal mode 关11兴, while in he
sine-Go don equa ion i has been shown ha he exci a ion o
ce ain phonons can imply oscilla ions in he shape o he
kink 关17,18兴.
FIG. 1. Sine-Go don kink dy-
namics in a slow expanding and
la e con ac ing space. Top-le
panel: scale ac o a共 兲, Eq. 共32兲
wi h ⌬a=1, 0=50, 1=150, and
⌬ =10. Top- igh panel: spa ial
p o ile o he kink
␾
共x, ix兲a
imes ix=0 共solid line兲, ix=100
共long-dashed line兲, and ix=200
共dashed line兲. Th ee ollowing
panels: posi ion o he cen e o
he kink Xphys共 兲, peculia kink e-
loci y Vpec共 兲, and wid h o he
kink lphys共 兲as unc ions o ime
gi en by he nume ical simula-
ions o he pa ial di e en ial
equa ion 关PDE兴共solid line兲, he
nume ical solu ions o he CCs
equa ion 共long-dashed line兲, and
he adiaba ic app oxima ion
共dashed line兲. Bo om- igh panel:
momen um o he kink P共 兲exac
solu ion. Ini ial condi ions: a kink
wi h X共0兲=10 and X
˙共0兲=0.1.
CAO, ZAMORA-SILLERO, AND QUINTERO PHYSICAL REVIEW E 73, 056603 共2006兲
056603-4
On he o he hand, in gene al, he applicabili y o he
pe u ba i e app oaches is es ic ed o slow and small ex-
pansions and/o con ac ions. The pe u ba i e app oaches
ew i e Eq. 共6兲in he o m
␾
−
␾
xx+dU/d
␾
=
⑀
⬅−H共 兲
␾
−关1/a2共 兲−1兴
␾
xx, whe e
⑀
is ea ed as a small pe u ba ion.
The e o e, hei ange o applicabili y is limi ed no only o
small Hbu also o small 共1/a2−1兲, i.e., slow and small
expansions and/o con ac ions.
B. Fas expansion and/o con ac ion (nonadiaba ic egime)
When he expansion and/o con ac ion is as e he adia-
ba ic app oxima ion b eaks, and nonadiaba ic e ec s appea
as adia ion 共compa e he kink p o iles in Figs. 1 and 2兲
and as a change in he inal wid h and eloci y o he kink
关 ha a e no longe hose p edic ed by Eqs. 共26兲and 共27兲兴.
共See Fig. 2, ha co esponds o he nonadiaba ic egime
H⬃⌬a/⌬ =2 and H
˙⬃⌬a/共⌬ 兲2=4.兲The CCs e olu ion
equa ions p edic s ong oscilla ions in he eloci y and he
wid h o he kink. In he nume ical in eg a ion o he com-
ple e e olu ion equa ions 关Eq. 共6兲兴 he oscilla ions a e indeed
p esen in he wid h. Howe e , he e a e o he deg ees o
eedom o which he ene gy can be ans e ed. This esul s
in he damping o he oscilla ions h ough adia ion emis-
sion. The loss o ene gy can also be easily shown no ing ha
a e an expansion-con ac ion cycle he kink mo es slowe
共see Fig. 2兲.
We would like o s ess ha all he p e ious commen s
abou slow and as expansions and/o con ac ions apply
bo h o he sine-Go don and he
␾
4po en ial. We ha e pe -
o med nume ical simula ions and compa isons o he adia-
ba ic app oxima ion, he CCs e olu ion equa ions, and he
ull e olu ion equa ion o bo h po en ials ob aining analo-
gous esul s.
V. CONCLUSIONS
We ha e s udied he e ec s on a kink o slow and as
expansions and/o con ac ions o he media, bo h o he
sine-Go don and o he
␾
4equa ions. We ha e used he Rice
Ansa z 关11兴in he con inui y equa ions o he momen um
and he ene gy, in o de o ob ain he e olu ion equa ions o
he wid h and he cen e o a kink in an expanding and/o
con ac ing space. In gene al, his se o ODEs canno be
FIG. 2. Sine-Go don kink dy-
namics in a as expanding and
la e con ac ing space. Top-le
panel: scale ac o a共 兲, Eq. 共32兲
wi h ⌬a=1, 0=40, 1=100 and
⌬ =0.5. Top- igh panel: spa ial
p o ile o he kink
␾
共x, ix兲a
imes ix=37.5 共solid line兲, ix
=75 共long-dashed line兲, and ix
=150 共dashed line兲. The h ee ol-
lowing panels: posi ion o he cen-
e o he kink Xphys共 兲, peculia
kink eloci y Vpec共 兲, and wid h o
he kink lphys共 兲as unc ions o
ime gi en by he nume ical simu-
la ions o he PDE 共solid line兲,
he nume ical solu ions o he
CCs equa ion 共long-dashed line兲,
and he adiaba ic app oxima ion
共dashed line兲. Bo om- igh panel:
momen um o he kink P共 兲exac
solu ion. Ini ial condi ions: a kink
wi h X共0兲=10 and X
˙共0兲=0.1.
