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0Russian Journal of Nonlinear Dynamics, 2024, vol. 20, no. 1, pp. 15-26. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd231206
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 35C08, 35Q51
Attractive Impurity as a Generator of Wobbling Kinks
M. I. Fakhretdinov, K. Y. Samsonov, S. V. Dmitriev, E. G. Ekomasov
The theory is widely used in many areas of physics, from cosmology and elementary particle physics to biophysics and condensed matter theory. However, in the ^>4 model, there are no spatially localized solutions in the form of breathers. Topological defects, or kinks, in this theory describe stable, solitary wave excitations. In practice, these excitations, as they propagate, necessarily interact with impurities or imperfections in the on-site potential. In this work, with the help of numerical calculations using the method of lines, the interaction of the kink in the ^>4 model with extended impurities is considered. The case of an attractive rectangular impurity is analyzed. It is found that after the kink-impurity interaction, an internal mode with
the help of kink-impurity interaction, an extended rectangular attracting impurity, as well as a point impurity, can be used as a generator for excitation of long-lived high-amplitude localized breather waves. The structure of the excited wobbling breather (or wobbler), which consists
Received July 02, 2023 Accepted December 02, 2023
The work of S.V.D. was supported by the Russian Science Foundation under Grant No. 21-12-00229. The work of E.G.E. and K.Yu.S. was supported by the Russian Foundation for Basic Research under Grant No. 20-31-90048.
Marat I. Fakhretdinov fmi106tf@gmail.com Evgenii G. Ekomasov ekomasoveg@gmail.com
Ufa University of Science and Technology ul. Zaki Validi 32, Ufa, 450076 Russia
Kirill Y. Samsonov k.y.samsonov@gmail.com
Tyumen State University
ul. Volodarskovo 6, Tyumen, 625003 Russia
Sergey V. Dmitriev dmitriev.sergey.v@gmail.com
Institute of Molecule and Crystal Physics Oktyabrya ave. 151, Ufa, 450075 Russia
and Breathers in the Model
frequency
becomes a wobbling kink. It is shown that with
of a compact core and an extended tail, is described. It is shown that the wobbler tail has the form of a spatially unbounded quasi-sinusoidal function with a classical frequency \/2. To determine the lifetime of the wobbler, the dependence of the amplitude of the impurity mode on time is found. For the case of small impurities, it turned out that it practically does not change for a long time. For the case of large impurities, the wobbler amplitude begins to noticeably decrease with time. The frequency of wobbler oscillations does not depend on the initial velocity of the kink. The dependence of the impurity mode oscillation amplitude on the initial kink velocity has minima and maxima. By changing the impurity parameters, one can also control the dynamic parameters of the wobbler. A linear approximation is considered that allows an analytical solution of the problem for localized breather waves, and the limits of its applicability for this model are found.
Keywords: p4 model, impurity, soliton theory, wobbling kink, wobbler
1. Introduction
One of the nonlinear differential equations frequently used in physics belonging to the class of Klein-Gordon equations (KGE) is the p4 model [1, 2]. It has been used in cosmology, quantum field theory, biophysics, and condensed matter physics [1, 3, 4]. A new impetus to the study of this equation in recent years has been given by its use to describe physical processes in graphene [5, 6].
In the p4 model, there are no spatially localized solutions in the form of breathers. The p4 kink differs from the kink of the sine-Gordon equation (SGE) [7] by the presence of an internal vibrational mode [1, p. 4]. This vibrational degree of freedom can accumulate energy and periodically release it, which leads to the appearance of resonances in kink - antikink [8-11] and kink-impurity [12, 13] interactions, as well as stimulate the formation of a kink-antikink pair [14]. The kink of the p4 model with excited internal vibrational mode is called a wobbling kink [15-17]. Internal vibrational degrees of freedom of particles often play an important role in the dynamics of their arrays [18-21]. In the p4 model, kinks and antikinks cannot simply pass through each other [2]. Numerically, it was found that at high velocities, the kink and antikink reflect inelastically from each other, losing energy. At lower velocities, the kink and antikink coupled into a long-lived oscillatory state, reminiscent of the SGE breather, but slowly decaying [1, p. 10].
