#### Scattering of the double sine-Gordon kinks

Eur. Phys. J. C
Scattering of the double sine-Gordon kinks
Vakhid A. Gani 1 2
Aliakbar Moradi Marjaneh 0
Alidad Askari 4
Ekaterina Belendryasova 2
Danial Saadatmand 3
0 Young Researchers and Elite Club, Quchan Branch, Islamic Azad University , Quchan , Iran
1 Theory Department, National Research Center Kurchatov Institute, Institute for Theoretical and Experimental Physics , Moscow 117218 , Russia
2 National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) , 115409 Moscow , Russia
3 Department of Physics, University of Sistan and Baluchestan , Zahedan , Iran
4 Department of Physics, Faculty of Science, University of Hormozgan , P.O.Box 3995, Bandar Abbas , Iran
We study the scattering of kink and antikink of the double sine-Gordon model. There is a critical value of the initial velocity vcr of the colliding kinks, which separates different regimes of the collision. At vin > vcr we observe kinks reflection, while at vin < vcr their interaction is complicated with capture and escape windows. We obtain the dependence of vcr on the parameter of the model. This dependence possesses a series of local maxima, which has not been reported by other authors. At some initial velocities below the critical value we observe a new phenomenon - the escape of two oscillons in the final state. Besides that, at vin < vcr we found the initial kinks' velocities at which the oscillons do not escape, and the final configuration looks like a bound state of two oscillons.
1 Introduction
The (1 + 1)-dimensional field-theoretical models
possessing the topologically non-trivial solutions – kinks – are of
special interest for modern physics. They arise in a vast
variety of models in quantum and classical field theory, high
energy physics, cosmology, condensed matter physics, and
so on, [
1–4
]. Firstly, the (1 + 1)-dimensional models can be
investigated analytically and numerically much easier than
(2+1) or (3+1)-dimensional. Because of that, some general
properties of topological defects can be studied within the
(1 + 1)-dimensional setups. Secondly, many physical
systems can be effectively described by the one-dimensional
structures. For example, a plane domain wall – a wall, which
separates regions with different vacuum states – in the
direction orthogonal to it, presents a kink. Surely, the topological
defects arise and in more complex models with two, three or
more fields. For example, in [
5–11
] the kink-like structures
were studied in models with two interacting real scalar fields,
for further information see also [
12–20
].
Kink–antikink collisions, as well as interactions of kinks
with impurities (spatial inhomogeneities), are of growing
interest since 1970s [
21,22
]. Nevertheless, today it is a very
fast developing area of research. For investigating of the
kink–antikink interactions various approximate methods are
widely used. Among them, the collective coordinate
approximation [
23–31
]. Withing this method a real field system “kink
+ antikink” is approximately described as a system with one
or several degrees of freedom. For instance, the kink-antikink
separation can be used as the only (translational) degree of
freedom. The more complicated modifications of the method
have also been elaborated, which include other degrees of
freedom (in particular, vibrational), see, e.g., [
24–26
].
Another approximate method for investigating the kinks
interactions is Manton’s method [3, Ch. 5], [
32–35
]. This
method is based on using of the kinks asymptotics, it enables
to estimate the force between the kink and the antikink at
large separations.
On the other hand, recently the numerical simulation
has become a powerful tool for studying the dynamics of
the one-dimensional field systems. Using various
numerical methods, many important results were obtained. In
particular, the resonance phenomena – escape windows and
quasi-resonances – have been found and investigated in the
kinks’ scattering [
23,36–48
]. Many important results have
been obtained for the models with polynomial potentials
of fourth, sixth, eighth, and higher degree self-interaction
[
23,37–42,44,45,49–56
]. One should note interesting results
on the long-range interaction between kink and antikink
[
40,44,45,57–60
]. The models with non-polynomial
potentials are also widely discussed in the literature, for
exam
∞
−∞
which is a vacuum manifold of the model, and V (φ) = 0
for all φ ∈ V. The energy functional corresponding to the
Lagrangian (1) is
(3)
(4)
(5)
(6)
(7)
(8)
ple, the modified sine-Gordon [
61
], the multi-frequency
sineGordon [
62
], the double sine-Gordon [
36,63–66
], and a
number of models, which can be obtained by using the
deformation procedure [
46,47,54,67,68
].
The impressive progress is achieved in the investigation
of domain walls, bubbles, vortices, strings [
69–77
], as well
as the embedded topological defects, e.g., a Q-lump on a
domain wall, a skyrmion on a domain wall, etc. [
78–88
].
Besides that, we have to mention various configurations of
the type of Q-balls [
89–96
]. Topologically non-trivial field
configurations could also lead to a variety of phenomena in
the early Universe [
97,98
].
In this paper we study the kink-antikink collisions within
the double sine-Gordon model [
36
], [
64–66
]. There is a
critical value of the initial velocity of the colliding kinks,
vcr, which separates different regimes of the collision. At
vin > vcr the kinks pass through each other and escape to
infinities, while at vin < vcr the kinks’ capture and a complex
picture of the so-called escape windows are observed, see,
e.g., [
21,22,64
]. We performed a detailed study of the
kinkantikink scattering at various values of the model parameter
R. We have found a series of local maxima of the
dependence of vcr on R, which has not been reported up to now.
Besides that, at some initial velocities of the colliding kinks
we observed final configuration of the type of two oscillons,
which form a bound state or could escape to spatial infinities
with some final velocities.
Our paper is organized as follows. In Sect. 2 we give some
general information about the (1 + 1)-dimensional models
with one real scalar field. Section 3 introduces the double
sine-Gordon model, describes its potential, kinks, and their
main properties. In Sect. 4 we study the scattering of the
kink and antikink. In this section we present our main results
related to the kink–antikink collisions. Finally, we
summarize and formulate prospects for future works in Sect. 5.
