Scattering of kinks of the sinhdeformed \(\varphi ^4\) model
Eur. Phys. J. C
Scattering of kinks of the sinhdeformed ϕ4 model
Dionisio Bazeia 2
Ekaterina Belendryasova 1
Vakhid A. Gani 0 1
0 Theory Department, National Research Center Kurchatov Institute, Institute for Theoretical and Experimental Physics , 117218 Moscow , Russia
1 National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) , 115409 Moscow , Russia
2 Departamento de Física, Universidade Federal da Paraíba , João Pessoa, Paraíba 58051900 , Brazil
We consider the scattering of kinks of the sinhdeformed ϕ4 model, which is obtained from the wellknown ϕ4 model by means of the deformation procedure. Depending on the initial velocity vin of the colliding kinks, different collision scenarios are realized. There is a critical value vcr of the initial velocity, which separates the regime of reflection (at vin > vcr) and that of a complicated interaction (at vin < vcr) with kinks' capture and escape windows. Besides that, at vin below vcr we observe the formation of a bound state of two oscillons, as well as their escape at some values of vin.

Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2 Topological solitons in (1,1)dimensional models . . .
3 The ϕ4 model . . . . . . . . . . . . . . . . . . . . . .
4 Deformation procedure and the sinhdeformed ϕ4 model
5 Kinkantikink collisions in the sinhdeformed ϕ4 model
6 Comments and conclusion . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
Topological defects arise in a diversity of contexts in high
energy physics, cosmology, quantum and classical field
theory, condensed matter, and so on. In high energy physics,
they are topologically nontrivial solutions of the equations
of motion and possess very interesting properties, which lead
to new physical phenomena [
1–4
].
Nowadays, the study of the topological defects is a very
fast developing area with significant effort being applied to
the investigation of domain walls, vortices, strings, as well as
embedded topological defects such as a Qlump on a domain
wall and a skyrmion on a domain wall, and so on [
5–17
]. It
is also of interest to mention the socalled Qballs and
similar configurations [
18–23
], which are charged and protected
against decaying into the elementary excitations supported
by the respective model. Also, it is worth mentioning other
possibilities, such as the study of solitons in fibers [24],
bubble collisions in cosmology [
25
], and localized excitations in
nonlinear systems [
26
].
Models in (1, 1) spacetime dimensions are of
special interest [
2,3,27,28
], since the dynamics of some
twoor threedimensional systems can be reduced to the
onedimensional models. For example, a planar domain wall,
which separates regions with different minima of the
potential, in the direction perpendicular to it can be interpreted
as a onedimensional topological configuration (a kink).
On the other hand, the (1, 1)dimensional fieldtheoretical
models can be a first step towards more complicated
higherdimensional models. Moreover, even in the (1,
1)dimensional case, topological defects may arise in more
complex models with two or more fields, see, e.g., Refs. [
29–59
].
In this more general context, several works have developed
analytical solutions, which, in turn, has allowed one to study
their stability and to use them in application of interest in
physics [
29–40
]. Other investigations have dealt with the
presence of junctions and/or intersections of defects [
41–
44
], and with issues related to compositekink internal
structures, twinlike models with several fields and scalar triplet
on domain walls [
45–57
], among other issues.
In the case of models described by real scalar fields
with standard kinematics, in the (1, 1) spacetime the
presence of interactions that develop spontaneous symmetry
breaking in general leads to localized topological structures
having the kinklike profile. The interactions of these
onedimensional topological structures with each other and with
spatial inhomogeneities (impurities) have attracted the
attention of physicists and mathematicians for a long time; see,
e.g., Ref. [
27,28
]. The first studies on this subject date back
to the 1970s and 1980s [
60–62
]. Nevertheless, forty years
later we see that it is still an actively developing area with
many new applications. Many important results have been
obtained by means of the numerical simulation, which is
one of the most powerful tools for studying the subject.
In particular, resonance phenomena – escape windows and
quasiresonances – were found in the kinkantikink
scattering process. A broad class of (1, 1)dimensional models
with polynomial potentials such as the ϕ4, ϕ6, ϕ8 models,
and those with higher degree polynomial selfinteraction has
been considered [
62–76
]. One should also mention the new
results on the longrange interaction between kinks [
74–79
].
Other models with nonpolynomial potentials are also being
discussed in the literature. For example, the modified
sineGordon [
80
], the double sineGordon [
81–83
], and a variety
of models which can be obtained using the deformation
procedure, which we explain below.
Apart from the numerical solving of the equation of
motion, other methods are widely used for investigating the
kinkantikink interactions. One of them is the collective
coordinate method [
64,84–91
]. Within this approximation a real
fieldtheoretical system (which formally has an infinite
number of degrees of freedom) is approximately described as a
system with one or a few degrees of freedom. For example,
in the case of the kinkantikink configuration one can use
the distance between the kink and the antikink as the only
degree of freedom (collective coordinate). In more
complicated modifications of this approach other degrees of freedom
(for instance, vibrational ones) can be involved, see, e.g., [
84–
86
]. Another approximation, which allows to estimate the
force between kink and antikink, is the Manton’s method [3,
Ch. 5], [
92–95
]. This method is based on using the kinks’
asymptotics in situations where the distance between the
kinks is large. However, one should mention that the
applicability of this method for kinks and solitons with powerlaw
asymptotics is not obvious.
An impressive progress has been achieved in the analytical
treatment of the (1, 1)dimensional fieldtheoretical models.
Among several possibilities to deal with the problem
analytically, the trial orbit method was suggested in [
29
] as a way
to solve the equations of motion in systems described by two
real scalar fields that interact nonlinearly. This method has
been used by others, and in [
48
] it was shown to be very
effective when the equations of motion can be reduced to
firstorder differential equations. Also, in [
49
] the authors
have used the integrating factor to solve the equations of
motion in the case of a very specific potential.
Another possibility of searching for models that
support analytical solutions appeared before in [
96
] and also
in Refs. [
73,97,98
]. It refers to the deformation procedure, a
method of current interest which helps us to introduce new
models, and solve them analytically. This will be further
(2.1)
(2.2)
(2.4)
reviewed below, and used to define the model [
99
] we want to
investigate in the current work. In particular, the new model
is somehow similar to the ϕ4 model with spontaneous
symmetry breaking, so we will compare its features with the ϕ4
case, in order to highlight the differences between the two
cases, and to see how the nonpolynomial interaction of the
new model modifies the behavior seen in the standard ϕ4
model.
In this work we focus our attention on the kinkantikink
scattering process and organize the investigation as
follows. In Sect. 2 we give general introduction to the (1,
1)dimensional fieldtheoretical models, which possess
topological solutions with the kink profile. In Sect. 3 we review
the ϕ4 model, briefly accounting for the kinkantikink
scattering within this model. Furthermore, in Sect. 4 we apply
the deformation procedure to the ϕ4 model in order to
introduce a model with nonpolynomial potential, which we call
the sinhdeformed ϕ4 model. In Sect. 5 we focus on the
collisions of the kink and the antikink of the sinhdeformed ϕ4
model. In this section we present our main results and
compare them with the results of the ϕ4 model. Finally, in Sect. 6
we conclude with a discussion of the results and the prospects
for future works.
2 Topological solitons in (1,1)dimensional models
Consider a fieldtheoretical model in the (1, 1)dimensional
spacetime with its dynamics defined by the Lagrangian
1
L = 2
∂ϕ
∂t
2
where ϕ(x , t ) is a real scalar field. The potential U (ϕ) is
supposed to be nonnegative function with two or more
degenerate minima, ϕ1(0), ϕ2(0), …, such that U (ϕ1(0)) = U (ϕ2(0)) =
· · · = 0. The Lagrangian (2.1) leads to the following equation
of motion for the field ϕ
∂2ϕ ∂2ϕ dU
∂t 2 − ∂ x 2 + dϕ = 0.
The energy functional corresponding to the Lagrangian (2.1)
is
E [ϕ] =
∞
−∞
For the energy of the static configuration to be finite, the two
following conditions must hold
where ϕi(0) and ϕ(0) are two adjacent minima of the
potenj
tial. These expressions (2.6) are necessary conditions for the
energy of a static configuration to be finite. If (2.6) hold,
then the second and the third terms in the integrand in (2.3)
fall off at x → ±∞, hence the integral (2.3) can be
convergent. Configurations with ϕi(0) = ϕ(j0) are called topological
and have a kinklike shape. In this sense, a conserved
topological current can be introduced, and for the models to be
investigated below one can use
μ 1 εμν ∂ν ϕ.
jtop = 2
The corresponding topological charge is
Qtop =
∞
−∞
1
jt0opd x = 2 [ϕ(+∞) − ϕ(−∞)] .
This charge is determined only by the asymptotics (2.6), so
it does not depend on the behavior of the field ϕ(x ) at finite
x .
For every nonnegative potential U (ϕ) we can introduce
a smooth function W (ϕ), called the superpotential, as
EBPS = W [ϕ(+∞)] − W [ϕ(−∞)].
From Eq. (2.10) one can see that the energy of any static
configuration belonging to a given topological sector is bounded
from below by EBPS. The configurations with the minimal
energy (2.11) are called BPS configurations, or BPS saturated
configurations [
100–102
]. From Eq. (2.10) it is easy to see
1
U (ϕ) = 2
d W
dϕ
1
E = EBPS + 2
where
2
.
∞
−∞
Using the superpotential we can rewrite the energy of a static
configuration ϕ(x ) in the following manner
dϕ d W
d x ± dϕ
2
d x ,
This equation can be easily transformed into the first order
differential equations
that any BPS configuration satisfies the first order differential
equations
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
dϕ d W
d x = ± dϕ ,
which coincide with (2.5).
Below we deal with kinks and antikinks – the BPS
saturated topological solutions of Eq. (2.5), which interpolate
between neighboring minima of the potential. The solution
with the asymptitics ϕ(+∞) > ϕ(−∞) is called kink, while
the term antikink stands for the solution with ϕ(+∞) <
ϕ(−∞). Sometimes we use the term kink for both kink and
antikink, for brevity.
Many phenomena observed in the kinkantikink scattering
can be explained by the presence of the vibrational mode(s) in
the kink’s excitation spectrum. In order to find the spectrum
of localized excitations of a kink, we have to add a small
perturbation δϕ(x , t ) to the static kink solution ϕk(x ),
ϕ(x , t ) = ϕk(x ) + δϕ(x , t ), δϕ
ϕk.
The substitution of ϕ(x , t ) into the equation of motion (2.2)
leads to the partial differential equation for the perturbation
δϕ(x , t ); after linearization one gets
∂2δϕ ∂2δϕ d2U
∂t 2 − ∂ x 2 + dϕ2 ϕk(x)
δϕ = 0.
Since the second derivative of the potential calculated at the
static solution ϕk(x ) depends only on x , we can assume that
δϕ has the form
δϕ(x , t ) = η(x ) cos ωt,
and this allows us to obtain the eigenvalue problem of the
type of the stationary Schrödinger equation,
Hˆ η(x ) = ω2η(x ),
d2
Hˆ = − d x 2 + u(x ),
with the potential
d2U
u(x ) = dϕ2
ϕk(x)
.
where the operator Hˆ (the Hamiltonian) is
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
For each state of the discrete spectrum, the corresponding
eigenfunction η(x ) is a twice continuously differentiable and
squareintegrable on the x axis. Kink and antikink have the
same excitation spectrum.
Fig. 3 The formation of a
bound state in the ϕ4
kinkantikink collision at
vin = 0.2541. Left panel – the
spacetime picture of a bion
formation. Right panel – the
time dependence of the field at
the origin
The discrete spectrum in the potential (2.18) always
possesses a zero (or translational) mode ω0 = 0. It can easily
be shown by differentiating Eq. (2.4) with respect to x , and
taking into account that ϕk (x ) is a solution of Eq. (2.4), i.e.
d2 dϕk d2U
− d x 2 d x + dϕ2 ϕk(x) d x
dϕk
= 0,
or
dϕk
Hˆ · d x
= 0.
So we see that dϕk is an eigenfunction of the Hamiltonian
d x
(2.17) associated with the eigenvalue ω0 = 0. The presence
of a zero mode in the kink’s excitation spectrum is a
consequence of the translational invariance of the Lagrangian.
(2.19)
Hˆ = A† A,
where A† and A are the first order differential operators
A†
d d2 W
= d x + dϕ2
ϕk(x)
. (2.21)
This factorization shows that the operator Hˆ is nonnegative,
so the static solution ϕk (x ) is linearly stable.
(2.20)
(3.1)
ϕk(x ) = tanh x , ϕk(x ) = − tanh x .
¯
(3.2)
In this section we recall some facts about kinks of the ϕ4
model. We use for simplicity dimensionless fields and
spacetime coordinates and write the potential of the ϕ4 model in
the form
1
U1(ϕ) = 2 (1 − ϕ2)2.
This potential possesses two degenerate minima ϕ1(0) = −1
and ϕ2(0) = 1, see Fig. 1a.
The Eq. (2.5) with the potential (3.1) can be easily
integrated, which yields the static topologically nontrivial
solutions, the kink and the antikink:
These kinks interpolate between the two minima of the
potential, as shown in Fig. 1b. The mass of the kink (antikink), i.e.
Fig. 5 The quantummechanical potential (4.6)
the energy E [ϕk (x )] of the static kink (antikink), is
4
Mk = 3 .
(3.3)
The moving kink (antikink) can be obtained from Eq. (3.2)
by the Lorentz boost.
The quantummechanical potential (2.18), which defines
the spectrum of the localized excitations of the ϕ4 kink, has
the form
It is the wellknown modified PöschlTeller potential [
103
].
Apart from the zero mode ω
√0 = 0, there is a vibrational mode
with the frequency ω1 = 3. As we explain below, the
presence of the vibrational mode leads to resonance phenomena
in the kinkantikink collisions.
As we informed in the Introduction, the scattering of the
ϕ4 kinks is wellstudied, so let us now briefly review the main
features of the collision processes in this case.
Consider the initial configuration in the form of kink and
antikink centered at the points x = −x0 and x = x0,
respectively, and moving towards each other with the initial
velocities vin in the laboratory frame, i.e.
⎛ x + x0 − vint ⎞
ϕ(x , t ) = tanh ⎝ ⎠
To find evolution of this initial configuration, we solved the
equation of motion (2.2) with the potential (3.1) numerically
using the standard explicit finite difference scheme,
∂2ϕ
∂t 2 =
∂2ϕ
∂ x 2 =
ϕij+1 − 2ϕij + ϕij−1
δt 2
ϕi+1 − 2φij + ϕi−1 ,
j j
δx 2
,
where (i, j ) number the x and t coordinates of the grid
points, (xi , t j ), on a grid with the steps δt = 0.008 and
δx = 0.01. We repeated selected computations with smaller
steps, δt = 0.004 and δx = 0.005, in order to check our
numerical results. We also checked the total energy
conservation. In all simulations of the ϕ4 kinks collisions we used
the initial halfdistance x0 = 5.
(3.5)
(3.6)
Depending on the initial velocity, the kinks scattering
looks differently. There is a critical value of the initial
velocity vcr ≈ 0.2598. At vin > vcr we observe kinks escape
after a collision, see Fig. 2. Some part of the energy is being
emitted in the form of small waves.
At vin < vcr the kinks collide and form a longliving
bound state, a bion, which is illustrated in Fig. 3. This bion
decays slowly, emitting its energy in the form of waves of
small amplitude. However the kinks capture appears not for
all vin < vcr, since there is a pattern of escape windows in
the collision processes. An escape window refers to a narrow
interval of initial velocities, within which kinks do not form a
bound state but escape to infinities. It is important point that,
unlike bouncing off at vin > vcr, within an escape window
the kinks escape to infinities after two, three or more
collisions. According to the number of collisions before escaping,
there are twobounce windows, threebounce windows, and
so on. See Fig. 4 for some illustrations of two, three and
fourbounce windows. The escape windows form a fractal
structure. Twobounce windows are the broadest, and near
each of them there is a series of threebounce windows. Near
(4.1)
(4.2)
We start from the ϕ4 model with the potential (3.1) and
use the deforming function f (ϕ) = sinh ϕ. Then we come
to the potential of the sinhdeformed ϕ4 model
1
U2(ϕ) = 2 sech2ϕ 1 − sinh2 ϕ
2
.
This potential has two degenerate minima, ϕ± = ±arsinh 1,
V (ϕ±) = 0, see Fig. 1a. The kinks of the sinhdeformed ϕ4
model are
ϕk(x ) = arsinh(tanh x ), ϕk(x ) = −arsinh(tanh x ), (4.4)
¯
see Fig. 1b. The mass of the sinhdeformed ϕ4 kink
(antikink), i.e. the energy E [ϕk(x )] of the static kink
(antikink), is
Mk = π − 2.
The excitation spectrum of the kink (antikink) (4.4) is
defined by the quantummechanical potential
8 tanh2 x − 4
u2(x ) = 2 tanh2 x + 1 + (1 + tanh2 x )2 ,
which is presented in Fig. 5. We performed a numerical
search of the discrete part of the excitation spectrum in the
potential (4.6). This problem was solved using the standard
shooting method. For various values of ω2 we integrated
Eq. (2.16) with the known asymptotic behavior η(x ) ∼
exp(−√4 − ω2x ) of its solutions at x → ±∞, starting
from a large negative x and from a large positive x . As a
result, we obtained two different solutions, the “left”
solution and the “right” solution, which were then matched at
some point x¯ close to the origin (the particular choice of x¯ is
not important). The Wronskian of the “left” and the “right”
solutions, calculated at the matching point, as a function of
ω2 turns to zero at eigenvalues of the Hamiltonian (2.17) with
the potential (4.6).
each threebounce window, in turn, there is a series of
fourbounce windows, and so on, see, e.g., [
27,28
].
The explanation of the appearance of the escape windows
is that they are related to the resonance energy exchange
between the kinetic energy (the translational mode) and the
vibrational mode of the kink (antikink). The mechanism
works as follows: consider, for example, the twobounce
window illustrated in Fig. 4a. At the first collision, some part of
the kinks kinetic energy is transferred into their vibrational
modes. As a result of the loss of the kinetic energy, the kink
and the antikink are not able to overcome mutual attraction,
and they return and collide again. However, if a certain
resonance relation between the time T12 between the first and
the second collisions and the frequency ω1 of the vibrational
mode holds, a part of the energy can be returned into the
kinetic energy, and the kinks are then able to escape from
each other.
ϕ(new)(x ) = f −1(ϕk(old)(x )).
k
4 Deformation procedure and the sinhdeformed ϕ4
model
The sinhdeformed ϕ4 model can be obtained from the
ϕ4 model by applying the deformation procedure used in
Refs. [
73,96–98,104
]. The potential U2(ϕ) of the new model
is related with the old model potential U1(ϕ) by a deforming
function f (ϕ),
U2(ϕ) =
U1(ϕ → f (ϕ))
(d f /dϕ)2
,
where “ϕ → f (ϕ)” means that one must substitute the field
ϕ by f (ϕ). At the same time, the kink of the new model,
ϕ(new)(x ), can be easily obtained from the kink of the old
k
model, ϕk(old)(x ), by the inverse deforming function f −1,
(4.3)
(4.5)
(4.6)
We found two levels in the potential (4.6): the zero mode
ω0 = 0, and the vibrational mode with the frequency ω1 ≈
1.89.
5 Kinkantikink collisions in the sinhdeformed ϕ4
model
We studied the collisions of the kink and the antikink of
the sinhdeformed ϕ4 model using the initial configuration
similar to that used in Sect. 3 in the case of the ϕ4 kinks
scattering, namely
⎡ ⎛ x + x0 − vint ⎞ ⎤
ϕ(x , t ) = arsinh ⎣ tanh ⎝ ⎠ ⎦
which corresponds to the kink and the antikink centered at
x = ± x0 and moving towards each other with the initial
velocities vin. We used 2x0 = 10 and the same parameters
(5.1)
of the numerical scheme as for the ϕ4 kinks in Sect. 3, see
Eq. (3.6) and the paragraph below this equation.
We found a critical value of the initial velocity vcr ≈
0.4639, which separates two different regimes of the kinks
scattering. At vin > vcr the kinks bounce off and escape to
infinities after one collision. This is illustrated in Fig. 6, and
the situation here is similar to that observed for the ϕ4 kinks
above the critical velocity, as depicted in Fig. 2.
At the initial velocities below the critical value, vin < vcr,
we observed the kinks’ capture and formation of their bound
state, Fig. 7, and a rich variety of resonance phenomena. First
of all, in this range of the initial velocities we found a
complicated pattern of escape windows, similar to the case of
the ϕ4 kinks. We identified many twobounce, threebounce,
etc., escape windows. The field dynamics within these
windows is similar to the case of the ϕ4 model. In Fig. 8 we
present examples of the field behavior within two and
threebounce windows. In a way similar to the case of the ϕ4 model,
the escape windows that appear in the sinhdeformed model
seem to form a fractal structure. We found several
threebounce escape windows near a twobounce window, see Fig.
9. Near one of the threebounce escape windows we observed
fourbounce escape windows. This behavior is similar to the
one found in the ϕ4 model, so it suggests that the escape
windows also form a fractal structure in this case.
At the same time, the bion formation in the range vin < vcr
outside the escape windows looks differently. In our
numerical experiments we observed new phenomena, which are
not typical for the ϕ4 kinks. In many cases the final
configuration looked like a bound state of two oscillons. These
oscillons oscillate around each other near the origin and, as a
consequence, the dependence on time of the field at the
origin has a lowfrequency envelope, as it is shown in Fig. 10.
One notes that the amplitude and frequency of oscillations of
these structures depend on the initial velocity of the
colliding kinks. Moreover, at some values of the initial velocity we
observed escape of the two oscillons with the final velocity
vos, which varies in a wide range, as one can see from Fig.
11. The situation can be interpreted as follows: at some
initial velocities of the colliding kinks the bion is formed, which
evolves rather fast into a bound state of two oscillons, which
can either oscillate around each other, or escape to infinities.
The intervals of the initial velocity of the colliding kinks, at
which the oscillons escape, form oscillons’ escape windows.
The frequency of the field oscillations is the same for all
oscillons, ωos ≈ 1.88, which is very close to ω1 = 1.89.
In Fig. 12 we show the dependence of the period of
oscillations on the initial velocity of the colliding kinks. The shaded
areas denote the escape windows for oscillons, i.e. the
intervals of the initial velocity, at which the two oscillons escape
to infinities. The widths of the escape windows are 0.00030,
0.00013, 0.00011, 0.00002, and ∼ 0.00001 for the 1st, 2nd,
3rd, 4th, and 5th windows, respectively.
The high frequency in Fig. 10 (right panels) is close to
the frequency of the vibrational mode ω1 ≈ 1.89 of the
sinhdeformed ϕ4 kink. For example, at the initial velocity
vin = 0.44183 the frequency is 1.86, at vin = 0.44188 it is
1.84, and at vin = 0.44190 it equals 1.83.
6 Comments and conclusion
In this work, we investigated the scattering of kinks of the
sinhdeformed ϕ4 model, obtained from the ϕ4 model by the
deformation procedure, and compared it with the same
process in the ϕ4 model. We showed that the two models
engender similar behavior in several aspects: they support similar
kinklike configurations, and their stability potentials present
almost the same profile, which gives rise to the zero mode
and the vibrational state with the frequency ω1 = √3 ≈ 1.73
in the case of the ϕ4 model, and ω1 ≈ 1.89 for the
sinhdeformed ϕ4 model.
Moreover, in the scattering of kinks, the two models also
admit a critical velocity vcr, which separates two different
regimes of the collisions. On the one hand, at vin < vcr we
observed the capture of kinks and the formation of bound
states and, on the other hand, for vin > vcr the kinks escape
to infinity after one collision. The value of the critical velocity
is vcr = 0.4639 for the sinhdeformed ϕ4 model and for the
ϕ4 model it is equal to 0.2598.
In the study of collisions of kinks in the sinhdeformed ϕ4
model, we observed that for velocities in the range vin < vcr,
there appeared several escape windows, which are also
specific for the ϕ4 and some other models. In particular, we have
found twobounce, threebounce, and fourbounce escape
windows; recall that within an nbounce window the kinks
escape to infinities after n collisions. The emergence of the
escape windows is related to the resonant energy exchange
between the translational and the vibrational modes of the
kink and the antikink.
denotes the twobounce escape window for kinks, while the pinkshaded
areas denote the fourbounce escape windows for kinks
The general results of the kink collisions in the
sinhdeformed model suggest that the model is not integrable, and
that its kinklike configuration is not a soliton. Interestingly,
at this point one can make a connection with the sineGordon
model, which is an integrable model [
105
]. This model can
also be obtained from the ϕ4 model with the same
deformation procedure: using the deformation function f (ϕ) = sin ϕ,
the potential U1(ϕ) in Eq. (3.1) transforms into
(6.1)
(6.2)
1
U3(ϕ) = 2 cos2 ϕ.
This is the potential of the sineGordon model, and its soliton
solution can be written in the form
ϕs(x ) = arcsin(tanh x ).
As is wellknown, the corresponding stability potential
supports the zero mode and no other bound state, and this helps
one to understand its integrability. The sineGordon potential
is periodic, in contrast to the ϕ4 model described by a
polynomial potential. In this sense, the sinhdeformed model which
we have studied in this work seems to be farther away from
the sineGordon and integrability, and hence it should present
information that is absent in the ϕ4 model.
With this motivation in mind, we then looked deeper
into the escape windows in the sinhdeformed model and
observed a new phenomenon, the conversion of the
kinkantikink pair into a complex oscillating structure at the
collision point at the origin. This structure can be interpreted as a
bound state of two individual oscillons. It is interesting that
at some initial velocities of the colliding kinks we observed
the escape of these two oscillons.
The interval of initial velocities of the kinks, in which the
kinks collide and form a bound state of two oscillons, which
then escape, can be called an oscillons’ escape window. In
our simulations the final velocity of the escaping oscillons
varies in a wide range from zero to ∼ 0.2. Near the oscillons’
(6.3)
(6.4)
escape window the period of the oscillons’ oscillations in
their bound state increases, see Fig. 12.
As shown in the recent work [
104
], we can introduce
other models using the deformation function of the
hyperbolic type. In particular, we can start with the ϕ6 model
studied before in [
63
], which supports no vibrational state. For
instance, we can use the potential
U4(ϕ) = 21 ϕ2(1 − ϕ2)2,
and the deformation function f (ϕ) = sinh ϕ to get to a new
model
1
U5(ϕ) = 2 tanh2 ϕ (1 − sinh2 ϕ)2.
We note that for ϕ very small, the above model (6.4) leads
us back to the model in (6.3). We call this model (6.4) the
sinhdeformed ϕ6 model. It would be interesting to study
the scattering of kinks in this model, to see how it can be
connected to the investigation [
63–66
], which revealed a
resonant scattering structure that provided a counterexample to
the belief that the existence of the vibrational bound state
is a necessary condition for the appearance of multibounce
resonances. Another issue is the study of the force between
two kinks, to see if it can be connected with the scattering of
kinks.
We can also consider models with modified kinematics,
as the ones recently investigated in [
106
], where one
considers the DiracBornInfeld case. This modification changes
the standard scenario and may contribute to add new
possibilities to the escape windows that appear in the standard
situation. Another route concerns models described by two
real scalar fields, as the one investigated in Refs. [
33, 45
].
In this case, the presence of the two fields leads to
analytical kinklike solutions whose internal structure can be used
to model Bloch walls. The scenario here is richer, and the
study of the kinks scattering in this model would allow one
to see how the internal structure contributes to the presence
of the escape windows, etc. These and other similar issues
are currently under consideration, and we hope to report on
them in the near future.
Acknowledgements The authors would like to thank Dr. Vadim
Lensky for reading the manuscript and for valuable comments.
This work was performed using resources of the NRNU MEPhI
highperformance computing center. The research was supported
by the Brazilian agency CNPq under contracts 455931/20143 and
306614/20146, and by the MEPhI Academic Excellence Project under
contract No. 02.a03.21.0005, 27.08.2013.
Open Access This article is distributed under the terms of the Creative
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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
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