Mo ing disc e e b ea he s in a Klein–Go don chain
wi h an impu i y
J Cue as†, F Palme o†, JFR A chilla†and FR Rome o‡
†ETS Ingenie ´ıa In o m´a ica. Uni e sidad de Se illa. A da Reina Me cedes s/n,
41012-Se illa, Spain
‡Facul ad de F´ısica. Uni e sidad de Se illa. A da Reina Me cedes s/n,
41012-Se illa, Spain
Abs ac . We analyze he influence o an impu i y in he e olu ion o mo ing
disc e e b ea he s in a Klein–Go don chain wi h non-weak nonlinea i y. Th ee
diffe en beha iou s can be obse ed when mo ing b ea he s in e ac wi h he
impu i y: hey pass h ough he impu i y con inuing hei di ec ion o mo emen ;
hey a e eflec ed by he impu i y; hey a e apped by he impu i y, gi ing ise
o chao ic b ea he s, as hei Fou ie powe spec a show. Resonance wi h a
b ea he cen ed a he impu i y si e is conjec u ed o be a necessa y condi ion o
he appea ance o he apping phenomenon. This pape es ablishes a diffe ence
be ween he esonance condi ion o he non-weak nonlinea i y app oach and he
esonance condi ion wi h he linea impu i y mode in he case o weak nonlinea i y.
PACS numbe s: 63.20.Pw 63.20.Ry 63.50.+x 66.90.+
Submi ed o: J. Phys. A: Ma h. Gen.
E-mail: [email p o ec ed]
1. In oduc ion
In insic localized modes o disc e e b ea he s esul s om he combina ion o
nonlinea i y wi h spa ial disc e eness. They can be ob ained in Klein–Go don la ices
as exac solu ions o dynamical equa ions [1, 2, 3]. In addi ion, hese localized
oscilla ions, unde ce ain condi ions, can mo e and hey a e usually called mo ing
b ea he s [4, 5, 6, 7, 8].
The in e ac ion o nonlinea localized oscilla ions wi h impu i ies in a sys em can
play an impo an ole in i s anspo p ope ies. This p oblem has been s udied
du ing he las decades wi hin diffe en amewo ks. The sca e ing o kinks and
en elope soli ons wi h impu i ies has been s udied in one–dimensional a omic la ices
wi h nonlinea in e ac ions [9]. Ano he app oach is conce ned wi h he in e ac ion
be ween high– equency con inuous b ea he s and impu i ies in he sine-Go don model
[10, 11]. This app oach has been ex ended o he low– equency case in [12], and o he
case o kinks in he con inuous sine-Go don and ϕ4models [13, 14]. The sca e ing
o a kink by an impu i y in he F enkel–Kon o o a model has been conside ed in
[15, 16]. Re e ences [9, 15] desc ibe soli ons ha can be eflec ed by he impu i y o
pass h ough i , depending on he eloci y. Howe e , in [10, 11, 13, 14], i is obse ed
soli ons can also be apped o in e media e eloci ies.
Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 2
The in e ac ion o a mo ing disc e e b ea he wi h an impu i y in a Klein–Go don
chain has been conside ed by Fo inash e al [17]. In his case, i is assumed ha he
sys em has weak nonlinea i y and h ee diffe en beha iou s can be obse ed: (a) he
mo ing b ea he passes h ough he impu i y, (b) i is eflec ed, o (c) i is apped
by he impu i y, o igina ing a deposi o y o ene gy. All hese effec s a e ela ed o
esonances wi h he impu i y modes.
In his pape , we a e in e es ed in he las app oach, i.e., o s udy he ea u es
o he in e ac ion o mo ing disc e e b ea he s wi h an impu i y a es in a Klein–
Go don chain o oscilla o s wi h non-weak nonlinea i y.
Al hough he beha iou in all o hese app oaches is quali a i ely simila , he e
a e some significa i e diffe ences conce ning o he appea ance o he apping
phenomenon. In he case o soli ons, he diffe en phenomena a e eloci y–dependen .
In disc e e la ices, he b ea he and he impu i y mode a e ela ed en i ies [18, 19], i.e.,
hey can be connec ed h ough a con inuous pa h. The e o e, he in e play be ween
a b ea he and an impu i y mode can be much s onge han be ween a soli on and
an impu i y. In Fo inash’s app oach, he necessa y condi ion o he appea ance o
he apping phenomenon is ha he b ea he equency esona es wi h he linea
impu i y mode one.
