Physics LettersA 159 (1991) 318—322
North-Holland
PHYSICS LETTERS A
Kink capture by a local impurity in the sine-Gordon model
Fei Zhang, Yuri S. Kivshar
~,
Boris A. Malomed 2 and Luis Vázquez
Departamento de Fisica Teórica I, Facultad de Ciencias FIsicas, Universidad Complutense, E-28040 Madrid, Spain
Received 16 June 1991; accepted for publication 14 August 1991
Communicated by AR. Bishop
We demonstrate analytically and numerically that the impurity mode plays a very important role in the kink scattering by a
local inhomogeneity, and a sine-Gordon kink may be trapped by an impurity due to loss of its kinetic energy to excite the impurity
mode. We develop a collective-coordinate approach to describe the kink—impurity interactions and calculate the threshold velocity ofkink capture. Our numerical simulation results are in good agreement with the theoretical prediction.
1. Introduction
approach, the only inelastic effect is emission of linear waves (radiative losses). Further, it was shown
It is well known that nonlinearity may drastically
change transport properties of disordered systems
when it contributes to create soliton pulses [1,2]. As
a first step to understand the soliton transmission
through disordered media, one has to study the ~
liton scattering by a single impurity. The study of the
so-called dynamical solitons has been carried out in
a number of papers (see, e.g., refs. [3—6]) taking into
account one or two impurities. In particular numerical and experimental investigations have shown that
due to the interaction with one impurity the amplitude of the soliton decreases a little, but the soliton
can pass the impurity almost freely. In such an interaction, the soliton loses its energy to generate a
reflected wave, and in the case of a light-mass impurity, to excite a localized mode in the vicinity of
the impurity site, the so-called impurity mode [3—5].
On the other hand, the kink—impurity interactions
have been explained, for a long time, by the wellknown model in which a kink moving in an inhomogeneous medium is considered as an effective
particle, the kink position being the particle coordinate (see ref. [7—91
and references therein). In this
by Malomed [10,11] that a sine-Gordon (sG) kink
may be trapped by an attractive impurity due to radiative losses and the threshold velocity was found
analytically. However, the previous theoretical studies [7—11]totally ignored the fact that the underlying nonlinear system supports a localized impurity
mode which may be excited due to the kink scattering. The spectrum of the linearized sG model has a
gap, so that the impurity modes in a linear system
are permitted either below or above the frequency
band, the latter is possible for a discrete model when
the cut-offfrequency exists. In the nonlinear case the
impurity modes may be treated as breather excitations, which are trapped by the impurities [12].
The importance of the impurity mode in the kink
scattering was firstly observed in numerical simulations of the 0” chain [13]. It has been demonstrated that an incident kink may excite the impurity
mode, so that, after the reflection or transmission of
the kink, the impurity continues to vibrate with undiminished amplitude. However, a kink cannot be
trapped by a heavy mass impurity investigated in ref.
[1 3].
In the present paper we will demonstrate that the
sG kink may be trapped by a local impurity due to
loss of its kinetic energy to excite the impurity mode,
and in this scattering process the impurity mode excitation is more important than radiative losses. We
I
2
OnleavefromlnstituteforLowTemperaturePhysicsandEngineering, 47 Lenin Avenue, 310164 Kharkov, USSR.
P.P. Shirshov Institute for Oceanology, 117218 Moscow,
USSR.
318
Elsevier Science Publishers B.V.
Volume 159, number 6,7
PHYSICS LETTERS A
develop a collective-coordinate approach to describe
the kink—impurity interactions and calculate the
threshold velocity of kink capture analytically and
numerically. We find that the perturbation theory
taking into account only radiativelosses is valid only
for very small impurity amplitudes, while the collective-coordinate method can give ~ good estimation of the threshold velocity for a wide range of impurity amplitudes. We also point out that the
collective-coordinate approach may predict a surprising new effect, i.e., the kink may be totally re,flected by an attractive impurity due to resonant energy exchange between the kink translational mode
and the impurity mode.