2 Topological defects in (1 + 1) dimensions
Consider a field-theoretical model in (1 + 1)-dimensional
space-time with a real scalar field φ (x , t ). The dynamics of
the field φ is described by the Lagrangian density
1
L = 2
∂φ
∂t
2
1
− 2
∂φ
∂ x
2
− V (φ),
where the potential V (φ) defines self-interaction of the field
φ. We assume that the potential is a non-negative function of
φ, which has a set of minima
V =
φ1(vac), φ2(vac), φ3(vac), . . . ,
In order for the energy of the static configuration to be finite,
it is necessary that
φ (−∞) = x →lim−∞ φ (x ) = φi(vac)
and
φ (+∞) = x →lim+∞ φ (x ) = φ(jvac),
(1)
(2)
where φi(vac), φ(jvac) ∈ V. If these two equalities hold, the
second and the third terms in square brackets in Eq. (3) fall
off at x → ±∞ (the first term turns to zero for all static
configurations), and the integral in Eq. (3) can be convergent.
If the vacuum manifold V consists of more than one point,
i.e the potential V (φ) possesses two or more degenerate
minima, the set of all static configurations with finite energy can
be split into disjoint equivalence classes (or topological
sectors) according to the asymptotic behaviour of the
configuration at x → ±∞. Configurations with φi(vac) = φ(jvac)
in Eqs. (7) and (8) are called topological, while those with
φi(vac) = φ(jvac) – non-topological. A configuration
belonging to one equivalence class (topological sector) can not be
transformed into a configuration from another class
(topological sector) through a continuous deformation, that is via
a sequence of configurations with finite energies.
To describe the topological properties of the
configurations, one can introduce a conserved topological current, e.g.,
μ 1 εμν ∂ν φ,
jtop = 2
here εμν stands for the Levi-Civita symbol, the indices μ and
ν take values 0 and 1 for a (1+1)-dimensional configuration,
and ∂0φ ≡ ∂∂φt , ∂1φ ≡ ∂∂ φx . The corresponding topological
charge does not depend on the behaviour of the field at finite
x ,
Qtop =
∞
The value of Qtop is determined only by the asymptotics
(7), (8) of the field. The topological charge (10) is conserved
during the evolution of the configuration. Nevertheless,
configurations from different topological sectors may have the
same topological charge. At the same time, configurations
with different topological charges necessarily belong to
different topological sectors.
Further, for the non-negative potential V (φ) we can
introduce the superpotential – a smooth (continuously
differentiable) function W (φ) of the field φ:
1
V (φ) = 2
d W
dφ
2
Using this representation of the potential, the energy of a
static configuration can be written as
(9)
(10)
(11)
(12)
(13)
(14)
(15)
urated) configurations. Note that Eq. (15) coincides with
Eq. (6).
A kink is a BPS saturated topological solution φk(x ) of
Eq. (6), which connects two neighboring vacua of the model,
i.e. for the kink solution the values φi(vac) and φ(jvac) in Eqs.
(7), (8) are adjacent minima of the potential V (φ). Below
we use the terms “kink” and “antikink” for solutions with
φ(jvac) > φi(vac) and φ(jvac) < φi(vac), respectively.
Nevertheless, in some cases we use “kink” for both solutions, just to
be brief.
3 The double sine-Gordon model
Consider the double sine-Gordon (DSG) model. The
potential of the DSG model can be written in several different
forms. Below in this section we briefly recall two of them,
and after that we give a detailed introduction to the properties
of the DSG model employed by us.
3.1 The η-parameterized potential
Recall that, e.g., in papers [
36,64
] the following
parameterization has been used:
4
Vη(φ) = 1 + 4|η|
φ
η (1 − cos φ) + 1 + cos 2
,
(16)
where η is a real parameter, −∞ < η < +∞. It is easy to
see that
(17)
⎧⎪ cos φ − 1 for η → −∞,
Vη(φ) = ⎪⎪⎨ 1 − cos φ fφor η → +∞,
⎪⎪⎪⎩ 4 1 + cos 2
for η = 0,
i.e. the potential V (φ) reduces to a sine-Gordon form for the
field φ at η → ± ∞, and for the field φ/2 at η = 0.
The shape of the potential (16) crucially depends on the
parameter η. Following [
64
], we can split all values of η into
four regions: η < − 41 , − 41 < η < 0, 0 < η < 41 , and
1
η > .
4
1. At η < − 41 the potential (16) has two distinct types of
minima, φn(vac) = 4π n + arccos 41η and φm(vac) = 4π m −
1
arccos , n, m = 0, ±1, ±2, . . ., degenerate in energy,
4η
V (φn(vac)) = V (φm(vac)) = 0, which are separated by
inequivalent barriers.
2. At − 41 < η < 0 the potential (16) has a single type of
minima at φn(vac) = (2n + 1)2π with V (φn(vac)) = 0.
where
EBPS =
W [φ (+∞)] − W [φ (−∞)] .
E ≥ EBPS,
dφ d W
d x = ± dφ ,
Here the subscript “BPS” stands for Bogomolny, Prasad,
and Sommerfield [
99–101
]. From Eq. (12) one can see that,
firstly, the energy of any static configuration is bounded from
below by EBPS,
and, secondly, the static configuration, which satisfies the
equation
saturates the inequality (14), i.e has the minimal energy (13)
among all the configurations within a given topological
sector. The solutions of Eq. (15) are called BPS (or BPS
sat(21)
(23)
(25)
(26)
(27)
(28)
or
and
3.3 The double sine-Gordon kinks
In terms of the parameter R the static kink (+) and antikink
(−) solutions can be written in a simple form,
sinh x
φk(k¯)(x ) = 4π n ± 4 arctan cosh R
The DSG kink (antikink) can also be expressed as a
superposition of two sine-Gordon solitons,
φk(k¯)(x ) = 4π n ± φSGK(x + R) − φSGK(R − x ) ,
(22)
φk(x ) = 2π(2n − 1) + φSGK(x + R) + φSGK(x − R)
φk(x ) = 2π(2n + 1) − φSGK(x + R) + φSGK(x − R) ,
¯
(24)
where φSGK(x ) = 4 arctan exp(x ) is the sine-Gordon soliton.
