PHYSICAL REVIEW E 94, 012221 (2016)
Kink a che induced by a ime-dependen symme ic ield po en ial
Be na do S´
anchez-Rey*
Depa amen o de F´
ısica Aplicada I, E.P.S., Uni e sidad de Se illa, Vi gen de ´
A ica 7, 41011 Se illa, Spain
Jes´
us Casado-Pascual†
F´
ısica Te´
o ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, 41080 Se illa, Spain
Niu ka R. Quin e o‡
Depa amen o de F´
ısica Aplicada I, E.P.S., Uni e sidad de Se illa, Vi gen de ´
A ica 7, 41011 Se illa, Spain
(Recei ed 28 Ap il 2016; published 22 July 2016)
The a che e ec o a sine-Go don kink is in es iga ed in he absence o any ex e nal o ce while he symme y
o he ield po en ial a e e y ime ins an is main ained. The di ec ed mo ion appea s by a ime shi o he
sine-Go don po en ial h ough a ime-dependen addi ional phase. A symme y analysis p o ides he necessa y
condi ions o he exis ence o ne mo ion. I is also shown analy ically, by using a collec i e coo dina e heo y,
ha he no el physical mechanism esponsible o he appea ance o he a che e ec is he coupled dynamics
o he kink wid h wi h he backg ound ield. Biha monic and dicho omic pe iodic a ia ions o he addi ional
phase o he sine-Go don po en ial a e conside ed. The p edic ions es ablished by he symme y analysis and he
collec i e coo dina e heo y a e e i ied by means o nume ical simula ions. In e sion and maximiza ion o he
esul ing cu en as a unc ion o he sys em pa ame e s a e in es iga ed.
DOI: 10.1103/PhysRe E.94.012221
I. INTRODUCTION
Soli ons a e localized nonlinea wa es ha beha e like
pa icles in many si ua ions, wi h hei own mass, eloci y,
and o he pa iclelike p ope ies [1]. By using a collec i e
coo dina e heo y, i is shown ha he soli on dynamics can
be educed o he s udy o a sys em o o dina y di e en ial
equa ions o collec i e a iables, such as he kink cen e
o mass, i s wid h, e c. This scheme cap u es and explains
he main ea u es o di e en phenomena, such as soli on
sca e ing and soli on di usion [2].
An in e es ing phenomenon which appea s in pa icle as
well as in ex ended sys ems is he so-called a che e ec ,
whe e pa icles o soli ons mani es a unidi ec ional mo ion,
gene ally due o he ac ion o pe iodic o ces o ze o mean.
In ac , in a simila way o ha o he ec i ica ion o
andom mo ion o B ownian pa icles in pe iodic po en ials [3],
unidi ec ional mo ion o soli ons is induced by b eaking
he spa io empo al and/o ield symme ies o he ex ended
sys em [4,5]. Fu he mo e, cha ac e is ic ea u es o he a che
phenomena in poin -pa icle sys ems also a ise in he case o
soli ons. Fo ins ance, cu en e e sals and esonance beha -
io s o he soli on a e age eloci y a e achie ed h ough pa-
ame e a ia ions o po en ials, o ces, and damping [4,6–8].