DOMAIN WALL DYNAMICS IN EXPANDING SPACES PHYSICAL REVIEW E 73, 056603 共2006兲
056603-5

sol ed analy ically and he solu ion mus be ob ained nu-
me ically. Howe e , his e ec i e desc ip ion in e ms o he
collec i e coo dina es usually gi es ele an in o ma ion and
insigh o he e olu ion and he phenomena in ol ed
关16,19–22兴. In his case he CCs app oxima ion is able o
desc ibe e y accu a ely he dynamics o he kink when he
p ocess is slow, e en when he a ia ions o aa e la ge. We
ha e ound ha he main e ec s o a slow expansion and/o
con ac ion a e he change o he wid h and he eloci y o
he kink, ollowing he adiaba ic ela ions 共26兲and 共27兲.In
addi ion, he collec i e coo dina e e olu ion equa ions o
he wid h and he cen e o he kink a e able o ake in o
accoun he small de ia ions om adiaba ici y 共excluding ex-
plici ly he adia i e e ec s, because hey a e ela ed o ans-
e ence o ene gy o o he deg ees o eedom兲.
An impo an consequence o he adiaba ic ela ions is
ha he speed o a kink can be uned by a slowly expansion
and/o con ac ion o space. Thus, his esul p o ides a
mechanism o con ol he speed o a kink whene e we can
expand and con ac he media whe e i p opaga es.
On he o he hand, as expansions and/o con ac ions
b eak he adiaba ic app oxima ion, gi ing ise o nonadia-
ba ic e ec s, as, o example, adia ion, ha imply a change
in he inal wid h and eloci y o he kink 共i.e., hey a e no
longe hose p edic ed by he adiaba ic ela ions兲. The collec-
i e coo dina e e olu ion equa ions p edic s ong oscilla-
ions in he wid h o he kink. In he ull p oblem, hese
oscilla ions a e also p esen . Howe e , he e a e o he de-
g ees o eedom o which he ene gy o hese oscilla ions
can be ans e ed. This esul s in emission o adia ion and
in he damping o hese oscilla ions. In his sense, he as
expansion and/o con ac ion o space p o ides a me hod o
implemen a as change in he speed o he kink, and also o
gene a e phonons.
ACKNOWLEDGMENTS
We acknowledge inancial suppo om he Minis e io de
Ciencia y Tecnología o Spain unde G an s No. BFM2003-
02547/FISI 共F.J.C.兲, No. NAN2004-09087-C0303 共F.J.C.兲,
and No. FIS2005-973 共E.Z.S., N.R.Q.兲, and om he Jun a
de Andalucía h ough he p ojec s FQM-0207 and 00481
共E.Z.S., N.R.Q.兲. In addi ion, E.Z.S. hanks he Uni e si y o
Se illa o i s inancial suppo .
APPENDIX A: EXPANDING SPACES
This appendix includes some echnical de ails ela ed o
expanding spaces.
1. Me ic
The me ic in an homogeneous and iso opous la space
is gi en by
ds2=d 2−a2共 兲dx2=g
␮
␯
dx
␮
dx
␯
,共A1兲
whe e a共 兲is he scale ac o o he expansion and/o con-
ac ion, he me ic enso is
g
␮
␯
=
冉
10
0−a2共 兲
冊
.共A2兲
I s in e se, g
␮
␯
, de ined by g
␮
␯
g
␯
␭=g
␮
␭=
␦
␮
␭共wi h
␦
␮
␭a K o-
necke del a兲is
g
␮
␯
=
冢
10
0−1
a2共 兲
冣
.共A3兲
The con ac ion wi h g
␮
␯
lowe s indices, and he con ac ion
wi h g
␮
␯
aises indices.