According to Eqs. (22)–(24), the DSG kink can be viewed
as the superposition of two sine-Gordon solitons, which are
separated by the distance 2 R and centered at x = ±R, see
Fig. 2. The energy of the static DSG kink (antikink) is a
function of the parameter R,
2 R
E (R) = 16 1 + sinh 2 R
,
this dependence is shown in Fig. 3. Below we study the
collisions of the DSG kinks. In such processes the kink’s internal
modes may be very important. Therefore, now we investigate
the spectrum of small localized excitations of the DSG kink
(antikink) using a standard method. Namely, we add a small
perturbation δφ (x , t ) to the static DSG kink φk(x ):
φ (x , t ) = φk(x ) + δφ (x , t ), |δφ|
|φk|.
Substituting this φ (x , t ) into the equation of motion (4), and
linearizing in δφ, we obtain the partial differential equation
for δφ (x , t ):
the minima of the potential (16) are φn = 4nπ
with V (φn) = 1 +84|η| , and φm(vac) = (2m + 1)2π with
V (φm(vac)) = 0, n, m = 0, ± 1, ± 2, . . ..
This our paper deals with the DSG model with the positive
values of η. In this case, as we show below, it is convenient
to introduce another positive parameter.
3.2 The R-parameterized potential
For positive values of η, it is convenient to introduce another
positive parameter, R, such that
1
η = 4 sinh2 R.
In terms of the parameter R the potential (16) of the DSG
model reads:
4
VR(φ) = tanh2 R (1 − cos φ) + cosh2 R
φ
1 + cos 2
Depending on the parameter R the shape of the potential (19)
looks different, see Fig. 1. At R = 0 we have a sine-Gordon
potential for the field φ/2, while at R → +∞ the potential
(19) reduces to a sine-Gordon form for the field φ:
⎧ φ
VR(φ) = ⎨⎪⎪⎪⎪⎪ 4 1 + cos 2
φ 2
⎪⎪ 4 − 1 − cos 2
⎪⎩⎪⎪ 1 − cos φ for R → +∞.
for R = 0,
for R = arcsinh 1,
(18)
Looking for δφ in the form
δφ (x , t ) =
ηn(x ) cos ωnt,
n
· δφ = 0.
It can be easily shown that the discrete spectrum in the
potential (31) always possesses a zero mode ω0 = 0.
Differentiating Eq. (5) with respect to x , and taking into account that
φk(x ) is a solution of Eq. (5), we see that
d2 dφk d2V
− d x 2 d x + dφ2 φk(x)
dφk
· d x
= 0,
(29)
(30)
(31)
(32)
or, in other words,
dφk
Hˆ · d x
dφk
= 0 · d x .
So dφk is really an eigenfunction of the Hamiltonian (30)
d x
associated with the zero frequency.
The potential U (x ) for the double sine-Gordon kink
(antikink) can be obtained by substituting Eqs. (19) and (21)
in (31),
8 tanh2 R
U (x ) = (1 + sech2 R sinh2 x )2
+
2(3 − 4 cosh2 R)
cosh2 R
1
1 + sech2 R sinh2 x + 1. (34)
The shape of the potential crucially depends on the parameter
R, see Fig. 4a. For R = 0 Eq. (34) gives the Pöschl-Teller
potential,
2
U0(x ) = 1 − cosh2 x ,
which corresponds to the case of the sine-Gordon model. On
the other hand, for R 1 from Eq. (34) we obtain
⎩
2
U∞(x ) ≈ ⎧⎨ 1 − cosh2(x − R) for ||x | − R|
1
for ||x | − R|
1,
1.
The discrete spectrum in the potential well (34) for
arbitrary value of R can be obtained numerically by using a
modification of the shooting method, see, e.g., [
38,44,45
]. First of
all, for all R there is the zero mode ω0 = 0. Apart from that,
we have found the vibrational mode ω1 with the frequency
that depends on R, see Fig. 4b.
At R → 0 the frequency ω1 goes to the boundary of the
continuum, which corresponds to the sine-Gordon case. With
(33)
(35)
(36)
increasing R the frequency ω1 decreases to zero. At large R’s
the levels ω0 and ω1 are the result of splitting of the zero mode
of each of the two potential wells (36).
4 Collisions of the double sine-Gordon kinks
We studied the collision of the DSG kink and antikink. In
order to do this, we used the initial configuration in the form
of the DSG kink and the DSG antikink, centered at x = −ξ
and x = ξ , respectively, and moving towards each other
with the initial velocities vin. We solved the partial
differential equation (4) with the R-parameterized potential (19)
∂φ (x , 0)
∂t
numerically, extracting the values of φ (x , 0) and
from the following initial configuration:
⎛ x + ξ − vint ⎞ ⎛ x − ξ + vint ⎞
φ (x , t ) = φk ⎝ 2 ⎠ + φNk ⎝ 2 ⎠ − 2π
1 − vin 1 − vin
= 4 arctan
−4 arctan
⎡
1
⎣ cosh R
⎛ x + ξ − vint ⎞ ⎤
sinh ⎝
2 ⎠ ⎦
1 − vin
⎡
1
⎣ cosh R
⎛ x − ξ + vint ⎞ ⎤
sinh ⎝ 2 ⎠ ⎦ − 2π.