In ou case, ha is, a Klein–Go don chain wi h non-weak nonlinea i y, he
necessa y condi ion o he appea ance o apping is ha he e mus exis s a b ea he
a he impu i y wi h a equency close o ha o he mo ing b ea he . This ac
gua an ees he exis ence o a wide ange o pa ame e s o which apping is possible.
Ne e heless, his condi ion is no sufficien , as he apping phenomenon does no
occu when he ails o he a b ea he cen ed a he impu i y si e and he linea
impu i y mode ha e diffe en ib a ion pa e ns. We p opose he hypo hesis ha
bo h ails mus ha e he same ib a ion pa e n in o de ha he apping occu s.
2. Model and solu ions gene a ion
2.1. Fo mula ion o he model
In o de o s udy he effec s o impu i ies on he mo emen o b ea he s, we conside
a simple model whe e mo ing b ea he s can be gene a ed, ha is, a Klein–Go don
chain wi h nea es neighbou s a ac i e in e ac ions [4, 5]. I s Hamil onian is gi en
by:
H=
N
∑
n=1 (1
2˙u2
n+Vn(un) + 1
2C(un−un−1)2),(1)
whe e un ep esen s he displacemen o he n- h pa icle wi h espec o i s
equilib ium posi ion, Cis a coupling cons an and Vn(un) is he subs a e po en ial a
he n- h si e. We choose Vas he Mo se po en ial, i.e., Vn(u) = Dn(e−u−1)2, which
p o es o be e y sui able o ob aining mo ing b ea he s [5, 6, 8]. Dn ep esen s he
well dep h in he n- h si e. In his model, he inhomogenei y is in oduced assuming
a diffe en well dep h in only one si e, i.e., Dn=Do(1 + αδn,0), hen we e e o he
pa icle loca ed a n= 0 as an impu i y. αis a pa ame e which unes he magni ude
o he inhomogenei y. I akes i s alues in he in e al [−1,∞). He ea e , we will
conside Do= 1/2.
The esul s p esen ed he e co espond o ee ends bounda y condi ions, al hough
pe iodic bounda y condi ions lead o he same esul s.
Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 3
This model has been used ex ensi ely in DNA dynamics; in his con ex i is
usually e e ed o as he Pey a d-Bishop model [20]. In he amewo k o his model,
he a iables un ep esen he s e ching o he hyd ogen bonds connec ing each pai
o bases, Dis he dissocia ion ene gy, and Cis he s acking coupling cons an .
The inhomogenei y can also be in oduced in a simila way a ying he mass o
a pa icle, o he coupling cons an . The esul s in he fi s case a e equi alen o
he ob ained o he inhomogenei y in he po en ial well. The e a e, howe e , se e al
diffe ences when he inhomogenei y is in oduced h ough he coupling cons an , and
we will make some commen s abou hem a he end o his pape .
The Hamil onian (1) leads o he dynamical equa ions
F({un})≡¨un+V′
n(un) + C(2un−un+1 −un−1) = 0.(2)
which ha e wo kinds o solu ions, linea ones, which co espond o oscilla ions
o small ampli ude, and nonlinea ones, which co espond o in insic localized modes
o disc e e b ea he s.
2.2. Linea modes
The dynamical equa ions can be linea ized i he ampli udes o he oscilla ions a e
small. Thus, he equa ions (2) a e ans o med in he sys em o coupled equa ions:
¨un+ω2
nun+C(2un−un+1 −un−1) = 0,(3)
whe e ωnis he na u al equency o he n- h oscilla o in he ha monic limi . I
is gi en by ωn=√2Dn, which implies ha ω2
n=ω2
o(1 + αδn,0), wi h ωo= 1, as Do
has been chosen o be 1/2.
These equa ions has N−1 non-localized solu ions (being N he numbe o
pa icles) co esponding o linea ex ended modes and one localized solu ion, which
co esponds o a linea impu i y mode. He ea e , unless s a ed o he wise, he e m
mode will be ese ed o linea modes.
The equencies o he ex ended modes can be calcula ed supposing ha hey a e
plane wa es (un( ) = uoexp(iω(q) −nq)) and ha he impu i y mode decays in he
space ollowing a dependence o he o m un( ) = uoexp(iωL ) |n|[17], whe e is a
spa ial decay pa ame e . Thus, he equencies o he ex ended modes a e gi en by:
ω(q, α) = √ω2
o+ 4Csin2q(α)
2,(4)
whe e q(α) is he ex ended modes wa e ec o , which depends in a non-
s aigh o wa d way on α.