2. In a ian elemen o space- ime olume
De ining
g⬅− De g
␮
␯
,共A4兲
i can be shown 共关10兴pages 98 and 99兲 ha
冑gd dx 共A5兲
is an in a ian olume elemen unde gene al coo dina e
ans o ma ions. 共d dx ans o ms wi h he Jacobian, while
冑g ans o ms wi h he in e se o he Jacobian.兲
The e o e he app op ia e ela ion be ween he ac ion S,
and he Lag angian densi y Lis
S=
冕
d dx冑gL.共A6兲
This de ini ion implies ha i he ac ion is in a ian unde
gene al coo dina e ans o ma ions he Lag angian densi y is
also in a ian . In ou case 冑g=a共 兲.
3. Ene gy-momen um enso
The ene gy-momen um enso is
T
␮
␯
=
⳵
␮
␾
⳵
␯
␾
−g
␮
␯
L.共A7兲
Some use ul ela ions o calcula e i s componen s a e
⳵
x
␾
=gx
␴
⳵
␴
␾
=gxx
⳵
x
␾
=−
␾
x
a2,共A8兲
⳵
␾
=
⳵
␾
=
␾
.共A9兲
The ime- ime componen o he ene gy-momen um enso o
ene gy densi y is
␳
E⬅T =
⳵
␾
⳵
␾
−g
冋
1
2
␾
2−1
2
␾
x
2
a2−U共
␾
兲
册
=1
2
␾
2+1
2
␾
x
2
a2+U共
␾
兲,共A10兲
he space- ime componen o momen um densi y is
␳
P⬅Tx =T x =
⳵
␾
⳵
x
␾
−g xL=− 1
a2
␾
␾
x,共A11兲
and he space-space componen o momen um cu en is
CAO, ZAMORA-SILLERO, AND QUINTERO PHYSICAL REVIEW E 73, 056603 共2006兲
056603-6
jP⬅Txx =1
a2
冉
1
2
␾
2+1
2
␾
x
2
a2−U共
␾
兲
冊
.共A12兲
APPENDIX B: EQUIVALENCE AMONG DIFFERENT
CC APPROACHES
In his appendix we show ha he e olu ion equa ions o
he collec i e coo dina es ob ained wi h he de i a ion o he
momen um and he ene gy 关13兴共 ha we used in Sec. III兲a e
he same as hose ob ained wi h he Lag angian me hod
关14,15兴and wi h he so-called GTWA 关12,16兴共based on p o-
jec ion echniques兲. In all cases, we use he Rice Ansa z as an
app oxima ed kinklike solu ion o he ollowing pe u bed
nonlinea Klein-Go don equa ion:
␾
−
␾
xx
a2共 兲=−dU
d
␾
−关H共 兲+
␤
共 兲兴
␾
+z共x, ,
␾
兲,共B1兲
whe e
␤
共 兲is a ime-dependen damping coe icien and
z共x, ,
␾
兲 ep esen s a gene ic pe u ba ion on he sys em.
No e ha o
␤
=0 and z共x, ,
␾
兲=0 we eco e he sys em
共6兲, in oduced in Sec. II.
1. Gene alized Lag angian o malism
In oducing a new ime a iable
␶
⬅
冕
0
d ⬘
c共 ⬘兲,共B2兲
whe e c共 兲is gi en by
c共 兲⬅exp
再
冕
0
关H共 ⬘兲+
␤
共 ⬘兲兴d ⬘
冎
=a共 兲
a共0兲exp
再
冕
0
␤
共 ⬘兲d ⬘
冎
,
共B3兲
Eq. 共B1兲becomes he dissipa ionless equa ion,
␾
␶␶
C2共
␶
兲−
␾
xx
A2共
␶
兲=−dU
d
␾
+Z共x,
␶
,
␾
兲,共B4兲
wi h C共
␶
兲=c关 共
␶
兲兴,A共
␶
兲=a关 共
␶
兲兴, and Z共x,
␶
,
␾
兲=z关x, 共
␶
兲,
␾
兴.
The e olu ion equa ion 共B4兲can be ob ained om he
ac ion
S=
冕
d
␶
dx L=
冕
d
␶
dx
再
1
2
␾
␶
2−1
2
C2共
␶
兲
A2共
␶
兲
␾
x
2
−C2共
␶
兲关U共
␾
兲+M共x,
␶
,
␾
,
␾
x兲兴
冎
共B5兲
Lbeing he Lag angian densi y, and M共x,
␶
,
␾
,
␾
x兲an
“Eule -Lag ange in eg al” o Z共x,
␶
,
␾
兲. Some o he possible
in eg a ions a e
共i兲 he
␾
unc ional in eg al
M共x,
␶
,
␾
兲=−
冕
D
␾
Z共x,
␶
,
␾
兲,共B6兲
and
共ii兲 he ollowing in eg a ion o he pa icula case
whe e Z共x,
␶
,
␾
兲=F共x,
␶
兲G共
␾
兲关15兴:
M共x,
␶
,
␾
,
␾
x兲=
␾
xG共
␾
兲
冕
x0
x
dx⬘F共x⬘,
␶
兲.共B7兲
Once we de ine he Lag angian
L=
冕
–⬁
+⬁
dx L,共B8兲
he Eule -Lag ange e olu ion equa ions o he collec i e
a iables a e ob ained inse ing a gi en Ansa z in his La-
g angian and using ei he exp ession 共B6兲o 共B7兲. This p o-
cedu e is he gene aliza ion o he Lag angian o malism de-
eloped in 关14兴.