The ele ance o hese phenomena co e s a wide ange o
a eas om biophysics [9] o possible echnological applica-
ions [10]. Speci ically, soli on a che s ha e been obse ed
expe imen ally in he damped Josephson junc ions d i en by
ex e nal asymme ic o ces [11,12]. In hese expe imen s, a
long quasi-one-dimensional Josephson junc ion is desc ibed
*[email p o ec ed]
†[email p o ec ed]
‡[email p o ec ed]
by he pe u bed sine-Go don equa ion
 (x, )−xx(x, )+U[(x, )]=−β (x, )+ (x, )
(1)
o he supe conduc ing phase di e ence (x, ) ac oss he
junc ion, whe e x≡∂/∂x, ≡∂/∂ ,β>0is he
damping coe icien , U(z) is he de i a i e wi h espec o
zo a cosine po en ial U(z), and (x, ) is an ex e nal
o ce. In his sys em, he a che e ec o kink (o an ikink)
exci a ions is induced: (i) by using a symme ic ield po en ial,
U()=1−cos(), oge he wi h an ex e nal pe iodic o ce
ha b eaks ei he empo al symme ies [4,7,11,13] o spa ial
symme ies [14]; (ii) by using an asymme ic saw oo h
po en ial o he ype U()=C−cos()+(λ/2) sin(2),
whe e Cand λa e cons an s, plus an ex e nal ac o ce
( )[6,8,14,15]; (iii) by conside ing local and pe iodic a ays
o inhomogenei ies U(,x)=1−cos()[1 +i,n δ(x−
xi−nL)] (mic osho s along he Josephson junc ions), o-
ge he wi h he ac ion o an ac o ce ( )[16,17]; and
inally, (i ) by modula ing he ield po en ial wi h an ac
o ce, U(, )=1−cos()[1 +1sin(ω1 )], oge he wi h
an addi i e ac signal ( )=2sin(ω2 )[18].
I is in e es ing o no e ha in all he cases men ioned
abo e, he a che mechanism is due o a combina ion o
a pe iodic po en ial wi h space- o ime-dependen ex e nal
o ces. Howe e , o an ensemble o B ownian pa icles, a
di ec ed cu en has also been ob ained solely by using a
symme ic pe iodic po en ial ha al e na es be ween wo s a es
ha di e only by a disc e e ansla ion [19]. I is he e o e
na u al o pose he ollowing ques ion: Can a di ec ed mo ion
o kinks be ob ained in he absence o any ex e nal o ce while
keeping he ield po en ial symme y a e e y ime ins an ? The
aim o his pape is o answe his ques ion by ex ending his
a che mechanism o he sine-Go don kink. The key idea is
o shi he sine-Go don po en ial o wa ds and backwa ds by
2470-0045/2016/94(1)/012221(7) 012221-1 ©2016 Ame ican Physical Socie y
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ANCHEZ-REY, CASADO-PASCUAL, AND QUINTERO PHYSICAL REVIEW E 94, 012221 (2016)
in oducing a ime-dependen addi ional phase. In con as o
he B ownian pa icle case, ne mo ion o kink is achie ed
in he absence o noise. He e a no el mechanism o soli on
a che s appea s, whe e, unlike o he models, he backg ound
ield plays a decisi e ole in he gene a ion o ne mo ion.
The ou line o he pape is as ollows. A ull desc ip ion
o he model unde conside a ion is p esen ed in Sec. II.
Necessa y condi ions o he occu ence o ne mo ion a e
es ablished by using a symme y analysis. In Sec. III,a
collec i e coo dina e app oach is de eloped in o de o
p o ide a physical insigh in o he a che mechanism. In
Sec. IV, he heo e ical esul s o he p e ious sec ions a e
hen compa ed wi h nume ical simula ions. Biha monic and
dicho omic pe iodic signals a e used o shi he sine-Go don
po en ial in ime and he dependence o he a che eloci y
on he sys em pa ame e s is in es iga ed. Finally, he main
con ibu ions o ou wo k a e summa ized in he las sec ion.
II. DESCRIPTION OF THE MODEL AND
SYMMETRY ANALYSIS
In his s udy, ou a en ion is ocused on a sine-Go don
sys em o he o m
 (x, )−xx(x, )+β (x, )+U[(x, ), ]=0(2)
wi h a ime-dependen po en ial
U(, )=1−cos [+θη( )],(3)
whe e η( ) is a pe iodic unc ion o pe iod Tand ze o ime-
a e age (such ha T
0d η( )/T =0), and θis a pa ame e
in oduced o adjus he ampli ude o η( ). Fo a ixed alue o
θη( )=κ, he po en ial conside ed abo e co esponds o ha
used o model wha is called in he li e a u e a κjunc ion [20],
ha is, a Josephson junc ion wi h an addi ional phase shi κ.