1 − vin
We discretized space and time using a grid with the spatial
step h, and the time step τ . We used the following discrete
expressions for the second derivatives of the field:
11φn, j+1 − 20φn, j + 6φn, j−1 + 4φn, j−2 − φn, j−3 ,
12τ 2
−φn−2, j + 16φn−1, j − 30φn, j + 16φn+1, j − φn+2, j ,
12h2
(37)
(38)
(39)
∂2φ
∂t2 =
∂2φ
∂ x2 =
where φn, j = φ (nh, j τ ), n = 0, ±1, ±2, . . ., and j =
−3, −2, −1, 0, 1, 2, . . ..
We performed the numerical simulations for the steps h =
0.025 and τ = 0.005, respectively, and for two different ξ : 10
and 20. We have also checked the stability of the results with
respect to decrease of the steps. Fixed boundary conditions
were used.
In the kink-antikink collisions there is a critical value of the
initial velocity, vcr, which separates two different regimes of
the collisions. At the initial velocities above the critical value,
vin > vcr, the DSG kinks pass through each other and escape
to infinities after one collision, see Fig. 5a. At vin < vcr one
observes the kinks’ capture and formation of their long-living
bound state – a bion, see Fig. 5b. At the same time, in the
range vin < vcr the so-called “escape windows” have been
found. An escape window is a narrow interval of the initial
velocities, at which the kink and the antikink escape after
two, three, or more collisions, see Fig. 5c, d.
4.1 The R-dependence of the critical velocity
First of all, we found the dependence of the critical velocity
vcr on the parameter R. Our results are shown in Fig. 6. One
can see a series of peaks on the curve vcr(R), see also Table
1. Note that at this point we have some discrepancy with the
results of [
64
]. The authors of [
64
] report only one maximum
of the dependence vcr(R) at R ≈ 1. Probably it can be a
consequence of small amount of experimental points in [
64
].
From Fig. 6 one can see that the critical velocity turns
to zero at R = 0, which corresponds to the integrable
sineGordon model, see Eq. (20). Besides that, vcr decreases to
zero with increasing R at large R. Remind here, that the limit
R → +∞ also corresponds to the case of the integrable
sine-Gordon model, as one can see from Eq. (20). Therefore
it is quite natural that the critical velocity has a maximum at
some R and tends to zero at R → 0 and R → +∞. The
presence of a series of local maxima on the curve is an
interesting fact that is observed for the first time. Apparently we
can assume that one of the maxima is the main [probably,
R(max)], while the other ones appear due to some change of
1
the kink–antikink interaction in the collision process with
increasing R. At large values of R the DSG kink splits into
two sine-Gordon solitons. Therefore, we can assume that in
the DSG kink–antikink collision the four sine-Gordon
solitons interact pairwise. This transition from the simple
collision of the DSG kinks to more complicated pairwise
interaction of the sine-Gordon solitons can lead, in particular, to the
non-monotonicity of the dependence vcr( R) at R > R(max).
1
4.2 Two oscillons in the final state
In the kink–antikink collisions below the critical velocity
we observed a phenomenon, which, to the best of our
knowledge, has not been reported for the double sine-Gordon model
before. At some initial velocities of the colliding kinks we
observed final configuration in the form of two escaping
oscillons. At the same time, at some initial velocities we
found formation of the configuration, which we can classify
as a bound state of two oscillons. In Fig. 7 we show some
typical scenarios of that kind.
For example, at the initial velocity vin = 0.1847 we
observe formation of a bound state of the kink and the
antikink (a bion), which then evolves into two oscillons.
These two oscillons are moving from each other, then stop,
and start moving back to the collision point. This repeats
several times, and after that the oscillons escape to infinities
with the final velocity vf ≈ 0.10, see Fig. 7a.
Formation of the bound state of oscillons and the escape
of oscillons are extremely sensitive to changes of the initial
velocity of the colliding kinks. For example, at the initial
velocity vin = 0.18467 the oscillons escape to infinities after
vcr
0.24
Fig. 6 The critical velocity vcr as a function of the parameter R. (The
initial half-separation is ξ = 20 in these calculations)
Table 1 Positions of the local
maxima of the dependence
vcr(R), which is shown in Fig. 6
fewer number of collisions, see Fig. 7b. The final velocity
of the escaping oscillons also differs substantially: at vin =
0.1847 we obtain vf ≈ 0.10 (Fig. 7a), while at vin = 0.18467
we have vf ≈ 0.03 (Fig. 7b), and at vin = 0.18470001 we
obtain vf ≈ 0.15 (Fig. 7c). At the initial velocity of the
colliding kinks vin = 0.18470003, Fig. 7d, the final velocity
of the escaping oscillons is vf ≈ 0.19.
In the kink–antikink collision at the initial velocity vin =
0.18472, Fig. 7e, we observe formation of two oscillons,
which are moving apart from each other, then approach and
collide. After that we have final configuration in the form of
the oscillating configuration of the type of bion at the origin.
Apparently this final configuration can be viewed as a bound
state of two oscillons, which oscillate around each other with
small amplitude.
At the initial velocity vin = 0.18473, Fig. 7f, we observe
even more complicated picture. The kinks collide, form a
bion, which, in turn, decays into two oscillons. These
oscillons escape at some distance and then collide again. After
that, for some time we observe the bound state of oscillons –
small amplitude oscillations of oscillons around each other.
Finally, the oscillons escape at some valuable distance,
collide for the last time, and escape to infinities with the final
velocities vf ≈ 0.07.
The obtained results show that in the DSG kink–antikink
scattering we found new phenomenon – formation of the pair
of oscillons, which can form a bound state or escape to
spatial infinities. Note that similar behaviour has been observed
recently in the collisions of kinks of another model with
nonpolynomial potential [
46,47
].
5 Conclusion
We have studied the scattering of kinks of the double
sineGordon model. Several different parameterizations of this
model are known in the literature. We used the so-called
Rparameterization, in which the potential of the model depends
on the positive parameter R, see Eq. (19).