Using he Rice Ansa z de ined by 共14兲and 共15兲共 o he
SG and
␾
4po en ials, espec i ely兲and a e some s aigh -
o wa d calcula ions, we ob ain he Lag angian as a unc ion
o ou collec i es a iables and hei de i a i es wi h espec
o
␶
L共X,X⬘,l,l⬘兲=M0l0
2l共X⬘兲2+
␣
M0l0
2l共l⬘兲2
−1
2M0C2共
␶
兲
冉
l
l0
+l0
A2共
␶
兲l
冊
−C2共
␶
兲
冕
−⬁
+⬁
dx M共x,
␶
,
␾
,
␾
x兲.共B9兲
Replacing L共X,X⬘,l,l⬘兲in o he Eule -Lag ange equa ions,
d
d
␶
冉
⳵
L
⳵
X⬘
冊
=
⳵
L
⳵
X,共B10兲
d
d
␶
冉
⳵
L
⳵
l⬘
冊
=
⳵
L
⳵
l,共B11兲
and ew i ing he equa ions o mo ion in he ime a iable ,
we ge
X
˙=a共 兲P共 兲l共 兲
M0l0
,共B12兲
dP
d =−共
␤
+2H兲P−1
a共 兲
再
冕
−⬁
+⬁
d
␪
z共X+
␪
l, ,
␾
兲
␾
␪
冎
,
共B13兲
␣
关l
˙2−2共
␤
+H兲ll
˙−2ll
¨兴=l2
l0
2
冉
1+a2共 兲P2
M0
2
冊
−1
a2
+2l2
M0l0
冕
−⬁
+⬁
d
␪
z共X+
␪
l, ,
␾
兲
␪␾
␪
,
共B14兲
whe e
␪
=关x−X共 兲兴/land he do s deno e he de i a i e wi h
espec o . These a e he e olu ion equa ions o he collec-
DOMAIN WALL DYNAMICS IN EXPANDING SPACES PHYSICAL REVIEW E 73, 056603 共2006兲
056603-7
i e a iables ob ained o ei he o he wo p e ious exp es-
sions o M共x, ,
␾
,
␾
x兲Eq. 共B6兲o Eq. 共B7兲共in hei espec-
i e egime o alidi y兲. Taking in hese equa ions
␤
=0 and
z=0, we ob ain he Eqs. 共19兲–共21兲de i ed in Sec. III.
2. GTWA
In o de o apply he GTWA we ew i e Eq. 共B1兲as
␾
˙=
␺
,
␺
˙=
␾
xx
a2共 兲−dU
d
␾
−关H共 兲+
␤
共 兲兴
␺
+z共x, ,
␾
兲.共B15兲
The p ocedu e o ob ain he CCs equa ions wi h he GTWA
consis s o inse ing ou speci ic unc ional o m o
␾
, Eq.
共14兲 o sine-Go don o 共15兲 o
␾
4po en ial, in o Eq. 共B15兲,
mul iplying he i s equa ion by
⳵
␺
/
⳵
Xand he second one
by
⳵
␾
/
⳵
X, aking hei di e ence and in eg a ing o e x,
equa ing he esul wi h ze o; and epea ing he same p oce-
du e wi h
⳵
␺
/
⳵
land
⳵
␾
/
⳵
l. This gi es us he same o dina y
di e en ial equa ions o X共 兲,P共 兲, and l共 兲as hose ob ained
in he p e ious subsec ion 关see Eqs. 共B12兲–共B14兲兴.
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关23兴The wid h and he cen e o he kink posi ion o he PDE
esul s a e ob ained i ing he Rice Ansa z, Eq. 共14兲o Eq.
共15兲, o he kink p o ile gi en by he PDE e olu ion 共6兲.
CAO, ZAMORA-SILLERO, AND QUINTERO PHYSICAL REVIEW E 73, 056603 共2006兲
056603-8