The paper is organized as follows. In section 2 we
discuss the model which supports an impurity mode.
In section 3 we develop a collective-coordinate approach and calculate the threshold velocity of kink
capture. Section 4 concludes the paper.
21 October 1991
fluxon, propagating along the junction (see, e.g., ref.
[8]).
To describe the motion of the kink (2) in the presence of a localized inhomogeneity, the adiabatic perturbation theory for solitons was usually used [7—9].
In the framework ofthis perturbation theory, the kink
coordinate X is considered as a collective variable,
and its evolution is described by a simple motion
equation ofa classical particle with mass m=8 placed
in the effective potential
2f
(3)
cosh2X~
For ~>0, the impurity in eq. (1) gives rise to an attractive potential U(X) to the kink. Since the partide (kink) conserves its energy, it cannot be trapped by the potential well if it has non-zero velocity
at infinity. However, according to the results of
Malomed [10] (see also ref. [111), the kink may be
U(X)
—
The model we deal with is the sG system including
a local inhomogeneity,
trapped by an attractive impurity due to radiative
losses and there exists a threshold velocity Vthr(~),
such that if the kink initial velocity is larger than
Vthr(~)it will pass through the impurity and escape
to infinity, otherwise it will be trapped in the potential well caused by the impurity [10]. The threshold
~
velocity was found to be
2. Model and the impurity mode
[l—~ô(x)] sin u=0,
(1)
where ô(x) is the Dirac ö-function. From the physical point of view the model (1) may correspond, for
example, to a long Josephson junction (LJJ) with a
point-like inhomogeneity, u being the normalized
magnetic flux. The constant parameter ~ is a
“strength” ofthe inhomogeneity, while the cases ~>0
and <0 correspond to the so-called microresistor
and microshunt respectively. There are many other
physical problems which may be modeled by the perturbed sG equation (1) (see refs. [8,9] and references therein).
When the perturbation is absent (~=0),the sG
model (I) supports a topological soliton, the socalled kink which is given by
Uk(X, t)=uk(z)=4
tan—’ exp(az)
,
(2)
V,h~22USJtI~’4~3/t
exp(—~J~7~)
.
By numerical simulations, we find that this formula is valid only for very small . The reason is that
the perturbation theory used in ref. [101 totally ignored the fact that the attractive impurity supports
a localized oscillating state, the so-called impurity
mode, which may be easily excited due to kink scattering (see fig. 1). Therefore, at the first interaction,
the kink kinetic energy is transferred not only to radiation but also to the impurity mode. By linearizing
eq. (1) in small u, the shape of the impurity mode
can be found analytically,
Uim(X, t)
=a(t) exp( —f xl /2)
—
,
(5)
where a(t)=a
0cos(Qt+00), Q is the frequency of
2, X= Vt+X
where z= (x—X)/~Jl V
0 is the kink
coordinate,
V is its loss
velocity,
and a=we
±1assume
is the
kink polarityk=
(without
of generality
that a= + 1). In the theory of LJJ, the kink solution
(2) describes a magnetic flux quantum, known as a
(4)
the impurity
2/4mode,
Q=\/1
(6)
f
and O~is an initial phase. As a matter offact, the impurity mode (5) can be considered as a small-am319
Volume 159, number 6,7
PHYSICS LETTERS A
and assuming a and f are small enough so that the
higher-order terms can be neglected, we may derive
7.0
5.0
d
21 October 1991
the following (reduced) effective Lagrangian,
2 + (1 / f) (a2 Q2a2)
Leff = 4X
—
3.0
1.0
—U(X)—aF(X),
50
~
(10)
where U(X) is given by eq. (3), and
200
T
Fig. 1. The impurity displacement u(0, 1) calculated by numerical simulation of eq. (1) where =0.5. It shows that above the
threshold velocity the kink passes the impurity inelastically losing part of its energy to excite the impurity mode, which forms a
long-lived oscillating state.
plitude breather trapped by the impurity [13], with
energy
Eim = 1
)~
[(ôuim\2
8t1
(~jm~~~2
\ax,
[1 fö(x)
]U~m]
=Q2a~/f.