The equency o he impu i y mode is gi en by he ela ion [17]:
ω2
L=ω2
o+ 2C+ sign(α)√α2ω4
o+ 4C2.(5)
The sign o indica es he ib a ion pa e n o he impu i y mode. Thus, i > 0,
he pa icles o he mode ib a e in phase and will ha e a wa e ec o q= 0. On he
con a y, i < 0, he mode will ha e a zigzag ib a ion pa e n and a wa e ec o
q=π. The pa ame e is gi en by:
=−sign(α)αω2
o+√4C2+α2ω4
o
2C.(6)
Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 4
−1 −0.5 0 0.5 1 1.5
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2α es αc
2ωb
ωb
Linea modes equencies
α
Figu e 1. F equencies o he linea modes e sus he pa ame e α, o C= 0.13.
The dependence is quali a i ely simila o e e y alue o C. A α=α es and
α=αc, wo diffe en bi u ca ions occu , being he fi s one due o he esonance
be ween he impu i y mode and he b ea he .
The e o e, αand has opposi e sign. This ac also implies ha , o he ex ended
modes, q∈(0, π] i α < 0 and q∈[0, π) i α > 0. Figu e 1 shows he dependence o
he equencies o he linea modes wi h α, whe e he isola ed equencies co esponds
o he impu i y modes.
As i is explained below, he alues o he equency and wa e ec o o he
impu i y mode will be he key o explain he occu ence o he apping phenomenon.
2.3. S a iona y and mo ing b ea he s
A s a iona y b ea he can be ob ained by sol ing he ull dynamical equa ions. I
can be achie ed using common me hods based on he an icon inuous limi [21]. The
implemen a ion o hese me hods basically consis s in calcula ing he o bi o an
isola ed oscilla o a fixed equency ωb, and using his solu ion as a seed o sol e he
comple e dynamical equa ions by means o a New on–Raphson con inua ion me hod.
I he oscilla o ini ially chosen is he co esponding o he impu i y, a s a ic
b ea he cen ed a he impu i y is ob ained. I will be called impu i y b ea he .
Once a s a iona y b ea he is ob ained, i can be mo ed unde ce ain condi ions.
The e exis s a sys ema ic me hod o calcula ing mo ing solu ions [4, 5] which consis s
in adding o he eloci ies o he s a iona y b ea he a pe u ba ion o magni ude λ
colinea o he di ec ion o he pinning mode and le ing he sys em e ol e in ime.
The pinning mode is an an i-symme ic linea localized mode, which may appea in
he se o linea pe u ba ions o he sys em p o ided he coupling is s ong enough
[6]. Thus, a pe u ba ion in i s di ec ion b eaks he ansla ional symme y o he
sys em and make he b ea he mo e.
The esul s p esen ed he e co espond o a equency ωb= 0.8 and a coupling
Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 5
Figu e 2. Diffe en egimes in he in e ac ion o a mo ing b ea he wi h an
impu i y in oduced as an inhomogenei y in he po en ial well dep h.
C= 0.13, al hough o he alues o he same o de gi e quali a i ely simila esul s.
In his way, we ob ain mo ing b ea he s wi h low phonon adia ion o alues o he
pe u ba ion λ.0.2. I is wo h emaking ha he nonlinea i y o ou sys em is
highe han he conside ed in [17], as, in ha pape , he equencies o he localized
exci a ions oscilla e be ween 0.922 and 0.980 (in ou equency uni s, i.e. no malized
o he linea equency a ze o coupling ωo= 1), which a e e y close o he linea
ones.
We ha e conside ed diffe en alues o he pa ame e αin he ange −1≤α≤1.
3. In e ac ion o mo ing b ea he s wi h an impu i y
3.1. Nume ical obse a ions
We ha e s udied he beha iou o mo ing b ea he s when hey in e ac wi h an
impu i y. The s udy has been pe o med a ying he alue o he inhomogenei y
pa ame e α. Fou diffe en egimes, sepa a ed by c i ical alues o he pa ame e α,
ha e been ound (see figu e 2):
(i) Ba ie . The impu i y ac s as a po en ial ba ie . I occu s ei he wi h α > 0
o α∈(−1, α1) wi h α1<0. As he mo ing b ea he eaches he impu i y, i
is gene ally eflec ed, lea ing he impu i y exci ed du ing a sho ime, whose
ampli ude dec eases wi h |α|. The only excep ion o his beha iou occu s o
α&0. In his case, he b ea he can pass h ough he impu i y p o ided he
ansla ional eloci y is high enough.
