VOLUME 84, NUMBER 5 PHYSICAL REVIEW LETTERS 31JANUARY 2000
Anomalous Resonance Phenomena o Soli a y Wa es wi h In e nal Modes
Niu ka R. Quin e o* and Angel Sánchez†
G upo In e disciplina de Sis emas Complicados (GISC), Depa amen o de Ma emá icas, Uni e sidad Ca los III de Mad id,
A enida de la Uni e sidad 30, E-28911 Leganés, Mad id, Spain
F anz G. Me ens‡
Physikalisches Ins i u , Uni e si ä Bay eu h, D-95440 Bay eu h, Ge many
(Recei ed 16 July 1999)
We in es iga e he nonpa ame ic, pu e ac d i en dynamics o nonlinea Klein-Go don soli a y wa es
ha ing an in e nal mode o equency Vi. We show ha he s onges esonance a ises when he d i ing
equency d苷Vi兾2, whe eas when d苷Vi he esonance is weake , disappea ing o nonze o damping.
A esonance, he dynamics o he kink cen e o mass becomes chao ic. As we iden i y he esonance
mechanism as an indi ec coupling o he in e nal mode due o i s symme y, we expec simila esul s
o o he sys ems.
PACS numbe s: 05.45.Y , 02.30.J , 03.50.–z, 63.20.Pw
An impo an pa adigm es ablished o e he las wo
decades is ha soli a y wa es o soli ons beha e e y much
like poin pa icles when subjec ed o (a la ge class o ) ex-
e nal o ces and pe u ba ions [1–3]. Howe e , many soli-
a y wa es possess one (some imes mo e han one) in e nal
o shape mode [4,5], and in ha case he pa icle pic u e o
hei dynamics may be o e simpli ied: Indeed, he in e -
nal mode can empo a ily s o e ene gy and elease i a a
la e s age, gi ing ise o esonance phenomena in soli a y
wa e collisions [5] o in soli a y wa e in e ac ions wi h in-
homogenei ies [6]. As in e nal modes a e qui e common
in nonlinea sys ems, ei he in insically o as a esul o
small pe u ba ions [7], he ques ion o hei in luence on
he dynamics o soli a y wa es is a e y gene al and ele-
an one.
One aspec o soli a y wa e dynamics ha has p o en
i sel di icul o unde s and is ha o opological soli a y
wa es o kinks subjec ed o pu e, i.e., nonpa ame ic ac
d i ing. Thus, only ecen ly [8] he ac d i en dynamics o
sine-Go don kinks ( ha do no possess an in e nal mode)
has been de ini ely cla i ied. Nai ely, he only new phe-
nomenon one expec s when a nonpa ame ic ex e nal d i -
ing ac s on soli ons wi h in e nal modes is a esonance
when i s equency, D, ma ches ha o an in e nal mode,
Vi. The aim o his Le e is o show ha , in ac , he ac ual
scena io is mos unexpec ed and highly non i ial. As we
will see below, a s ong, anomalous esonance a ises when
d苷Vi兾2, whe eas he no mal esonance a d苷Viis
de ini ely weake , only possible a exac ly ze o damping,
and e en hen i can be supp essed by app op ia e choices
o o he pa ame e s. We expec his esul o be gene ic,
because ou analy ical app oach allows us o iden i y he
mechanism o such a peculia phenomenon: The ac o ce
does no ac di ec ly on he in e nal mode (because o sym-
me y easons), bu a he , hey in e ac indi ec ly ia he
ansla ional mo ion which couples o he in e nal mode.
Ou p edic ions a e ully con i med by nume ical simu-
la ions, which in addi ion show he implica ions o hese
esonances o he kink dynamics.
As a speci ic example o a kink wi h in e nal mode, we
ake he well known [1] 4equa ion, which, when d i en
wi h an ac o ce 共 兲苷esin共d 1d
0兲, eads
2
xx 1U0共 兲苷2b 1 共 兲,(1)
whe e U共 兲苷共 221兲2兾4and bis a damping coe i-
cien . P e ious ela ed wo ks on his sys em a e [9], whe e
esonances in he p esence o an ex e nal ( ime indepen-
den ) po en ial ha e been conside ed, and [10], which deal
wi h non esonan , high equency pa ame ic ac d i ings.
Fo ou p oblem, ou analy ical app oach will be he well
known collec i e coo dina e (CC) me hod [2,3]. A i s
o de o app oxima ion is gi en by he McLaughlin-Sco
me hod [11]: We assume ha he solu ion o (1) is o he
o m
共x, 兲苷 anh"x2X共 兲
l0p12V共 兲2#,(2)
whe e l0苷p2. The cen e o he kink X共 兲and i s e-
loci y V共 兲a e ela ed by X共 兲苷R
0d 0V共 0兲1X共0兲, and
bo h a e unknown unc ions desc ibing he mo ion o he
kink as a cohe en en i y. By means o a s anda d p oce-
du e [2,3,8,11] in ol ing conse a ion laws, an o dina y
di e en ial equa ion o mo ion o V共 兲can be ob ained,
linea ized [8], and sol ed, yielding
V共 兲苷 共 兲
p11 共 兲2,(3)
共 兲⬅¯ce2b 23p2e
2共b21d
2兲
3关bsin共d 1d
0兲2dcos共d 1d
0兲兴 ,(4)
¯c苷g0V共0兲13p2e
2共b21d
2兲关bsin共d0兲2dcos共d0兲兴 ,
0031-9007兾00兾84(5)兾871(4)$15.00 © 2000 The Ame ican Physical Socie y 871
VOLUME 84, NUMBER 5 PHYSICAL REVIEW LETTERS 31J
ANUARY 2000
whe e g0⬅1兾p12V共0兲2. Fo he undamped case
(b苷0), we see ha he kink oscilla es i g0V共0兲苷
3p2ecos共d0兲兾共2d兲; o he wise, he kink will mo e ei he
o he igh o o he le , depending on he ela ion be ween
he pa ame e s o he ac o ce and he ini ial eloci y.
This dc mo ion is absen when bfi0, as has been
nume ically con i med o sine-Go don kinks in [8].
In o de o include in e nal mode e ec s, we p oceed as
ollows: We ew i e Eq. (1) as
ᠨ
c苷2dH
d 2bᠨ
1 共 兲,ᠨ
苷dH
dc ,(5)
whe e c苷ᠨ
, he do meaning de i a i e wi h espec o
ime, and
H苷Z1`
2`
dx Ω1
2c211
2 2
x1U共 兲æ(6)
is he Hamil onian o he sys em when e苷b苷0.
We now make he ansa z 共x, 兲苷 关x2X共 兲,l共 兲兴,
whe eas om he de ini ion o cwe ha e ha c共x, 兲苷
c关x2X共 兲,l共 兲,ᠨ
X,ᠨ
l兴. As in he McLaughlin-Sco
me hod, X共 兲 ep esen s he kink cen e posi ion, bu now
we in oduce a second collec i e a iable l共 兲 ha will
s and o he kink wid h, i.e., he in e nal mode exci a ion,
below.
The p ocedu e o ob ain he CC equa ions co espond-
ing o his gene alized a eling wa e ansa z has been pu
o wa d in [12,13]. Basically, i consis s o inse ing ou
ansa z in o (5), mul iplying he i s equa ion by ≠ 兾≠X
and he second one by ≠c兾≠X(≠ 兾≠land ≠c兾≠l), ak-
ing hei di e ence, and in eg a ing o e x. This yields a
a he cumbe some, o dina y di e en ial equa ion o X共 兲
[l共 兲], which we omi he e o b e i y. The nex s ep is
o choose a speci ic unc ional o m o , which we do
ollowing he wo k o Rice [14], and le
关x2X共 兲,l共 兲兴 苷 0∑x2X共 兲
l共 兲∏,(7)
whe e 0关共x2X兲兾l0兴is he s a ic kink solu ion, which is
an odd unc ion (wi h espec o i s cen e ) o he 4and
o any o he e en po en ial. Upon pa icula iza ion o he
CC equa ions o his o m o we inally ob ain
M0l0
¨
X
l2M0l0
ᠨ
Xᠨ
l
l2
苷2bM0l0
ᠨ
X
l22 共 兲,(8)
aM0l0
¨
l
l1M0l0
ᠨ
X2
l2
苷Kin 共l,ᠨ
l,ᠨ
X兲2 baM0l0
ᠨ
l
l;
(9)
whe e Kin 苷2≠E兾≠land
E苷1
2
l0
lM0
ᠨ
X21l0
2laM0
ᠨ
l211
2M0µl0
l1l
l0∂
(10)
is he kink ene gy. Impo an ly, in ob aining Eq. (9) a e m
o he o m 共 兲R`
2` dx ≠ 0兾≠l, coming om he cou-
pling o he ac d i ing o he in e nal mode, has anished
because o symme y. Fo he 4equa ion, 0苷 anhxin
Eq. (7), which yields a苷共p226兲兾12 and M0苷2p2兾3.
Le us now simpli y hese exp essions in o de o make
i s physical signi icance mo e anspa en . To begin wi h,
Eq. (8) can be sol ed o ᠨ
X兾l, yielding
P⬅M0l0
ᠨ
X
l
苷22ebsin共d 1d
0兲2dcos共d 1d
0兲
共b21d
2兲
1e2b ∑2ebsin共d0兲2dcos共d0兲
共b21d
2兲1M0l0V共0兲
ls∏,
(11)
whe e ls苷l0兾g0. Inse ing Eq. (11) in o Eq. (9), we ind
a关ᠨ
l222l¨
l22blᠨ
l兴苷l2
l2
0∑11P2
M2
0∏21. (12)
Al hough Eqs. (11) and (12) a e qui e complica ed, choos-
ing V共0兲and he phase d0so ha he exponen ial e ms in
Eq. (11) anish, we can use a change o a iables (p o-
posed in [15] o he dc d i ing case) o ans o m he
Eq. (12) in o a Pinney-like equa ion (see [16] and e e -
ences he ein), which can be sol ed in e ms o Ma hieu
unc ions [17] i b苷0(when bfi0we ha e no been
able o sol e his p oblem analy ically).
In any e en , we do no need he analy ical exp essions
o he solu ion o he sys em (11) and (12) o unde s and
he physics p edic ed by ou app oach. The e m P2in
Eq. (12) is an oscilla o y unc ion wi h equency 2d[see
Eq. (11)]; hence, we can immedia ely expec a esonance
when he ex e nal equency dis hal he equency o he
in e nal mode, VR苷1兾pal0in he Rice app oxima ion.
Fo he 4model VRo e es ima es Vi苷p3兾2by 1.7%
[14]. The analy ical solu ion o b苷0con i ms his ex-
pec a ion, while nume ical in eg a ion o Eq. (12) p o es
ha he beha io o lis ha o a esonan , damped oscilla-
o when bfi0. When d苷VR, inspec ion o Eqs. (11)
and (12) leads o he conclusion ha ano he esonance
should be ound a d苷VRonly i b苷0(o he wise i
is a ansien phenomenon o li e ime b21); e en hen, by
choosing V共0兲and d0 o cancel he nonoscilla o y e ms in
Eq. (11) he esonance is comple ely supp essed. Fu he -
mo e, bo h analy ically and nume ically we ha e e i ied
ha , a away om he esonances, he beha io o he kink
cen e , X共 兲, is p ac ically he same as he one p edic ed by
he McLaughlin-Sco app oach, Eq. (3). This means ha ,
wi hin he CC amewo k, he beha io o ac d i en 4
kinks is desc ibed by he McLaughlin-Sco ansa z, and
only o d i ings close o VR兾2(and VRi b苷0) such
app oach ails and esonan phenomena a ise.
A his poin , wo key issues mus be add essed: Fi s ,
unde lying CC me hods is he assump ion ha no (o a
negligible amoun o ) adia ion is gene a ed by he pe u -
ba ion, an assump ion whose alidi y can be assessed only
h ough compa ison wi h he co esponding PDE. Sec-
ond, e en i ha is he case, he CC equa ions p edic
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VOLUME 84, NUMBER 5 PHYSICAL REVIEW LETTERS 31J
ANUARY 2000
an unbounded g ow h o l共 兲a esonance, and i is di -
icul o unde s and wha ha means in physical e ms
o he kink o he ull PDE, whose wid h is con olled
by he p ope ies o he equa ion. In iew o his, we
ha e compu ed he nume ical solu ion o he PDE (1)
by using he conse a i e S auss-Vázquez scheme [18]
wi h Dx苷0.1,D 苷0.01, and a o al sys em leng h o
L苷400. Ou ini ial condi ion was a kink a es and
d0苷p兾2, a pai o alues o which we should no see a
esonance a Vi. We come back o his below. We moni-
o ed he posi ion and he eloci y o he kink cen e as
well as he o al ene gy in he sys em, compu ed om he
Hamil onian (6). We also ied o measu e di ec ly he kink
wid h, bu we ound ha i is qui e complica ed o es ima e
i om he nume ics, his being he eason why we ha e
eso ed o less di ec measu emen s.
Figu e 1 shows examples o he kink cen e dynamics
and i s ene gy e olu ion bo h close o and away om he
p edic ed esonance. O - esonance, he beha io o bo h
magni udes is pe iodic, whe eas a he esonance i be-
comes chao ic. Speci ically, he ene gy inc eases wi h
ime: The close o he esonance alue, he as e he
inc emen . The cen e mo ion is ini ially pe iodic, un il
he in e nal mode ampli ude has inc eased oo much and
s o ed oo much ene gy, subsequen ly eleasing i h ough
i s coupling wi h he ansla ion mode (which we know ex-
is s om [5]), e en ually yielding he kink mo ion e a ic
as his p ocess is epea ed once and again. This is a clea
e idence in a o o he esonance p edic ed om he CC
ea men . We ha e e i ied ha a esonance he kink mo-
ion is e y sensi i e o changes in he ini ial condi ions,
hence ou claim o he appea ance o chaos.
Figu e 2 depic s he esonance as seen h ough he mean
ene gy, compu ed as a ime a e age om 苷10 000 (a -
e ansien s ha e died ou in he damped case) o he end
o he un a 苷25 000. Figu e 2(a) shows he beha io
o his magni ude a ound Vi兾2. I is clea om he plo
ha he e is a s ong esonance a d艐0.6102 艐Vi兾2,
bo h wi h and wi hou dissipa ion. In ac , some poin s
a e missing in he b苷0line because he co esponding
kinks, in hei appa en ly andom mo ion, le he sys em
be o e he end o he un. On he o he hand, in spi e o ou
choice o ini ial condi ions, o which no esonance is p e-
dic ed a Vi o b苷0, a weak esonance can be seen a
d艐1.225 艐Viin Fig. 2(b). We no e ha he o he wo
small peaks a e spu ious, since hey appea , disappea , o
change loca ion depending on he choice o he leng h o
he nume ical simula ion. We belie e ha his disc epancy
migh come om he di icul y o nume ically uning he
condi ion o i s supp ession. In any e en , ha would be
a e y special case, and he beha io we ind o hese pa-
ame e s is ep esen a i e o wha occu s o o he choices
o he ini ial eloci y and he phase. Those gi e ise o a
simila beha io , while, ema kably, he esonance a Viis
always weake and na owe han ha a Vi兾2(c . Fig. 2).
We also see ha he p edic ion ha he esonance a Viis
supp essed by he dissipa ion is also con i med, he small
0 5000 10000 15000 20000 25000
0.9
1.1
1.3
1.5
1.7
Ene gy
0 2500 5000
0.9
1.0
1.1
1.2
Ene gy
(a)
0 5000 10000 15000 20000 25000
−100
−60
−20
20
60
100
X( )
0 2500 5000
−20
−10
0
10
X( )
(b)
FIG. 1. Resul s o he PDE (1) wi h b苷0,e苷0.01.
(a) To al ene gy when d苷0.6100 (uppe line), d苷0.6080
(lowe line). (b) Same o he kink cen e , X共 兲. Inse s show
he same o d苷0.6102, a alue o which he kink eaches
he bounda y o he nume ical sys em be o e 苷25 000.
peak in he lowe line s emming om he ansien s, which
a e no exac ly ze o o 10 000 # #25 000. Ano he in-
e es ing ema k is ha Fou ie analysis shows ha he mo-
no onous inc easing o he ene gy ha appea s in Fig. 2(b)
comes om he ac ha , when d*1.1, he lowes phonon
mode (wi h equency p苷p2) begins o be exci ed,
he ampli ude o i s exci a ion mono onically inc easing as
d!p2. No e idence o lowes phonon mode exci a ion
is seen o he esonance a Vi兾2; he e o e, his is indeed
a phenomenon a ising om he coupling o he ansla ion
and he in e nal mode as p edic ed by he CC calcula ion.
In summa y, we ha e s udied how ac o ces a ec
soli a y wa es o kink ype wi h an in e nal mode o
equency Vi. Speci ically, we ha e clea ly shown
ha he beha io o 4kinks unde ac d i ing is e y
well desc ibed by he wo- a iable CC heo y we ha e
de eloped he e. The main ea u e o he nonpa ame i-
cally, ac d i en 4kink dynamics is ha he s onges
esonance occu s a d苷Vi兾2, and no a he equency
one would expec , d苷Vi. We emphasize ha his no el
esonance phenomenon is o ally unexpec ed om he
knowledge o he in e nal mode equency, and a ises
873
VOLUME 84, NUMBER 5 PHYSICAL REVIEW LETTERS 31J
ANUARY 2000
0.59 0.60 0.61 0.62 0.63
δ
0.92
1.00
1.08
1.16
1.24
1.32
1.40
Mean Ene gy
(a)
1.21 1.22 1.23 1.24 1.25 1.26
δ
0.98
1.08
1.18
1.28
1.38
Mean Ene gy
(b)
FIG. 2. Resul s o he PDE (1). Mean alue o he ene gy
(see ex o he way i is compu ed) s d i ing equency d,
(a) close o Vi兾2, (b) close o Vi. In bo h cases, he uppe line
co esponds o b苷0, and he lowe line o b苷0.001.
om he indi ec in e ac ion o he ex e nal o ce wi h he
in e nal mode ia he ansla ional mo ion. Al hough ou
esul s ha e been ob ained o a speci ic example, he 4
equa ion (1), o he models wi h e en on-si e po en ials and
in e nal modes, such as he double sine-Go don equa ion,
o ins ance, will beha e simila ly because a CC app oach
will lead o analogous esul s. The esonance ound a
he CC le el mani es s i sel a he PDE le el as e a ic
o chao ic (s ongly dependen on he ini ial condi ions)
mo ion o he kink as he kine ic ene gy o he cen e o
mass is s o ed in o, and eco e ed om, he in e nal mode.
To conclude, we no e ha he anomalous esonances de-
sc ibed he e a e impo an on hei own, as examples o
he highly non i ial beha io o nonlinea sys ems and as
hin s abou he mechanisms go e ning kink dynamics. In
addi ion, we hink ha his phenomenon should be e y
gene al, in iew o he ecen inding [7] ha pe u ba-
ions o kink-bea ing nonlinea sys ems o en lead o he
de elopmen o an in e nal mode. As his occu s in he
disc e e sine-Go don model [7], a esonance like he one
discussed he e could be ele an o he mode locking phe-
nomena epo ed o ha sys em in [19]. Finally, by using
his disc e eness induced in e nal mode, we poin ou ha
he esonance we ind could be obse ed in expe imen s
by using Josephson junc ion a ays as in [20].
We hank Yu i Gaididei, F ancisco Domínguez-Adame,
and JoséCues a o discussions. Wo k a GISC (Leganés)
has been suppo ed by DGESIC (Spain) G an No. PB96-
0119. T a el be ween Bay eu h and Mad id has been
suppo ed by “Acciones In eg adas Hispano-Alemanas,”
a join p og am o DAAD (Az. 314-AI) and DGESIC.
*Elec onic add ess: [email p o ec ed]
†Elec onic add ess: [email p o ec ed]
‡Elec onic add ess:
[email p o ec ed].uni-bay eu h.de
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