The scattering of the DSG kink and antikink looks as
follows. There is a critical value of the initial velocity vcr such
that at vin > vcr the kinks pass through each other and then
escape to infinities. At vin < vcr one observes formation of a
bound state of the kinks – a bion. Besides that, at some narrow
intervals of the initial velocity (which are called “escape
windows”) from the range vin < vcr the kinks escape to infinities
after two or more collisions.
We have obtained the dependence of the critical velocity
vcr on the parameter R. The curve vcr(R) has several
wellseen local maxima, see Fig. 6 and Table 1. Note some
discrepancy between our results and the results of [
64
]. The authors
of [
64
] reported only one maximum of the curve vcr(R). This
could be a consequence of small number of experimental
points between R = 1.8 and R = 2.4 presented in [
64
].
Apart from the previously known bions and escape
windows, in the range vin < vcr in our numerical experiments
we observed a new phenomenon, which could be classified
as formation of a bound state of two oscillons, and their
escape in some cases. So at some initial velocities of the
colliding kinks, in the final state we observed two oscillons
escaping from the collision point. The time between the first
kinks impact and the beginning of the oscillons escaping can
be rather big. The field evolution during this time is quite
complicated. First, we observe formation of a bion. After
a short time, this bion evolves into a configuration, which
can be identified as a bound state of two oscillons
oscillating around each other. The amplitude of these oscillations
can vary substantially. After that the oscillons either remain
bound or escape to spatial infinities, depending on the initial
velocity of the colliding kinks. It is interesting that
formation of a bound state of two oscillons, as well as escape of
oscillons, has been found recently in the collisions of kinks
of the sinh-deformed ϕ4 model [
46,47
]. We think that this
new phenomenon can be a part of new interesting physics
within a wide class of non-linear models.
We can assume that the escape of oscillons is a kind of
resonance phenomena, i.e. it is a consequence of the resonant
energy exchange between oscillon’s kinetic energy and its
internal vibrational degree(s) of freedom. A detailed study
of such exchange could be a subject of future work.
In conclusion, we would like to mention several issues
that we think could become a subject of future study.
– First, it would be interesting to explain the behaviour of
the dependence vcr( R) with a series of local maxima.
This non-monotonicity could be a consequence of the
kink’s shape changing with increasing of the
parameter R. So at large R’s the interaction of the DSG kinks
could be reduced to pairwise interaction of the subkinks,
which are the sine-Gordon solitons separated by the
distance 2 R. Note that the authors of [
26
] observed the
non-monotonic dependence of vcr on the model
parameter in the parametrically modified ϕ6 model. In order to
explain the phenomenon, they applied the collective
coordinate approach. We believe that similar analysis could be
applied to the double sine-Gordon kink-antikink system.
– Second, the oscillons escape in the final state, as well
as formation of a bound state of two oscillons, are new
interesting phenomena, which have to be explained
qualitatively and probably quantitatively.
– Third, it would be very interesting to study multikink
collisions within the DSG model in the spirit of [
42
].
Due to complex internal structure of the DSG kinks, the
multikink collisions could result in a rich variety of new
phenomena.
Answers to these questions would substantially improve
our understanding of the DSG kinks dynamics.
Acknowledgements This research was supported by the MEPhI
Academic Excellence Project (Contract No. 02.a03.21.0005, 27.08.2013).
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
1. R. Rajaraman , Solitons and instantons: an introduction to solitons and instantons in quantum field theory (North-Holland, Amsterdam, 1982 )
2. A. Vilenkin , E.P.S. Shellard , Cosmic strings and other topological defects (Cambridge University Press, Cambridge, 2000 )
3. N. Manton , P. Sutcliffe , Topological solitons (Cambridge University Press, Cambridge, 2004 )
4. T. Vachaspati , Kinks and domain walls: an introduction to classical and quantum solitons (Cambridge University Press, Cambridge, 2006 )
5. D. Bazeia , L. Losano , J.R.L. Santos , Kinklike structures in scalar field theories: from one-field to two-field models . Phys. Lett. A 377 , 1615 ( 2013 ). arXiv: 1304 . 6904
6. A. Alonso-Izquierdo , D. Bazeia , L. Losano , J. Mateos Guilarte , New models for two real scalar fields and their kink-like solutions . Adv. High Energy Phys . 2013 , 183295 ( 2013 ). arXiv: 1308 . 2724
7. H. Katsura , Composite-kink solutions of coupled nonlinear wave equations . Phys. Rev. D 89 , 085019 ( 2014 ). arXiv: 1312 . 4263
8. R.A.C. Correa , A. de Souza Dutra , M. Gleiser , Informationentropic measure of energy-degenerate kinks in two-field models . Phys. Lett. B 737 , 388 ( 2014 ). arXiv: 1409 . 0029
9. D. Saadatmand , A. Moradi Marjaneh , M. Heidari , Dynamics of coupled field solitons: a collective coordinate approach . Pramana J Phys 83 , 505 ( 2014 )
10. A. Alonso-Izquierdo , Kink dynamics in a system of two coupled scalar fields in two space-time dimensions . Phys. D 365 , 12 ( 2018 ). arXiv: 1711 . 08784
11. A. Alonso-Izquierdo , Reflection, transmutation, annihilation and resonance in two-component kink collisions . Phys. Rev. D 97 , 045016 ( 2018 ). arXiv: 1711 . 10034
12. V.A. Lensky , V.A. Gani , A.E. Kudryavtsev , Domain walls carrying a U(1) charge . Sov. Phys. JETP 93 , 677 ( 2001 ). arXiv:hep-th/0104266
13. V.A. Lensky , V.A. Gani , A.E. Kudryavtsev , Domain walls carrying a U(1) charge . Zh. Eksp. Teor. Fiz . 120 , 778 ( 2001 ). arXiv:hep-th/0104266
14. V.A. Gani , N.B. Konyukhova , S.V. Kurochkin , V.A. Lensky , Study of stability of a charged topological soliton in the system of two interacting scalar fields . Comput. Math. Math. Phys. 44 , 1968 ( 2004 ). arXiv: 0710 . 2975
15. V.A. Gani , N.B. Konyukhova , S.V. Kurochkin , V.A. Lensky , Study of stability of a charged topological soliton in the system of two interacting scalar fields . Zh. Vychisl. Mat. Mat. Fiz . 44 , 2069 ( 2004 ). arXiv: 0710 . 2975
16. D. Bazeia , A.S. Lobão Jr. , L. Losano , R. Menezes , First-order formalism for twinlike models with several real scalar fields . Eur. Phys. J. C 74 , 2755 ( 2014 ). arXiv: 1312 . 1198
17. V.A. Gani , M.A. Lizunova , R.V. Radomskiy , Scalar triplet on a domain wall: an exact solution . JHEP 04 , 043 ( 2016 ). arXiv: 1601 . 07954
18. V.A. Gani , M.A. Lizunova , R.V. Radomskiy , Scalar triplet on a domain wall . J. Phys. Conf. Ser . 675 , 012020 ( 2016 ). arXiv: 1602 . 04446
19. S. Akula , C. Balázs , G.A. White , Semi-analytic techniques for calculating bubble wall profiles . Eur. Phys. J. C 76 , 681 ( 2016 ). arXiv: 1608 . 00008
20. J. Ashcroft et al., Head butting sheep: kink collisions in the presence of false vacua . J. Phys. A Math. Theor . 49 , 365203 ( 2016 ). arXiv: 1604 . 08413
21. T.I. Belova , A.E. Kudryavtsev , Solitons and their interactions in classical field theory . Phys. Usp . 40 , 359 ( 1997 )
22. T.I. Belova , A.E. Kudryavtsev , Solitons and their interactions in classical field theory . Usp. Fiz. Nauk 167 , 377 ( 1997 )
23. V.A. Gani , A.E. Kudryavtsev , M.A. Lizunova , Kink interactions in the (1+1)-dimensional ϕ6 model . Phys. Rev. D 89 , 125009 ( 2014 ). arXiv: 1402 . 5903
24. H. Weigel, Kink-antikink scattering in ϕ4 and φ6 models . J. Phys. Conf. Ser . 482 , 012045 ( 2014 ). arXiv: 1309 . 6607
25. I. Takyi, H. Weigel , Collective coordinates in one-dimensional soliton models revisited . Phys. Rev. D 94 , 085008 ( 2016 ). arXiv: 1609 . 06833
26. A. Demirkaya et al., Kink dynamics in a parametric φ6 system: a model with controllably many internal modes . JHEP 12 , 071 ( 2017 ). arXiv: 1706 . 01193
27. H.E. Baron, G. Luchini, W.J. Zakrzewski , Collective coordinate approximation to the scattering of solitons in the (1+1) dimensional NLS model . J. Phys. A Math. Theor . 47 , 265201 ( 2014 ). arXiv: 1308 . 4072
28. K. Javidan , Collective coordinate variable for soliton-potential system in sine-Gordon model . J. Math. Phys. 51 , 112902 ( 2010 ). arXiv: 0910 . 3058
29. I. Christov, C.I. Christov , Physical dynamics of quasi-particles in nonlinear wave equations . Phys. Lett. A 372 , 841 ( 2008 ). arXiv:nlin/0612005
30. V.A. Gani , A.E. Kudryavtsev , Collisions of domain walls in a supersymmetric model . Phys. Atom. Nucl . 64 , 2043 ( 2001 ). arXiv:hep-th/9904209. arXiv:hep-th/9912211
31. V.A. Gani , A.E. Kudryavtsev , Collisions of domain walls in a supersymmetric model . Yad. Fiz . 64 , 2130 ( 2001 ). arXiv:hep-th/9904209. arXiv:hep-th/9912211
32. J.K. Perring , T.H.R. Skyrme , A model unified field equation . Nucl. Phys 31 , 550 ( 1962 )
33. R. Rajaraman , Intersoliton forces in weak-coupling quantum field theories . Phys. Rev. D 15 , 2866 ( 1977 )
34. N.S. Manton , An effective Lagrangian for solitons . Nucl. Phys. B 150 , 397 ( 1979 )
35. P.G. Kevrekidis , A. Khare , A. Saxena , Solitary wave interactions in dispersive equations using Manton's approach . Phys. Rev. E 70 , 057603 ( 2004 ). arXiv:nlin/0410045
36. V.A. Gani , A.E. Kudryavtsev , Kink-antikink interactions in the double sine-Gordon equation and the problem of resonance frequencies . Phys. Rev. E 60 , 3305 ( 1999 ). arXiv:cond-mat/9809015
37. P. Dorey , K. Mersh , T. Romanczukiewicz , Y. Shnir , Kink-antikink collisions in the φ6 model . Phys. Rev. Lett . 107 , 091602 ( 2011 ). arXiv: 1101 . 5951
38. V.A. Gani , V. Lensky , M.A. Lizunova , Kink excitation spectra in the (1+1)-dimensional ϕ8 model . JHEP 08 , 147 ( 2015 ). arXiv: 1506 . 02313
39. V.A. Gani , V. Lensky , M.A. Lizunova , E.V. Mrozovskaya , Excitation spectra of solitary waves in scalar field models with polynomial self-interaction . J. Phys. Conf. Ser . 675 , 012019 ( 2016 ). arXiv: 1602 . 02636
40. R.V. Radomskiy , E.V. Mrozovskaya , V.A. Gani , I.C. Christov , Topological defects with power-law tails . J. Phys. Conf. Ser . 798 , 012087 ( 2017 ). arXiv: 1611 . 05634
41. A. Moradi Marjaneh , D. Saadatmand , K. Zhou , S.V. Dmitriev , M.E. Zomorrodian , High energy density in the collision of N kinks in the φ4 model . Commun. Nonlinear Sci. Numer . Simul. 49 , 30 ( 2017 ). arXiv: 1605 . 09767
42. A. Moradi Marjaneh , V.A. Gani , D. Saadatmand , S.V. Dmitriev , K. Javidan, Multi-kink collisions in the φ6 model . JHEP 07 , 028 ( 2017 ). arXiv: 1704 . 08353
43. A. Moradi Marjaneh , A. Askari , D. Saadatmand , S.V. Dmitriev , Extreme values of elastic strain and energy in sine-Gordon multikink collisions . Eur. Phys. J B 91 , 22 ( 2018 ). arXiv: 1710 . 10159
44. E. Belendryasova , V. A. Gani , Scattering of the ϕ8 kinks with power-law asymptotics , arXiv:1708.00403
45. E. Belendryasova , V.A. Gani , Resonance phenomena in the ϕ8 kinks scattering . J. Phys. Conf. Ser . 934 , 012059 ( 2017 ). arXiv: 1712 . 02846
46. D. Bazeia , E. Belendryasova , V. A. Gani , Scattering of kinks of the sinh-deformed ϕ4 model . arXiv:1710.04993
47. D. Bazeia , E. Belendryasova , V.A. Gani , Scattering of kinks in a non-polynomial model . J. Phys. Conf. Ser . 934 , 012032 ( 2017 ). arXiv: 1711 . 07788
48. D. Saadatmand , S.V. Dmitriev , P.G. Kevrekidis, High energy density in multisoliton collisions . Phys. Rev. D 92 , 056005 ( 2015 ). arXiv: 1506 . 01389
49. M. A. Lohe, Soliton structures in P(ϕ)2 . Phys. Rev . D 20 , 3120 ( 1979 )
50. A. Khare , I.C. Christov , A. Saxena , Successive phase transitions and kink solutions in φ8, φ10, and φ12 field theories . Phys. Rev. E 90 , 023208 ( 2014 ). arXiv: 1402 . 6766
51. H. Weigel, Emerging translational variance: vacuum polarization energy of the ϕ6 kink . Adv. High Energy Phys . 2017 , 1486912 ( 2017 ). arXiv: 1706 . 02657
52. H. Weigel, Vacuum polarization energy for general backgrounds in one space dimension . Phys. Lett. B 766 , 65 ( 2017 ). arXiv: 1612 . 08641
53. P. Dorey et al., Boundary scattering in the φ4 model . JHEP 05 , 107 ( 2017 ). arXiv: 1508 . 02329
54. D. Bazeia , M.A. González León , L. Losano , J. Mateos Guilarte , Deformed defects for scalar fields with polynomial interactions . Phys. Rev. D 73 , 105008 ( 2006 ). arXiv:hep-th/0605127
55. S. He , Y. Jiang , J. Liu, Toda chain from the kink-antikink lattice . arXiv:1605.06867
56. S. Snelson, Asymptotic stability for odd perturbations of the the stationary kink in the variable-speed ϕ4 model . arXiv:1603.07344
57. L.E. Guerrero , E. López-Atencio , J.A. González , Long-range selfaffine correlations in a random soliton gas . Phys. Rev. E 55 , 7691 ( 1997 )
58. B.A. Mello , J.A. González , L.E. Guerrero , E. López-Atencio , Topological defects with long-range interactions . Phys. Lett. A 244 , 277 ( 1998 )
59. L.E. Guerrero , J.A. González , Long-range interacting solitons: pattern formation and nonextensive thermostatistics . Physica A 257 , 390 ( 1998 ). arXiv:patt-sol/9905010
60. A.R. Gomes , R. Menezes , J.C.R.E. Oliveira , Highly interactive kink solutions . Phys. Rev. D 86 , 025008 ( 2012 ). arXiv: 1208 . 4747
61. M. Peyrard , D.K. Campbell, Kink-antikink interactions in a modified sine-Gordon model . Physica D 9 , 33 ( 1983 )
62. G. Delfino, G. Mussardo, Non-integrable aspects of the multifrequency sine-Gordon model . Nucl. Phys. B 516 , 675 ( 1998 ). arXiv:hep-th/9709028
63. D.K. Campbell , M. Peyrard , Solitary wave collisions revisited . Physica D 18 , 47 ( 1986 )
64. D.K. Campbell , M. Peyrard , P. Sodano , Kink-antikink interactions in the double sine-Gordon equation . Physica D 19 , 165 ( 1986 )
65. Yu . S. Kivshar , B.A. Malomed , Radiative and inelastic effects in dynamics of double sine-Gordon solitons . Phys. Lett. A 122 , 245 ( 1987 )
66. B.A. Malomed , Dynamics and kinetics of solitons in the driven damped double sine-Gordon equation . Phys. Lett. A 136 , 395 ( 1989 )
67. D. Bazeia , L. Losano , J.M.C. Malbouisson , Deformed defects . Phys. Rev. D 66 , 101701 ( 2002 ). arXiv:hep-th/0209027