(ii) Exci a ion. The impu i y is exci ed and he b ea he is eflec ed. I occu s o
α∈(α1, α2). The ene gy o he exci ed impu i y is la ge han he ene gy o he
impu i y b ea he . Thus, he exci ed impu i y ib a es wi h a equency lowe
han ωbas he on–si e po en ial is so . This beha io is shown in figu e 3.
(iii) T apping. The b ea he is apped by he impu i y. I occu s in he in e al
α∈(α2, α3). When he mo ing b ea he is close o he impu i y, i becomes
apped while i s cen e oscilla es be ween he neighbou ing si es, as figu e 4
shows. Fu he mo e, he apped b ea he emi s a g ea amoun o phonon
adia ion and seems o be chao ic, as can be app ecia ed om i s Fou ie powe
spec um (Figu e 5).
(i ) Well. The impu i y ac s as a po en ial well. I occu s o α∈(α3,0) and
consis s o an accele a ion o he b ea he as i app oaches o he impu i y, and
a decele a ion a e he impu i y has been passed h ough.

Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 6
−20
−10
0
10
20 0
50
100
150
200
250
0
0.2
0.4
Pe iods
Posi ion
Ene gy
0 50 100 150 200 250
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
Ene gy densi y a he impu i y si e
Pe iods
EN
Figu e 3. In e ac ion o a b ea he wi h an impu i y o α=−0.52 and λ= 0.1,
which co esponds o he impu i y exci a ion case. Top: E olu ion o he mo ing
b ea he . Bo om: E olu ion o he ene gy densi y o he impu i y. ENis he
ene gy o he impu i y b ea he .
The ansi ion be ween he diffe en egimes is somehow diffuse, ha means, he
alues o he c i ical alues o αcanno be exac ly de e mined. Fu he mo e, hey a e
sligh ly dependen on he b ea he eloci y. An es ima ion o he c i ical alues o
ωb= 0.8, C= 0.13 and λ= 0.1 leads o: α1≈ −0.54, α2≈ −0.49 and α3≈ −0.02.
As commen ed abo e, hese egimes ha e also been ound o diffe en alues o he
b ea he equency. Fo s onge coupling, he phonon adia ion is significan and
could mask some o he desc ibed effec s.
Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 7
−20 −10 010 20 0
100
200
300
400
500
0
0.2
0.4
Pe iods
Posi ion
Ene gy
Figu e 4. E olu ion o he mo ing b ea he o α=−0.3 and λ= 0.1, which
co esponds o he apping case. The mo ing b ea he becomes apped by he
impu i y; a e wa ds, he b ea he emi s phonon adia ion and i s ene gy cen e
oscilla es be ween he si es adjacen o he impu i y,
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1
0
1
2
3
log|S(ω)|
ω
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
1
2
3
log|S(ω)|
ω
Figu e 5. Fou ie powe spec a o he apped b ea he o figu e 4 ( op) and a
s a iona y b ea he wi h he same alues o Cand ωb(bo om). The spec um
on he op panel is cha ac e is ic o chaos.
Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 8
−1 −0.8 −0.6 −0.4 −0.2 0
−1
−0.5
0
0.5
1
1.5
2
uo( =0)
α
Figu e 6. Pi ch o k bi u ca ion o ωb= 0.8 and C= 0.13. α es =−0.5328.
3.2. Discussion
Some o he esul s in he las subsec ion can be explained om he p ope ies o he
impu i y b ea he . Pa icula ly, i a con inua ion o a s a iona y b ea he is pe o med
a ying he pa ame e α, a bi u ca ion appea s o α=αc>0 and ano he one o
α=α es <0 (see Figu e 1). The fi s one is o igina ed by a localized Floque
eigenmode which abandons he uni ci cle and leads o a b ea he ex inc ion, i.e.,
he impu i y b ea he does no exis o α > αc. In he second case, he b ea he
bi u ca es wi h he ze o solu ion h ough a pi ch o k bi u ca ion (see figu e 6) in he
space o ime– e e sible solu ions o equency ωb‡. In his bi u ca ion, he ou e
b anches co espond o impu i y b ea he s ei he wi h un(0) >0 o un(0) <0, while
he cen al b anch co esponds o he ze o solu ion, i.e. all he oscilla o s a e a es .
Fo α=α es he equency o he impu i y mode is he same as he equency o
he impu i y b ea he wi h he mo ing b ea he equency, i.e., ωL=ωb.α es can be
calcula ed om he equa ion (5) as a unc ion o ωband C:
α es =−√(ω2
b−ω2
o)(ω2
b−ω2
o−4C)
ω2
o
(7)
Thus, o C= 0.13 and ωb= 0.8, α es =−0.5628, which is lowe han α1.