Ou model is inspi ed by expe imen al obse a ions o an-
si ions om posi i e (“0-phase s a e”) o nega i e (“π-phase
s a e”) coupling be ween he supe conduc o s o a junc ion
wi h a g aphene in e laye , which can easily be con olled by a
ga e ol age [21]. Simila ansi ions, induced, o ins ance,
by empe a u e a ia ions, ha e also been obse ed wi h
a e omagne ic in e laye [22]. The po en ial (3) can be
conside ed a heo e ical gene aliza ion o such obse a ions.
To ully speci y he ma hema ical p oblem, he pa ial
di e en ial equa ion (2)–(3) mus be amended by bo h ini ial
and bounda y condi ions. Since we a e in e es ed in s udying
solu ions o Eqs. (2)–(3) wi h only one kinklike s uc u e
p esen and, consequen ly, wi h opological cha ge 2π,we
conside ape iodic bounda y condi ions o he o m [23]
lim
x→+∞ (x, )=lim
x→−∞ (x, )+2π, (4)
lim
x→+∞ x(x, )=lim
x→−∞ x(x, ).(5)
Addi ionally, he ollowing ini ial condi ions a ime 0a e
assumed:
(x, 0)=4a c an(ex),(6)
 (x, 0)=0,(7)
which co espond o an unpe u bed kink cen e ed a x=0
and a es .
The cen e o mass o he kink and i s ime-a e age eloci y
can be espec i ely calcula ed om he exp essions
X( )=1
2π+∞
−∞
dx x x(x, )(8)
and
V=lim
 →∞
1
  0+
0
d X ( )=lim
 →∞
X( 0+ )
 ,(9)
whe e, acco ding o Eq. (6), i has been used ha X( 0)=0.
Le us now examine he condi ions unde which a ne
mo ion o he kink may be expec ed o occu . To his end, le
(x, ;θ, 0) be he solu ion o he p oblem de ined by Eqs. (2)–
(7) whe e, o con enience, i s dependence on he pa ame e s θ
and 0has been explici ly indica ed. I is hen s aigh o wa d o
e i y ha he unc ion 2π−(−x, ;−θ, 0) is also a solu ion
o he same p oblem. Consequen ly, om he uniqueness o he
solu ion o he p oblem (2)–(7), i ollows ha (x, ;θ, 0)=
2π−(−x, ;−θ, 0) and, aking in o accoun Eqs. (8) and (9),
ha
V(θ, 0)=−V(−θ, 0).(10)
Now le us assume ha he pe iodic unc ion η( ) sa is ies he
ollowing ime-shi symme y:
η( )=−η( +T/2).(11)
In his case, i is easy o show ha he unc ion (x, +
T/2; −θ, 0+T/2) is a solu ion o he p oblem (2)–(7).
Thus, om he uniqueness o he solu ion, i ollows ha
(x, ;θ, 0)=(x, +T/2; −θ, 0+T/2) and, bea ing in
mind Eqs. (8) and (9), ha
V(θ, 0)=V(−θ, 0+T/2).(12)
I can be seen ha he ime-a e age eloci y is independen o
he ini ial ime 0, i.e., V(θ, 0)=V(θ). Consequen ly, in o de
o gene a e a ne mo ion, he symme y (11) mus be b oken,
since o he wise om Eqs. (10) and (12) i would ollow ha
V(θ)=0.
III. COLLECTIVE COORDINATE APPROACH
Physical insigh in o he appea ance o ne kink anspo
can be gained by means o a collec i e coo dina e app oach.
To his end, le us de ine he “naked” kink ield as (x, )=
(x, )−ϕ( ), whe e ϕ( )=limx→−∞ (x, ) is he back-
g ound ield. This backg ound ield sa is ies he di e en ial
equa ion
ϕ ( )=−βϕ ( )−U[ϕ( ), ]
=−βϕ ( )−sin[ϕ( )+θη( )],(13)
wi h he ini ial condi ions ϕ( 0)=ϕ ( 0)=0.