(7)
For a given parameter f, the energy in the impurity
mode depends on the maximum amplitude a
0. It is
clear that this amplitude should be a function of the
initial velocity of the kink, which excites the impurity oscillation. In the next section we are going to calculate the amplitude of the impurity mode and estimate the threshold velocity of kink capture.
3. Collective-coordinate approach and the threshold
velocity
We shall analyse the kink—impurity interactions
by the so-called collective-coordinate approach taking into account two dynamical variables, namely,
the kink coordinate X(t) (see eq. (2)) and the amplitude
the impuritythe
mode
oscillation a ( t) (see
eq.
(5)).ofSubstituting
ansatz
exp[x—X(t)]
u=uk+u~m=4 tan
+a(t) exp( —f lxi /2)
into the Lagrangian of the system,
L=
320
{ u~ ~
~
—
—
[1—eó(x) ] (1
—
cos u) },
X
F(X)=—2f tanh
cosh x~
(11)
The equations of motion for the two dynamical vanables are
81+U’(X)+aF’(X)=O,
ä+Q2a+~fF(X)=0.
(12)
The
system (12)
a particle
with pocoordinate
anddescribes
massand
8 placed
in an(kink)
attractive
tential U(X(t)
X) (f>
0),
“weakly”
coupled
with a
harmonic oscillator a(t) (the impurity mode). Here
we say “weakly” because the coupling term aF(X)
is of order 0(f) and it falls off exponentially. The
system (12) is a generalization of the well-known
equation 8X= U’(X) describing the kink—impurity interactions in the adiabatic approximation (see,
e.g., ref. [9]).
We find that the dynamical system (12) can describe all features of the kink—impurity interactions
—
(for f> 0). Firstly, it may be used to calculate the
threshold velocity of kink capture. The main idea is
to use an energy transfer argument. Let us consider
a kink with initial velocity V> 0, coming from
then in the first-order approximation the equation of
motion for the kink coordinate X(t) takes the form
8)~+U’(X) =0, which has an exact solution
—
X(t)=sinh~2[A+sinh(
~f/V. Vt)]
Now we may insert (13) (13)
into
where
A = ,.,/equation
V
the second
of (12) and consider
,
(8)
f2A sinh( Vt)
(14)
f(t)
~F(X(t) = 1 +A2 sinh2( Vt)
as a pulse force acting on the harmonic oscillator. In-
(9)
troducing the complex variable c~(t)=â+iQa, we
obtain an equation
iQ~=f(t), which has the
solution
—
~—
Volume 159, number 6,7
~(t)
Jf(r) e
=e~
PHYSICS LETTERS A
dr,
—~
~ 1
~
t)
e
—
~ dt
f
=
~
J
1 +A2 sinh2( Vt)
2[QZ(V)/2V]
=27t 2f 2 sinh
cosh2(Qit/2V)
=
dos
-‘
f2V2—f\
(~v2 +
/
/
/
/
/
_—~
0.0
o.o
0.2
0.4
*
0.6
0.8
Fig. 2. The threshold velocity of kink capture as a function of
determined by eq. (18) (solid line) as well as eq. (4) (dashed
line). The dotted line is obtained by numerical simulation of the
re-
suits of numerical simulations of eq. (1).
2
dt
(16)
where
Z( V)
/
collective-coordinate equations (12). The asterisks are the
J
1 f 2A sinh( Vt) sin(Qt)
—
/
/
/
0.4
—
2
—
1.2
0.8
(15)
where we have used the initial conditions
a ( ao) = a ( x) = 0. The total energy transferred
from the particle (kink) to the oscillator (impurity
mode) may be calculated as
2a2)
I ~( + ~) 12
E0~~
=
(a +Q
f
—
21 October 1991
(17)
At the threshold velocity V= Vthr, the particle must
transfer all of its kinetic energy, Ek=4V2, to the oscillator. Thus the threshold velocity of kink capture
should be determined by
Jtf sinh[QZ( Vthr)/2Vthr]
Vthr
h(Q/2V)
(18)
For a given f> 0, this equation can be solved by
Newton iteration to obtain the threshold velocity
V,hr(f). In fig. 2 we have plotted the threshold yelocity determined by eq. (18) as well as by eq. (4).