Howe e , he apped b ea he does no exis o α > 0. I indica es ha he condi ion
α∈(α es,0) migh hold in o de ha he apped b ea he exis s.
The scena io o he apped b ea he s when α < 0 is he ollowing: in his
case, he impu i y mode has q= 0, and also all he pa icles o he impu i y b ea he
ib a es in phase; his ib a ion pa e n indica es ha he impu i y b ea he bi u ca es
om plane wa es wi h q= 0 [22], i.e., he impu i y bi u ca es om he impu i y
mode and i will be he only localized mode ha exis s when he impu i y is exci ed
‡No e ha he dynamical equa ions (2) do no co espond o a s anda d dynamical sys em
¨x= (x, ), whe e usually pi ch o k bi u ca ions a e desc ibed
Mo ing disc e e b ea he s in a Klein–Go don chain wi h an impu i y 9
o α > α es. Thus, when he mo ing b ea he eaches he impu i y, i can exci e
he impu i y mode. In ac , we ha e pe o med a success ul con inua ion om he
impu i y b ea he o he impu i y mode a cons an ac ion and α[18, 19], and a ying
a pa ame e swhich unes he nonlinea i y o he sys em. This pa ame e is in oduced
by changing he on–si e po en ial o he exp ession V∗
n(un) = Dn(u2
n−su2
n)+sVn(un),
whe e Vn(un) is he o iginal po en ial (1).
When α es < α < α2, he impu i y b ea he is unable o c ea e a apped en i y.
The ene gy o he impu i y mode dec eases wi h |α|, hus he e mus be a minimum
alue o he impu i y b ea he ene gy o he exis ence o apping. The na ow
window o impu i y exci a ions obse ed in he in e al (α1, α2) can be due o a
esonance o he mo ing b ea he wi h he impu i y b ea he wi h a equency sligh ly
smalle han ωb.
Fo α < α es, he apped b ea he canno be gene a ed, and he mo ing b ea he
is always eflec ed. In addi ion, he impu i y b ea he does no exis . The e o e, he e
migh be a connec ion be ween bo h ac s, i.e., he exis ence o he impu i y b ea he
seems o be a necessa y condi ion in o de o ob ain a apped b ea he .
I α > 0, he scena io is diffe en . In his case, he impu i y mode has q=π
bu he impu i y b ea he si es ib a e again in phase, ha is, he impu i y b ea he
does no bi u ca e om he impu i y mode. Thus, he e a e wo diffe en localized
exci a ions o α > 0: he (linea ) impu i y mode and he (nonlinea ) impu i y
b ea he . Bu , ac ually, he equa ions ha go e n he sys em a e nonlinea , so he
linea modes can only co espond o low-ampli ude oscilla ions. In he case o he
impu i y b ea he , he linea egime co esponds o he ails. Thus, i he mo ing
b ea he eaches he impu i y si e, i will exci e he impu i y b ea he and he ails
o he impu i y mode. Bu he la e ib a es in zigzag. As a consequence, he e will
be wo diffe en linea localized en i ies: he ails o he impu i y mode ( ib a ing
in zigzag) and he ails o he impu i y b ea he ( ib a ing in phase). The e o e, we
conjec u e ha he exis ence o bo h linea localized en i ies a he same ime may be
he eason why he impu i y is unable o ap he b ea he when α > 0.
I can be hough whe he esonances wi h he ha monics o he b ea he
equency could ha e consequences in he in e ac ion o he mo ing b ea he wi h
he impu i y. In ou sys em, hese esonances occu o α > αc, and he e o e, he
impu i y b ea he does no exis , so ha no apping effec s ake place, leading only
o b ea he eflec ions.
We summa ize he e ou hypo hesis o he exis ence o apping:
T apping hypo hesis:The exis ence o an impu i y b ea he o a gi en alue
o αis a necessa y condi ion o he exis ence o apped b ea he s. Howe e , i he e
exis s an impu i y mode wi h a diffe en ib a ion pa e n o he impu i y b ea he one,
he apped b ea he does no o exis .
4. Inhomogenei y in he coupling pa ame e
In o de o check whe he he hypo hesis p oposed in he las sec ion holds o diffe en
si ua ions, we conside a chain o oscilla o s o which he imhomogenei y is in oduced
h ough he coupling cons an s. In his case, he Hamil onian can be w i en as:
H=∑
n(1
2˙u2
n+V(un) + 1
4Cn[(un−un−1)2+ (un−un−1)]),(8)
which leads o he ollowing dynamical equa ions: