INTERACTION OF MOVING BREATHERS WITH AN
IMPURITY
J CUEVAS, F PALMERO, JFR ARCHILLA AND FR ROMERO
Nonlinea Physics G oup o he Uni e si y o Se illa, Depa men o Applied
Physics I
ETSI In o m´a ica, A da Reina Me cedes s/n, 41012, Se illa, Spain
E-mail: [email p o ec ed]
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. 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.
1. In oduc ion
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, e.g.
he sca e ing o kinks wi h impu i ies in he con inuous sine-Go don and
ϕ4models 1and in he F enkel–Kon o o a model 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 3. In his case, i is assumed ha he
sys em has weak nonlinea i y. He e, we a e in e es ed in he s udy o 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. We
also es ablish a hypo hesis o he appea ance o apping o a b ea he by
an impu i y.
1
2
2. The Model
We conside a Klein–Go don chain wi h nea es neighbou s a ac i e in-
e ac ions wi h Hamil onian gi en by:
H=
N
∑
n=1 (1
2˙u2
n+Vn(un) + 1
2C(un−un−1)2),(1)
whe e Vn(un) = Dn(e−un−1)2is he subs a e po en ial a he n- h si e.
The inhomogenei y is in oduced assuming a diffe en well dep h a 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. α∈[−1,∞) is a pa ame e which unes he
magni ude o he inhomogenei y.
This Hamil onian leads o he dynamical equa ions which ha e s a ion-
a y and mo ing localized solu ions (i.e., s a iona y and mo ing b ea he s).
The o me a e calcula ed using he me hods based in he an i–con inuous
limi 5and he la e a e calcula ed using he ma ginal mode me hod 4.
The dynamical equa ions can be linea ized i he ampli udes o he os-
cilla ions a e small. These equa ions ha e N−1 non-localized solu ions
(linea ex ended modes) and one localized solu ion, (linea impu i y mode).
Thei equencies, ωEand ωL, espec i ely, a e gi en by:
ω(q, α) = √ω2
o+ 4Csin2q(α)
2, ω2
L=ω2
o+ 2C+ sign(α)√α2ω4
o+ 4C2,(2)
whe e q∈(0, π] i α < 0 and q∈[0, π) i α > 0. Figu e 1 shows he
dependence on α.
3. Nume ical simula 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 a ying he alue o he inhomogenei y pa ame e α. We ha e
ound ou diffe en egimes, sepa a ed by c i ical alues o he pa ame e
α6:
•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. I α&0, he b ea he
can pass h ough he impu i y p o ided he ansla ional eloci y
is high enough 7.
•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). This beha io is shown in figu e 2.
•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
3
(a) (b)
−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. (a) F equencies o he linea modes e sus he pa ame e α. 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 . (b) 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.
(a) (b)
−20
−10
0
10
20 0
50
100
150
200
250
0
0.2
0.4
Pe iods
Posi ion
Ene gy
−20 −10 010 20 0
100
200
300
400
500
0
0.2
0.4
Pe iods
Posi ion
Ene gy
Figu e 2. (a) In e ac ion o a b ea he wi h an impu i y o α=−0.52, which co -
esponds o he impu i y exci a ion case. (b) E olu ion o he mo ing b ea he o
α=−0.3, 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.
neighbou ing si es, as figu e 2 shows. The apped b ea he emi s
a g ea amoun o phonon adia ion and seems o be chao ic.
•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.
4
4. Discussion
I is obse ed ha he b ea he bi u ca es wi h he ze o solu ion a α=
α es. Tha is, o αsmalle han his alue, no impu i y b ea he exis s.
A α=α es, he equency o he impu i y mode coincides wi h he mo ing
b ea he equency, i.e., in (2), ωL=ωb.
The scena io o he apped b ea he s when α < 0 is he ollowing: he
impu i y mode has q= 0, and also all he pa icles o he impu i y b ea he
ib a e in phase; his ib a ion pa e n indica es ha he impu i y b ea he
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 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.
Fo α < α es, 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 impu i y mode has q=πbu he impu i y b ea he ’s 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. The e a e wo diffe en localized exci a ions: he
ails o he (linea ) impu i y mode and he impu i y b ea he . Thus, i he
mo ing b ea he eaches he impu i y si e, i will exci e hese localized exci-
a ions. 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.
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 ib a ion pa -
e n diffe en om he impu i y b ea he one’s, he apped b ea he does
no o exis .
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5
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