To analytically find the p4 breather, there have been attempts to use the series expansion in terms of the small parameter e [22, 23]. It was found that in the continuum theory there are only nanopteron solutions (oscillating solutions with infinite energy with oscillating low-amplitude tails going to infinity). Radiation of energy by the wobbling kinks can be compensated for by the external driving [24]. It is currently believed [1, p. 163] that the p4 breather is just one of the examples of weakly nonlocal solitary waves. Nonlocal, because the spatial localization inherent in the classical definition of a solitary wave is violated by quasi-sinusoidal radiation, unlimited in space. Weakly, because the amplitude of the radiation is very small. There is a core of the p4 breather — this is the central part of the wave with a large amplitude. Everywhere outside the core there are wings dominated by tiny sinusoidal ripples. The reason for the nonlocality of such a wave is the resonance between a solitary wave and a sine wave of infinitely small amplitude and a certain wavenumber [25]. A long-lived, nonlinear metastable bound state of three kinks, the triton, was also found [26]. This object is the product of a symmetrical collision of two kinks and
an antikink. Stability and the energy radiation rate by the wobbling kinks have been studied in the works [27-32].
For practical applications, the p4 model is often modified by considering the coefficients as functions of coordinates and time [2, 33-36]. Spatial modulation of the potential is often called an impurity. For the p4 model with impurities, only the case of point impurities has been considered in detail so far [2, 11-13]. It was shown that single point impurities can scatter or capture kinks and also generate a localized impurity mode [2, 13]. In [37], the dynamics of kinks was considered for a model with single extended impurities having spatial profiles of Gauss or Lorentz. Interaction of SGE and p4 kinks and breathers with a parity-time-symmetric defects was studied in the works [38-42]. Kink-kink and multisoliton collisions were studied in [43-45]. In discrete SGE and p4 models wobbling kinks can have different symmetry [46]. Qualitative agreement of the obtained results with the case of point impurities and a significant quantitative effect of the impurity profile on the shape of the localized impurity mode and kink scattering on impurities are shown. It was found in [47] that oscillons naturally arise as a result of kink -antikink collisions in the presence of an extended Gaussian impurity. In [48], the p4 model with an impurity described by a hyperbolic function was considered. Kink dynamics for the case of small perturbations of the p4 model using a deformed potential was considered in [49]. In this paper, we consider the possibility of using an extended impurity as a generator for excitation of a wobbling kink and a localized breather-type wave (wobbler) as a result of a kink-impurity interaction. This article is a continuation of our previous publication [50].
2. Impurity modes. Numerical calculation results
Consider a scalar field u(x, t) for which the equation of motion in the one-dimensional case has the form
utt - uxx + K(x) (u2 - 1) u = 0, (2.1)
where K(x) is a function of the coordinate x that takes into account the presence of impurity in the system. When K(x) = 1, Eq. (2.1) is a nonlinear differential equation of the p4 model and has a solution in the form of a kink [2]:
u(x, t) = tanh X ~Vot I , (2.2)
/2(l-t,g)
where 0 ^ v0 < 1 is the parameter determining the initial velocity of the kink. For the case of point impurity, when the K(x) function is represented using the Dirac delta function, Eq. (2.1) can be solved using the approximate method of collective coordinates [2, 13]. For the case of extended impurities and an arbitrary K(x) function, Eq. (2.1) can only be solved numerically. Let us consider, for certainty, the K(x) function in the form of a single extended impurity of rectangular form well studied for the case of SGE [51-53]:
{W W
1, for x <--and x > —,
w w (2.3)
1 - AK, for--^ x ^ —,
2 2
where AK and W are some constants. Similar to the case of point impurities [13], at AK > 0 the impurity is an effective potential well for the kink (see Fig. 1). From the physical point
u(x,t = ta), K{x)
Fig. 1. Scheme of the kink (at some point in time t = ta) and the impurity. Impurity parameters are AK = 0.5, W = 1.0
of view, the choice of the impurity form of the form (2.3) is justified to describe the cases of inhomogeneity having sharp boundaries of changes in the system parameters.
Equation (2.1) was solved numerically by the method of lines [54] for an interval x e e [-60, 60], t e [0, 2000]. The spatial coordinate step is Ax = 0.025. A kink of the form (2.2) with coordinate x0 = -10 was taken as the initial solution at t = 0. It was launched with different initial velocities v0 in the impurity direction and the dynamics of the field u(x, t) was observed. Neumann boundary conditions were used. At the boundaries of the computational cell, strong friction was included in order to absorb radiation due to the kink-impurity interaction.
For AK > 0, if the initial kink velocity v0 is greater than some critical velocity vcrit, the kink passes through the impurity. High-amplitude nonlinear oscillations (or impurity modes) of breather type arise in the impurity region. Since strong friction is set on the boundaries, after some time, starting from about t > 800, the radiation from the collision of the kink with the impurity disappears and only oscillations localized on the impurity remain (see Fig. 2a). The amplitude and frequency of the impurity mode can be studied by studying the dynamics of the impurity center coordinate u(x = 0, t) (see Fig. 2b). The oscillation amplitude A at the point x = 0 was calculated using the formula max(u(x = 0, t)) — min(u(x = 0, t)) for t > 500, and the frequency oscillations of Q, by the Fourier method. The Fourier expansion of the dependence u(x = 0, t) for t > 500 shows that there is one main oscillation frequency of the impurity mode, in our case it is equal to 1.344. The Fourier spectrum for u(x, t) was found for different values of x = —1, —0.5, —0.25, 0, 0.25, 0.5, and 1. It turned out that all points lying in the impurity region and near it oscillate with the same frequency. A nonlinear wave localized on an impurity can be considered a breather-type wave or wobbler. We will further speak of this oscillation frequency as the oscillation frequency of the wobbler.
To determine the lifetime of the wobbler, we consider the dependence of its amplitude on time. At small values of the parameters describing the impurity K(x), for example, for AK = 0.5, it turned out that for a long time (t0 & 5000) the wobbler amplitude practically does not change (Fig. 3, curves 4-6), i.e., the resulting localized wave can be considered long-lived [1, p. 25]. For the case of AK = 1.0, the wobbler amplitude begins to noticeably decrease with time (see Fig. 3, curves 1-3).
— A (a)
~ 1 1 V i i
-40 -20 0 20 40
x
_ -0.99975
to
W -1.00000
s -1.00025
1000 1050 1100 1150 1200 t
Fig. 2. Dynamics of the wobbler (impurity mode) excited as a result of the kink-impurity interaction. The initial kink velocity is v0 = 0.4. Impurity parameters are AK = 0.5, W = 1.0. Shown are: (a) u(x) at different values of t, (b) u(0, t), and (c) u(-20, t)
Let us now consider the structure and properties of the wobbler tail (see Fig. 2c). Fourier analysis shows the presence of two frequencies — 1.414 and 2.690. One of them is very close to \/2, which describes simple vacuum oscillations of the LpA equation. The second frequency is related to the waves emitted by the breather and is equal to twice the frequency of the wobbler. Thus, we can assume that the localized wave is a wobble of the <pA model [1, p. 187].
The oscillation frequency of the wobbler does not depend on the initial velocity of the kink, just as in the case of an impurity described by a Gaussian function [37]. The dependence of the oscillation amplitude of the impurity mode on the initial velocity of the kink has minima and maxima (see Fig. 4), in contrast to the SGE case, where this dependence had only one maximum [53]. For the case of a point impurity, an analytical expression was found for the dependence of the oscillation frequency of the impurity mode on the impurity amplitude e as follows Q2 = 2 — e2 [13]. For our case of an extended impurity, the dependence of the impurity mode frequency Q on AK is close to this dependence. As the parameter W increases, the oscillation frequency of the impurity mode gradually decreases. The analysis of these dependencies
0.150
0.125
^ 0.100 0.075
0.050
0.025
Fig. 3. Dependence of the wobbler amplitude on time for different kink initial velocities v0. The impurity is characterized by the parameters W = 1 and AK = 0.5 (solid lines) and AK =1.0 (dashed lines)
0.16 0.14 0.12 0.10 ^ 0.08 0.06 0.04 0.02 0.00
Fig. 4. Dependence of the amplitude of the impurity mode on the initial kink velocity, AK = 0.5
shows that the functional dependencies of Q on each of the parameters AK, W are close to each other. So for the same well area s = AK x W they differ from each other only slightly.