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The momen um and he ene gy o he “naked” kink a e
espec i ely gi en by he exp essions
P( )=−+∞
−∞
dx  (x, )x(x, ) (14)
and
E( )=+∞
−∞
dx[ (x, )]2
2+[x(x, )]2
2+
U[(x, ), ],
(15)
wi h 
U[(x, ), ] being he new po en ial U[(x, )+
ϕ( ), ]−U[ϕ( ), ]. In o de o ob ain a ini e esul o
E( ), he ze o o his new po en ial has been chosen so ha
limx→±∞ 
U[(x, ), ]=0. By di e en ia ing wi h espec o
ime Eqs. (14) and (15), and using Eqs. (2), (3), and (13), i is
easy o show ha
P ( )=−βP( )−2πsin [ϕ( )+θη( )](16)
and
E ( )=[ϕ ( )+θη ( )]+∞
−∞
dx {sin[(x, )+ϕ( )+θη( )]
−sin[ϕ( )+θη( )]}+sin[ϕ( )+θη( )]
×+∞
−∞
dx  (x, )−β+∞
−∞
dx [ (x, )]2.(17)
Le us now conside an ansa z o he “naked” kink ield o
he o m
(a)(x, )=4a c anexp x−X( )
L( ),(18)
whe e X( ) and L( ) a e, espec i ely, he cen e o mass and
he wid h o he kinklike s uc u e.
By inse ing his ansa z in o Eqs. (14) and (15), one ob ains
ha
P( )=8X ( )
L( ),(19)
whe e 8/L( ) plays he ole o an e ec i e mass, and
E( )=π2[L ( )]2+12{1+[X ( )]2+[L( )]2cos[ϕ( )+θη( )]}
3L( ).(20)
F om Eqs. (13) and (16), i is easy o see ha he unc-
ion ( )=P( )−2πϕ ( ) sa is ies he di e en ial equa ion
( )=−β ( ). In addi ion, om Eqs. (7) and (14), i is clea
ha ( 0)=0, and hence ( )=0∀ ⩾ 0. Consequen ly, i
is ob ained ha P( )=2πϕ ( ). Thus, acco ding o Eq. (19),
he kink eloci y can be exp essed as
X ( )=πL( )ϕ ( )
4.(21)
Rema kably, his exp ession shows ha a ne mo ion o he
kink appea s due o he coupling be ween he backg ound
ield and he wid h o he kink. In con as o o he soli on
a che mechanisms, whe e he backg ound ield is used only
o imp o e he collec i e coo dina e heo y [24], he e ϕ( )
plays an essen ial ole in he a che e ec .
A di e en ial equa ion o he ime e olu ion o L( ) can be
ob ained by eplacing Eqs. (18) and (20)inEq.(17) and using
Eq. (21). A e leng hy calcula ions, one inds
L ( )=[L ( )]2
2L( )−3L( )[ϕ ( )]2
8−βL ( )
+6
π2L( ){1−[L( )]2cos[ϕ( )+θη( )]},(22)
which has o be sol ed wi h he ini ial condi ions L( 0)=1
and L ( 0)=0. The ime-a e age eloci y can be calcula ed
om Eqs. (9) and (21) a e nume ically sol ing he di e en ial
equa ions (13) and (22).
A u he simpli ica ion o he collec i e coo dina e ap-
p oach can be ob ained by linea izing he nonlinea di e en ial
equa ions (13) and (22). To his end, le ∞
n=0θnϕ(n)( )/n!
and ∞
n=0θnL(n)( )/n! be he powe expansion in θo
he backg ound ield and he kink wid h, espec i ely. I is
hen easy o show ha ϕ(1)( ) and L(2)( ) sa is y he linea
di e en ial equa ions
ϕ(1)
( )+βϕ
(1)
( )+ϕ(1)( )=−η( ) (23)
and
L(2)
( )+βL
(2)
( )+12
π2L(2)( )
=6
π2[ϕ(1)( )+η( )]2−3
4ϕ(1)
( )2,(24)
and ha ϕ(0)( )=L(1)( )=0 and L(0)( )=1. Acco ding o
Eqs. (23) and (24), i is clea ha a e a ansien ime ϕ(1)( )
and L(2)( ) become pe iodic unc ions o wi h pe iod T.
The e o e, om Eqs. (9) and (21) one can see ha he ime-
a e age eloci y is app oxima ely gi en by he exp ession
V≈πθ3
8TT
0
d ˜ϕ(1)
( )˜
L(2)( ),(25)
whe e ˜ϕ(1)( ) and ˜
L(2)( ) a e, espec i ely, he pe iodic solu-
ions o Eqs. (23) and (24).
IV. NUMERICAL SIMULATIONS
In o de o check he exis ence o ne kink mo ion when
symme y condi ions a e b oken, we ha e pe o med nume -
ical simula ions o he damped sine-Go don equa ion (2)–(3)
o wo pa icula choices o he unc ion η( ). The ini ial and
bounda y condi ions a e gi en by Eqs. (6)–(7) and (4)–(5),
espec i ely. The algo i hm used is a Runge-Ku a-Ve ne
i h-o de me hod wi h space s ep x =0.02 and adap i e
s ep size in ime.
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-0.0004
-0.0002
0
0.0002
0.0004
0π/2 π 3π/2 2π
(b)
(a)
V
δ
-0.008
-0.004
0
0.004
0.008
−π −π/2 0 π/2 π
V
θ
FIG. 1. (a) Kink eloci y e sus he ampli ude pa ame e θ o
ixed δ=0.8. The ci cles a e he esul s ob ained by simula ion o
he sine-Go don equa ion (2)–(3), he solid line ep esen s he a e age
eloci y ob ained by using he collec i e coo dina e Eq. (21), and he
dashed line co esponds o i s linea app oxima ion (25). (b) Kink
eloci y e sus he phase di e ence δ o ixed θ=1. In bo h panels,
β=0.1andω=0.1.
A. Biha monic case
Fo a i s nume ical es , we ha e chosen he biha monic
unc ion η( )=cos(ω )+cos(2ω +δ), since i is a p o o-
ypical pe iodic unc ion ha b eaks he ime-shi symme y
gi en by Eq. (11)[25,26].
In Fig. 1(a), he dependence o he a e age eloci y on he
ampli ude θis shown. The ci cles ep esen he simula ion
esul s while he solid line co esponds o he collec i e
coo dina e app oach ob ained om sol ing he di e en ial
equa ions (13) and (22). No ice he excellen ag eemen
be ween he simula ions and he collec i e coo dina e app ox-
ima ion e en o la ge alues o θ. Wi h a dashed line, he
a e age eloci y ob ained using he linea app oxima ion (25)
o he collec i e coo dina e equa ions has also been plo ed.
As expec ed, he linea app oxima ion goes well only o
small alues o he pe u ba ion ampli ude θ. In his egime,
V∼Aθ3, whe e Ais independen o θ. This unc ional
dependence on he pe u ba ion ampli ude has been p o ed o
occu in a e y gene al amewo k, independen ly o he sys em
de ails, by using simple symme y conside a ions [27,28]. The
linea collec i e coo dina e equa ions allow us o calcula e he
dependence on he es o he pa ame e s o he p e ac o ha
mul iplies he θ3 e m.
0
0.0001
0.0002
0.0003
0.0004
0.0005
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
V
ω
FIG. 2. Kink eloci y e sus equency ω o ixed θ=0.1, δ=
0.8, and β=0.2. The ci cles a e he esul s ob ained by simula ion o
he sine-Go don equa ion (2)–(3), he solid line ep esen s he a e age
eloci y ob ained by using he collec i e coo dina e Eq. (21), and he
dashed line co esponds o i s linea app oxima ion (25).
Equally, o su icien ly small pe u ba ion ampli udes, he
gene al o malism de eloped in Re s. [27] and [28] oge he
wi h Eq. (10) lead o V∼Bcos(δ+δ0), whe e Band δ0
a e independen o δ. This dependence o Von he phase
di e ence δis displayed in Fig. 1(b) o ixed θ=1. Fo
he chosen pa ame e s, clea ly δ0≈π/2. Once again he
ag eemen be ween he collec i e coo dina e heo y (solid line)
and he simula ion esul s (ci cles) is excellen . The sligh
de ia ion o he linea collec i e coo dina e app oxima ion
(dashed line) is due o he ela i ely la ge alue o θused.
Finally, in Fig. 2, he dependence o Von he equency
ωis shown. The alue o he ampli ude employed, θ=0.1,
is a he small and o ha eason he linea app oxima ion
(dashed line) closely ma ches he collec i e coo dina e esul s
(solid line). The collec i e coo dina e heo y i s he simula ion
esul s (ci cles) e y well, bu only o low equencies. Fo
equencies ω0.45, signi ica i e disc epancies appea due
o he ac ha he equency ωapp oaches ωph =1( he
lowes equency o he phonons) and he e o e he phonons
can become exci ed. Consequen ly, his ou come indica es ha
he collec i e coo dina e heo y has an “adiaba ic” na u e and
i s alidi y equi es ha pe u ba ions mus be applied in a
su icien ly slow way [29].
B. Dicho omic case
In his sec ion, η( ) is a dicho omic pe iodic unc ion o
ime ha successi ely akes he alues +1 and −1 du ing
ime in e als o leng hs τ+1and τ−1, espec i ely, he eby
p o iding he pe iod T=τ+1+τ−1. The e o e, acco ding
o Eq. (3), he ime-dependen po en ial U(, ) can only
be in one o wo possible s a es, U+1()=1−cos(+
θ)o U−1()=1−cos(−θ), which di e me ely by a
ansla ion o 2θ. I is no di icul o see ha his unc ion can
be ep esen ed as
η( )=sgnsin ωτ
4−sin(ω ),(26)
whe e ω=2π/T is he equency and τ =τ+1−τ−1∈
[−T,T].
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The choice (26) in oduces a new symme y p ope y no
p esen in he biha monic case. Indeed, le (x, ;θ,τ, 0)be
he solu ion o he p oblem de ined by Eqs. (2)–(7) and (26),
hen (x, +T/2; −θ, −τ, 0+T/2) is also solu ion o he
same p oblem. Taking in o accoun ha he a e age eloci y
is independen o he ini ial ime 0, i ollows ha
V(θ,τ)=V(−θ, −τ).(27)
F om Eqs. (10) and (27) one ob ains
V(θ,τ)=−V(θ, −τ),(28)
which implies V(θ,0) =0. In e ec , i τ+1=τ−1, he unc ion
η( ) gi en by (26) sa is ies he ime-shi symme y (11) and
consequen ly he a e age eloci y is ze o. The e o e, in he
dicho omic case, a necessa y condi ion o a di ec ed kink
mo ion is ha he di e ence be ween he esidence imes in
each po en ial s a e, τ, has o be nonze o. Fu he mo e, in he
case whe e ne mo ion exis s, he lux can be e e sed h ough
he ope a ion τ →−τ.
In ou nume ical simula ions, we ha e i s e i ied ha o
τ = 0 a nonze o a e age eloci y is obse ed. In Fig. 3, he
ime e olu ions o he kink cen e , X( ), o τ =−2 (open
ci cles) and τ =0 ( ull ci cles) a e compa ed. Only when
he ime symme y (11) is no sa is ied by η( ) does ne mo ion
appea .
In Fig. 4, he a e age kink eloci y compu ed om simula-
ions (ci cles) is compa ed wi h he a e age eloci y ob ained
om he collec i e coo dina e heo y by using Eq. (21) (solid
line). Despi e conside ing small alues o θand a slow
undamen al equency ω=0.1, he ag eemen is e y poo .
The discon inuous cha ac e o he dicho omic unc ion η( )is
decisi e in his poo ag eemen due o he adiaba ic na u e o
he collec i e coo dina e app oxima ion.
No ice ha he e he pa ame e θplays a a he di e en
ole han in he p e ious sec ion, since now he po en ial
U(, )is2πpe iodic in θ. The dependence o he kink
eloci y on he pa ame e θo e he whole ange [−π,π]
is shown in Fig. 5. The pe u ba ion on he sys em is e y
s ong o θπ/2. Fo his eason, i is necessa y o apply
-1
-0.5
0
0.5
1
1.5
2
0 20 40 60 80 100 120 140
X( )
FIG. 3. Time e olu ion o he kink cen e wi h a wo-s a e
po en ial gi en by (3)and(26). No a che e ec is obse ed when
η( ) sa is ies he ime-shi symme y (11) (open ci cles, τ =0).
Ne kink mo ion appea s b eaking ha symme y by se ing τ =−2
( ull ci cles). The emaining pa ame e s a e θ=0.8, ω=π/3, and
β=0.8.
-0.0008
-0.0006
-0.0004
-0.0002
0
0.04 0.08 0.12 0.16 0.2
V
θ
FIG. 4. Compa ison o he a e age kink eloci y (ci cles) wi h
he a e age eloci y ob ained using he collec i e coo dina e app ox-
ima ion (solid line) o β=0.05, τ =−2, and ω=0.1.
a su icien ly la ge dissipa ion o p e en he kink om being
des oyed. The ull ci cles co espond o a damping coe icien
β=0.8, while o he open ci cles, β=1. In ag eemen wi h
ou symme y analysis, i can be clea ly app ecia ed ha V
is odd in θ[Eq. (10)]. Fu he mo e, i is πpe iodic. In
o de o unde s and his new symme y, no ice ha gi en a
solu ion (x, ;θ,τ, 0) o he p oblem de ined by Eqs. (2)–
(7) and (26), hen π+(x, ;θ−π,τ, 0)isalsosolu ion
o he same p oblem excep o he ini ial condi ion (6).
The e o e, i i is addi ionally assumed ha he a e age kink
eloci y is independen o he ini ial condi ions, we ob ain
V(θ,τ)=V(θ−π,τ).(29)
This ela ion, oge he wi h V(0,τ)=0, also implies
V(±π,τ)=0,(30)
o any alue o τ. This p ope y i ially ollows om he
ac ha he wo s a es o he po en ial U+1and U−1coincide
when θ=nπ, wi h nbeing any in ege numbe . Mo eo e , by
se ing θ=π/2in(29) and bea ing in mind Eq. (10), i is easy
o conclude ha
V(±π/2,τ)=0.(31)
-0.01
-0.005
0
0.005
0.01
−π −π/2 0 π/2 π
V
θ
FIG. 5. Kink eloci y e sus he ansla ion pa ame e θ o ixed
τ =−2andω=π/3. The open ci cles co espond o a damping
coe icien β=1, while ull ci cles co espond o β=0.8.
012221-5

S´
ANCHEZ-REY, CASADO-PASCUAL, AND QUINTERO PHYSICAL REVIEW E 94, 012221 (2016)
-0.01
-0.005
0
0.005
0.01
-1 -0.5 0 0.5 1
V
Δτ/T
FIG. 6. Kink eloci y e sus τ/T o ixed Tand θ.Open
squa es: T=4. Full squa es: T=6. Open ci cles: T=10. Full
ci cles: T=14. In all cases, θ=0.8andβ=0.8.
The p ope ies (30) and (31) a e isible in Fig. 5. The exis ence
o maxima and minima ollows di ec ly om he h ee abo e
equa ions.
A simila nonmono onic beha io o he kink eloci y is
ound when i is plo ed e sus τ/T o ixed Tand θ,
as shown in Fig. 6. The maxima and minima can be easily
unde s ood aking in o accoun ha no ne mo ion is possible
i no empo al symme y is b oken (τ =0) and ha nei he is
any ne mo ion possible i no al e na ion be ween he po en ial
s a es occu s (τ/T =±1). I should also be bo ne in mind
ha he e is cu en e e sal due o he symme y (28).
Figu e 6also p o ides he in ui ion ha Vmus display
ano he maximum i i is plo ed e sus T o ixed τ. Such
beha io is shown in Fig. 7as a unc ion o he undamen al
equency ω=2π/T . The ull and open ci cles ep esen
esul s ob ained om nume ical simula ion o τ =−2 and
τ =−0.5, espec i ely. On he one hand, in he limi ω→0,
he a che e ec disappea s. In his limi , he ime in e als
τ+1and τ−1a e much longe han β−1, which gi es oughly
he ime scale o he elaxa ion p ocess ha akes place each
ime we swi ch he po en ial s a e. As a consequence, he kink
mo es a he beginning o he esidence imes τ±1,bu i
-0.002
0.002
0.006
0.01
0.4 0.8 1.2 1.6 2
V
ω
FIG. 7. Kink eloci y e sus ω=2π/T o ixed τ. Full ci cles
co espond o τ =−2, while open ci cles co espond o τ =
−0.5. The emaining pa ame e s a e θ=0.8andβ=0.8.
s ops long be o e hose esidence imes inish. Hence, he
dis ance a eled by a kink in one o he po en ial s a es
is comple ely eco e ed when he po en ial swi ches o he
o he s a e and, consequen ly, no ne displacemen is achie ed
o each pe iod. On he o he hand, nei he does he a che
e ec exis when ω→∞. In his case, τ±1β−1, ha is, he
esidence imes a e so sho ha he kink is unable o espond
o he pe u ba ion. As a esul , be ween hese wo limi s, V
has o show a leas one maximum o minimum. Addi ionally,
one can obse e se e al cu en in e sions ha appea in he
low- equency egion.
V. CONCLUSIONS
The a che dynamics o sine-Go don kinks induced by
phase pe u ba ions has been in es iga ed. Symme y analysis
shows ha ne mo ion can be gene a ed when he phase
pe u ba ion o he po en ial, θη( ), b eaks he ime-shi
symme y (11). Rema kably, he kink mo es wi h a nonze o
a e age eloci y in he absence o any ex e nal o ce and
main aining he ield po en ial symme y a e e y ime ins an .
An app oxima ed heo y, wi h an ansa z wi h h ee collec-
i e coo dina es, namely, he cen e o he soli on, i s wid h,
and he backg ound ield, has been de eloped in o de o shed
ligh on he a che mechanism o he kink. Using his ansa z, i
is assumed ha he unc ional o m o he soli on is p ese ed
al hough he collec i e coo dina es become ime dependen . In
con as o o he soli on a che s [4,7,24,30], no a che e ec
is p edic ed he e in absence o he backg ound.
Mo eo e , when a biha monic phase pe u ba ion is used,
he ag eemen be ween he collec i e coo dina e heo y and
he simula ions o he sine-Go don sys em is excellen , e en
o ela i ely la ge pe u ba ion ampli udes. Howe e , when a
dicho omic pe u ba ion is employed, he ag eemen is poo ,
he eby c ea ing he challenge o inding a be e heo y o
discon inuous pe u ba ions.
The dependence o he kink a e age eloci y on he
sys em pa ame e s has been explo ed in de ail. The ich phe-
nomenology obse ed can be unde s ood h ough symme y
conside a ions ha allow ce ain ea u es o be explained, such
as he supp ession o anspo o pa icula alues o he
pa ame e s, nonmono onic beha io s, and cu en in e sions.
Al hough we speci ically in es iga e he exis ence o his
no el soli on a che mechanism wi hin he amewo k o
he sine-Go don equa ion, he ob ained esul s can easily be
gene alized o o he models wi h opological soli on solu ions,
such as he double sine-Go don and φ4sys ems [30,31].
ACKNOWLEDGMENTS
We acknowledge inancial suppo om he Minis e io de
Ciencia e Inno aci´
on o Spain h ough G an No. FIS2008-
02873 (B.S.-R. and J.C.-P.), om he Minis e io de Econom´
ıa y
Compe i i idad o Spain h ough G an No. FIS2014-54497-P
(N.R.Q.), and om he Jun a de Andaluc´
ıa. N.R.Q. also
acknowledges inancial suppo om he Alexande on
Humbold Founda ion o Ge many h ough he Resea ch
Fellowship o Expe ienced Resea che s SPA No. 1146358
STP and om he Jun a de Andaluc´
ıa h ough G an No.
P11-FQM-7276.
012221-6
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