68. C. A . Almeida , D. Bazeia , L. Losano , J.M.C. Malbouisson , New results for deformed defects . Phys. Rev. D 69 , 067702 ( 2004 ). arXiv:hep-th/0405238
69. L. Campanelli , P. Cea , G.L. Fogli , L. Tedesco, Charged domain walls . Int. J. Mod. Phys. D 13 , 65 ( 2004 ). arXiv:astro-ph/0307211
70. V.A. Gani , V.G. Ksenzov , A.E. Kudryavtsev , Example of a selfconsistent solution for a fermion on domain wall . Phys. Atom. Nucl . 73 , 1889 ( 2010 ). arXiv: 1001 . 3305
71. V.A. Gani , V.G. Ksenzov , A.E. Kudryavtsev , Example of a selfconsistent solution for a fermion on domain wall . Yad. Fiz . 73 , 1940 ( 2010 ). arXiv: 1001 . 3305
72. V.A. Gani , V.G. Ksenzov , A.E. Kudryavtsev , Stable branches of a solution for a fermion on domain wall . Phys. Atom. Nucl . 74 , 771 ( 2011 ). arXiv: 1009 . 4370
73. V.A. Gani , V.G. Ksenzov , A.E. Kudryavtsev , Stable branches of a solution for a fermion on domain wall . Yad. Fiz . 74 , 797 ( 2011 ). arXiv: 1009 . 4370
74. V.A. Gani et al., On decay of bubble of disoriented chiral condensate . Phys. Atom. Nucl . 62 , 895 ( 1999 ). arXiv:hep-ph/9712526
75. V.A. Gani et al., On decay of bubble of disoriented chiral condensate . Yad. Fiz . 62 , 956 ( 1999 ). arXiv:hep-ph/9712526
76. T.I. Belova , V.A. Gani , A.E. Kudyavtsev , Decay of a largeamplitude bubble of a disoriented chiral condensate . Phys. Atom. Nucl . 64 , 140 ( 2001 ). arXiv:hep-ph/0003308
77. T.I. Belova , V.A. Gani , A.E. Kudyavtsev , Decay of a largeamplitude bubble of a disoriented chiral condensate . Yad. Fiz . 64 , 143 ( 2001 ). arXiv:hep-ph/0003308
78. M. Nitta , Josephson vortices and the Atiyah-Manton construction . Phys. Rev. D 86 , 125004 ( 2012 ). arXiv: 1207 . 6958
79. M. Nitta , Correspondence between Skyrmions in 2 + 1 and 3 + 1 dimensions . Phys. Rev. D 87 , 025013 ( 2013 ). arXiv: 1210 . 2233
80. M. Nitta , Matryoshka Skyrmions . Nucl. Phys. B 872 , 62 ( 2013 ). arXiv: 1211 . 4916
81. M. Kobayashi , M. Nitta , Sine-Gordon kinks on a domain wall ring . Phys. Rev. D 87 , 085003 ( 2013 ). arXiv: 1302 . 0989
82. P. Jennings, P. Sutcliffe , The dynamics of domain wall Skyrmions . J. Phys. A 46 , 465401 ( 2013 ). arXiv: 1305 . 2869
83. S.B. Gudnason , M. Nitta , Domain wall Skyrmions . Phys. Rev. D 89 , 085022 ( 2014 ). arXiv: 1403 . 1245
84. N. Blyankinshtein , Q-lumps on a domain wall with a spin-orbit interaction . Phys. Rev. D 93 , 065030 ( 2016 ). arXiv: 1510 . 07935
85. A. Yu. Loginov , Q kink of the nonlinear O(3) σ model involving an explicitly broken symmetry . Phys. Atom. Nucl . 74 , 740 ( 2011 )
86. A. Yu. Loginov , Q kink of the nonlinear O(3) σ model involving an explicitly broken symmetry . Yad. Fiz . 74 , 766 ( 2011 )
87. D. Bazeia , A. Mohammadi , Fermionic bound states in distinct kinklike backgrounds . Eur. Phys. J. C 77 , 203 ( 2017 ). arXiv: 1702 . 00891
88. D. Bazeia , A. Mohammadi , D. C. Moreira , Fermion bound states in geometrically deformed backgrounds . arXiv:1706.04406
89. M. Mai , P. Schweitzer , Energy momentum tensor, and the D-term of Q-balls . Phys. Rev. D 86 , 076001 ( 2012 ). arXiv: 1206 . 2632
90. M. Mai , P. Schweitzer , Radial excitations of Q-balls, and their D-term . Phys. Rev. D 86 , 096002 ( 2012 ). arXiv: 1206 . 2930
91. M. Cantara , M. Mai , P. Schweitzer , The energy-momentum tensor and D-term of Q-clouds . Nucl. Phys. A 953 , 1 ( 2016 ). arXiv: 1510 . 08015
92. I.E. Gulamov, EYa. Nugaev, M.N. Smolyakov , Analytic Q-ball solutions and their stability in a piecewise parabolic potential . Phys. Rev. D 87 , 085043 ( 2013 ). arXiv: 1303 . 1173
93. D. Bazeia , M.A. Marques , R. Menezes , Exact solutions, energy and charge of stable Q-balls . Eur. Phys. J. C 76 , 241 ( 2016 ). arXiv: 1512 . 04279
94. D. Bazeia et al., Compact Q-balls . Phys. Lett. B 758 , 146 ( 2016 ). arXiv: 1604 . 08871
95. D. Bazeia , L. Losano , M.A. Marques , R. Menezes , Split Q-balls . Phys. Lett. B 765 , 359 ( 2017 ). arXiv: 1612 . 04442
96. V. Dzhunushaliev , A. Makhmudov , K.G. Zloshchastiev , Singularity-free model of electrically charged fermionic particles and gauged Q-balls . Phys. Rev. D 94 , 096012 ( 2016 ). arXiv: 1611 . 02105
97. V. A. Gani , A. A. Kirillov , S. G. Rubin , Classical transitions with the topological number changing in the early universe . JCAP 04 , 042 ( 2018 ). arXiv: 1704 . 03688
98. V.A. Gani , A.A. Kirillov , S.G. Rubin , Transitions between topologically non-trivial configurations . J. Phys. Conf. Ser . 934 , 012046 ( 2017 ). arXiv: 1711 . 07700
99. E.B. Bogomolny , Stability of classical solutions . Sov. J. Nucl. Phys . 24 , 449 ( 1976 )
100. E.B. Bogomolny , Stability of classical solutions . Yad. Fiz . 24 , 861 ( 1976 )
101. M.K. Prasad , C.M. Sommerfield , Exact classical solution for the 't Hooft Monopole and the Julia-Zee Dyon . Phys. Rev. Lett . 35 , 760 ( 1975 )