Next, we study the structure of the kink (2.2) after its interaction with an impurity, taking into account that an impurity mode emits waves. Let us consider the case of resonant reflection of a kink from an impurity (for example, v0 = 0.12875). In this case, after the reflection
-V - \ V3 \ N. N \ 1, v0 = QA2 2, = 0.45 3, ^0 = 0.50 4, Wo = 0.42 5, = 0.45 6, v0 = 0.50
6
5
4
| |
2000 4000
-15
Fig. 5. Function u(x, t) showing the kink's internal vibrational mode excited due to the interaction with the impurity. The initial kink velocity is v0 = 0.12875 and the parameters of the impurity are AK = 0.5, W = 1.0
from the impurity, the kink moves for a rather long time before reaching the boundary of the computational cell. It can be seen from Fig. 5 that low-amplitude periodic oscillations are excited on the kink. Fourier analysis of these oscillations shows the presence of two peaks. The large
peak corresponds to the frequency ^pulse = 1.225, which is approximately equal to which
corresponds to the internal oscillation mode of the wobbling kink of the p4 model. The small peak has frequency = 1.343, which is associated with the waves emitted by the wobbler, since in the case under consideration, the kink has not yet moved far enough away from the impurity. Thus, it can be argued that after the kink-impurity interaction we are dealing with a wobbling kink [1, p. 187].
3. Impurity modes. Analytical results for linear approximation
Equation (2.1) has static vacuum solutions u = —1, u = 0, and u = +1. The numerical calculations performed show that the core of the wobbler excited during the kink-impurity interaction lies in a small range along the x coordinate, related to the impurity width and centered at the impurity center at x = 0. Let us consider localized oscillations excited on impurities for the case when the kink, after the interaction with the impurity, has gone far to the right from the point x = 0. Then, for a low-amplitude localized wave, the solution can be taken in the form [13]:
u = —1 + e exp(—¿Qteor t)^(x), (3.1)
where Qteor is the wobbler frequency (theoretical), e is a small value (wobbler amplitude), and ^(x) is a function that determines the spatial form of the wobbler. Substituting (3.1)
into (2.1) and expanding it into a series in small e, we get an equation of the form
d2 0(x)
dx2
+ (Qteor - 2K)0(x) = 0. (3.2)
Equation (3.2) has three solutions depending on the region of the x coordinate. In region I, x < AK = 0 and the solution to (3.2) has the form
= cx exp (y2 - fiteora:). (3.3)
In region III, x > -y, AK = 0 and the solution to (3.2) has the form
03 = c3 exp (-v/2 - fyeorz) • (3-4)
In domains I and III we have discarded the solutions diverging as x — ±o. In region II, — -5- ^ x ^ K = 1 — AK and the solution to (3.2) is
02 = C2 siM A/^2eor - 2(1 - AK )x + 6). (3.5)
Applying the conditions of smoothness and continuity of the solution at the points x = — -y-and x = we can obtain 5 = and the dispersion relation of the form
\J'2 - fiLr = \/«t2eor - 2(1 - AK) cot (l -- 2(1 - Ail)). (3.6)
From (3.6) at AK 0 or W 0 we obtain the value of the maximum frequency of the impurity mode equal to f2teor = \/2. From the plot of f2teor = Qteor(AK) for different W (see Fig. 6), it is clear that the linear approximation (3.1) describes with good accuracy the dependence of the impurity oscillation frequency on the parameters AK and W for small impurities.
4. Conclusions
The interaction of kinks with an extended attractive impurity described by a rectangular function is studied numerically for the p4 model. It is shown that such an impurity is a generator for excitation of long-lived wobblers (breather-like impurity modes). A linear approximation is considered that admits an analytical solution of the problem. The limits of its applicability for this model are found. The tail of the breather has the form of a quasi-sinusoidal function unlimited in space and with a classical frequency It is found how, by changing the impurity parameters, it is possible to control the dynamic parameters of the wobbler. It is shown that,
after the kink-impurity interaction, an internal mode with a classical frequency equal to \J\ is excited on the kink and it becomes a wobbling kink.
In the future we would like to study similar phenomena in equations like p6 and p8.
Conflict of interest
The authors declare that they have no conflict of interest.
G 1.3 -
• 1, W = 0.3
2, W = 0.5
3, VF = 0.8 i 4, W= 1.0 I 5, W = 1.2 « 6, W = 1.5
Fig. 6. Dependence of the oscillation frequency l of the wobbler on the impurity parameter AK for different values of the imputify width W (the kink's initial velocity is v0 = 0.4). The dots show the frequency obtained numerically and the lines show the analytical solutions for the linear approximation
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