Comparing the analytical results with the direct numerical simulations (asterisks in fig. 2), we find that
the perturbation theory used in ref. [10] is valid only
for very small f (f ~ 0.05), while formula (18) gives
good estimations of V,h. ( f) for f over the region (0.2,
0.7).
Furthermore, eqs. (12) can be easily solved numerically. We discretize eqs. (12) by a conservative
scheme [14] and perform the numerical simulations
under initial conditions X(0)= —7, X(0)= V
1>0,
a (0) = 0, a (0) = 0. We find that, for a given E>-0,
there exists a threshold velocity Vlhr(f) such that if
the initial velocity of the particle is larger than
Vthr(f), then the particle will pass the attractive potential well U(X) and escape to +~, with final velocity Vf< V
part ofWe
its kinetic
energythe
is
transferred 1tobecause
the oscillator.
have plotted
threshold velocity determined by numerical simulations of eqs. (12) (dotted line in fig. 2). It is clear
that the results agree with the numerical simulation
of eq. (1) for f over the interval (0, 0.7).
Below the threshold velocity, the behaviour of the
particle is very interesting. More precisely, ifthe mitial velocity of the particle is smaller than the threshold velocity, the particle cannot escape to +oo after
the first interaction with the oscillator, but it will return to interact with the oscillator again. Usually the
particle can be trapped by the potential well, which
corresponds to a kink trapped by the attractive impurity. However, for some special initial velocities,
the second interaction may cause the particle to escape to x with final velocity Vf< 0. This resonance phenomenon can be explained by the mechanism of resonant energy exchange between the
particle and the oscillator [15,16]. The resonant reflection of the particle by the potential well corresponds to the reflection of the kink by the attractive
impurity. Recently, by direct numerical simulations
of eq. (1), we have indeed observed such resonance
phenomena, i.e., below the threshold velocity, a kink
is not necessarily to be trapped by the impurity, instead it may still escape (but to the opposite direction) from the attractive impurity, if its initial ye—
locity is situated in some well-defined narrow
intervals (resonance windows). This effect is quite
similar to the resonance phenomena in kink—antikink collisions in some non-integrable Klein—
321
Volume 159, number 6,7
PHYSICS LETTERS A
21 October 1991
Gordon equations [16—19]. The detailed results
concerning the resonance structure will be published
elsewhere [15].
Kivshar thanks Michel Peyrard and Rick Boesch for
valuable discussions, and he acknowledges the financial support of Complutense University through
a sabbatical program.
4. Conclusion
References
In conclusion, we have demonstrated analytically
and numerically that the impurity mode plays a very
important role in the kink scattering by a local inhomogeneity. In particular, we have shown that a
sine-Gordon kink may be trapped by an attractive
impurity due to loss of its kinetic energy to excite the
impurity mode. We have developed a collective-coordinate approach to describe the interaction and
calculated the threshold velocity of kink capture. By
comparing the analytical results with numerical simulations, we have found that the perturbation theory
used in refs. [10,11], which took into account only
radiative losses of the kink, is valid only for very
small , while the collective-coordinate approach can
give good estimation of the threshold velocity of kink
capture for a wide region of the impurity amplitude.
Moreover, by taking into account the energy exchange between the kink translational mode and the
impurity mode we may predict a possible resonance
phenomenon in the kink—impurity interactions.
Acknowledgement
This work is partially supported by the Direccion
General de Investigacion Cientifica y Tecnica
(Spain) under Grant No. TIC 73/89. One of us
(Zhang Fei) is grateful to the Ministry of Education
and Science of Spain for a research fellowship. Yu.S.
322
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